## Past events

### Wednesday, May 1, 2013 5-7 PMStudent Talks!RLM 7.104

Network decomposition: What is a cluster? by Maggie Miller

How can we logically define a "cluster" of well-connected vertices in a single network? We'll talk about desirable attributes in a partitioned network and how a computer might be told to detect these qualities. We will also discuss a few popular methods of network bisection and derive the maximum partition efficiency algorithm.

Fermat's Last Theorem by Dean Menezes

Two of the most important areas of number theory are analytic and algebraic number theory. A famous discovery in algebraic number theory is Wiles's Proof of Fermat's Last Theorem, which states that for any n > 2 and natural numbers x, y, z, n \in {\bf N}, x^n + y^n = z^n iff xyz = 0. This talk will provide a proof of Fermat's Last Theorem in the case where n = 3 based on Eisenstein integers. Unfortunately, the talk is not long enough to allow for a proof of the complete theorem for general n.

Fractional Calculus by Corey Howell

A brief introduction to Fractional Calculus.

### Wednesday, April 24, 2013 5-7 PMQuadratic forms and local-to-global principles by Professor Mirela CiperianiRLM 7.104

When trying to find rational solutions to an algebraic equation, a favorite trick is to first look for local solutions. That is, we seek solutions over the reals and modulo all powers of all primes. Sometimes local solutions exist if and only if a rational solution exists. This is when we say that the local-to-global principle holds. We will illustrate this principle by discussing the problem of representing integers by quadratic forms over the rationals.

### Wednesday, April 17, 2013 5-7 PM Newton's Problem of the Body of Minimal Resistance by Professor Francesco MaggiRLM 7.104

We shall discuss the state of the art about this old problem left open by Sir Isaac Newton

### Wednesday, April 3, 2013 5-7 PMBezout's Theorem: Intersection of Curves in the Plane by Graduate Student Pavel SafronovRLM 7.104

Algebraic geometry studies curves and surfaces given as zero sets of polynomials of several variables. After introducing some basic notions, I will describe Bezout's theorem which gives the number of solutions of a system of polynomial equations.

A video of this lecture may be found here.

### Wednesday, March 27, 2013 5-7 PMThe Brunn-Minkowski inequality: On the Volume of the Sum of Two Sets by Professor Alessio FigalliRLM 7.104

A video of this lecture may be found here.

### Wednesday, March 20, 2013 5-7 PMThe Boltzmann Transport Equation by Graduate Student Sona AkopianRLM 7.104

How can we describe the invisible, complex motion of the air that surrounds us? We will talk about this 140 year old open problem and discuss the first ever proposed proof of existence and uniqueness of solutions to the BTE.

A video of this lecture may be found here.

### Wednesday, March 6, 2013 5-7 PM2+2=4: Seeing 4-dimensional manifolds via Lefschetz fibrations by Professor Timothy PerutzRLM 7.104

Topologists study mathematical spaces called "manifolds", which have a notion of "dimension" that extends the idea of 1 dimension (the single degree of freedom when you move along a curved path), 2 dimensions (the two degrees of freedom when you travel on the earth's surface) and the familiar 3 dimensions of space. The possibilities for manifolds of dimensions 1 and 2 have long been understood. More surprisingly, a big class of manifolds of dimension 5, 6, or any higher number are also classified (in an abstract way), so by 1970, the only dimensions where we didn't understand the rules were 3 and 4. In 2012 only dimension 4 remains mysterious. I don't know how to resolve the mystery, but I'll explain a way to describe many 4-dimensional manifolds using surfaces and their symmetries.

A video of this lecture may be found here.

### Wednesday, February 27, 2013 5-7 PMArithmetic Geometry: From Circles to Circular Counting by Dr. Adriana SalernoRLM 7.104

In this talk, I will show you a glimpse of one of the most exciting facets of research in modern number theory: arithmetic geometry. We will start with a (gentle) introduction to this area of research through some familiar examples. Then we will move on to a not so familiar example where we count solutions of equations mod p. I will end by answering two of the oldest and most mystifying questions in mathematics: how does this work fit into the bigger picture, and who cares?

### Wednesday, February 20, 2013 5-7 PMThe ABC Conjecture by Professor Jeffrey VaalerRLM 7.104

In the early 1980's W. Stothers and (independently) R. Mason discovered a new and simple inequality about the zeros of polynomials. The proof uses only basic facts about derivatives. As a corollary Mason obtained a very simple proof of "Fermat's Last Theorem" for polynomials. Inspired by Mason's observations, Masser and Oesterl e proposed an analogous inequality for integers, which has come to be known as the ABC conjecture. In this talk I will discuss Mason's inequality, some of its applications, and the resulting ABC conjecture for integers. The talk should be accessible to a general mathematical audience.

A video of this lecture may be found here.

### Wednesday, February 13, 2013 5-7 PMKnots, Seifert surfaces, and the Slice-Ribbon Conjecture by Graduate Student Jeff MeierRLM 7.104

A video of this lecture may be found here.

### Wednesday, February 6, 2013 5-7 PMThe Banach-Tarski Paradox by Dr. Lewis BowenRLM 7.104

Is it possible to cut the unit sphere into a finite number pieces then rearrange them by rigid motions to form two spheres each of the same size and shape as the original? According to Banach and Tarski the answer is "yes". Their proof leads to questions about the nature of area, the axioms of set theory and a fundamental dichotomy in group theory between amenable and non-amenable groups.

### Wednesday, January 16, 2013 5-7 PMThe Virtual Haken Conjecture and CAT(0) Cube Complexes by Dr. Michah SageevRLM 9.166

We will tell the tale of the Virtual Haken Conjecture and the Virtual Fibered Conjecture. We will assume no prior knowledge of 3-manifold theory or CAT(0) cube complexes.

A video of this lecture may be found here.

### Wednesday, December 5, 2012 5-7 PMStudent Talks!!!RLM 12.166

Dustan Levenstein: Tiling Invariants

Christopher Miller: Optimal Stopping Problems

Nicole Yamzon: Establishing the Existence of Transcendental Numbers

### Wednesday, November 28, 2012 5-7 PMProfessor Karen UhlenbeckRLM 12.166

This week, we will be having Dr. Uhlenbeck as our speaker.

### Wednesday, November 14, 2012 5-7 PMA Goldbach Theorem for Polynomials by Graduate Student Cory ColbertRLM 12.166

Many classical questions in number theory can be restated in terms of polynomials. Strangely, the analogous question is often easy to resolve. In 1965 D. R. Hayes discovered that every polynomial over a PID $R$ with infinitely many primes may be written as a sum of two irreducible polynomials. We will prove his result and briefly discuss a generalization. Introductory courses in abstract algebra and number theory are desirable but not at all required.

A video of this lecture may be found here.

### Wednesday, November 7, 2012 5-7 PMMy Pet Field Theory by Graduate Student Aaron FenyesRLM 12.166

Field theories are some of the most fascinating and beautiful creatures in physics. Unfortunately, most of them are so complicated that it's hard to see the basic outlines of what they are and how they work. In this talk, I'll introduce you to a very small and well-behaved field theory that, despite its simplicity, shares many essential features with its less domesticated cousins. By playing with this theory, we'll learn about how field theories can be used to describe the physical world.

A video of this lecture may be found here.

### Wednesday, October 31, 2012 5-7 PMThe Based Loop Space and Groups up to Homotopy by Dr. Andrew BlumbergRLM 12.166

For a topological space X with a basepoint, the based loop space (the space of maps from [0,1] to X which take 0 and 1 to the basepoint) is a topological analogue of a group. The double loop space is a topological analogue of an abelian group. I'll discuss how this works and gently introduce the subject of the algebra of groups up to homotopy.

### Wednesday, October 24, 2012 5-7 PMMaking proof-based verified computation almost practical by Dr. Michael WalfishWAG 201

How can a machine specify a computation to another one and then, without executing the computation, check that the other machine carried it out correctly? And how can this be done without assumptions about the performer (replication, trusted hardware, etc.) or restrictions on the computation? This is the problem of _verified computation_, and it is motivated by the cloud and other third-party computing models. It has long been known that (1) this problem can be solved in theory using probabilistically heckable proofs (PCPs) coupled with cryptographic tools, and (2) the performance of these solutions is wildly impractical (trillions of CPU-years or more to verify simple computations).

I will describe a project that challenges the second piece of this folklore. We have developed an interactive protocol that begins with work by Ishai et al. (CCC '07) and incorporates new theoretical work to improve performance by 20 orders of magnitude. In addition, we have broadened the computational model from Boolean circuits to a general-purpose model. We have fully implemented the system, accelerated it with GPUs, and developed a compiler that transforms computations expressed in a high-level language to executables that implement the protocol entities.

The resulting system is not quite ready for the big leagues, but it is close enough to practicality to suggest that in the near future, PCPs could be a real tool for building actual systems.

Joint work with Andrew Blumberg, Ben Braun, Srinath Setty, and Victor Vu.

### Wednesday, October 17, 2012 5-7 PMGraduate School Application PanelRLM 12.116

If you are at all interested in applying to graduate school soon or at some point in the (not so) far-off future of your senior year, then you would be well-advised to attend our panel on applying to graduate school this week. Even if you are not applying to graduate school for mathematics but in one of the sciences, you will still get some valuable information.

There will be five panelists: three faculty members and two first year graduate students.

A video of this lecture may be found here.

### Wednesday, October 10, 2012 5-7 PMSubgroups of Products by Graduate Student Itamar GalRLM 12.166

Abstract: Arguably the most fundamental construction used to understand algebraic objects is the direct sum. In group theory, direct sum decompositions reduce the problem of understanding the structure of a large (and possibly very complicated) group, to the problem of understanding the structure of some collection of its subgroups (which are smaller, and hopefully simpler). A more general (but more complicated) construction is the semi-direct product. Given a direct (or semi-direct) product decomposition G = NK, we will show how to determine the subgroups of G from the subgroups of N and K.

A video of this lecture may be found here.

### Wednesday, October 3, 2012 5-7 PMSudoku and Graphs by Graduate Student Claudia RaithelRLM 12.166

We will begin by reformulating Sudoku as a coloring problem on graphs. Using this framework we will draw some conclusions as to the number of solutions for a given Sudoku. Time permitting, we will then talk about the relationship between Latin squares and Sudoku squares.

A video of this lecture may be found here.

### Wednesday, September 26, 2012 5-7 PMMaking Tracks: A Mathematician Plays with Trains by Dr. David RusinRLM 6.118

Suppose we are given a set of segments of track from a toy train, and wish to make interesting designs which are simple closed curves. What options are available to us? Surprisingly, even a simple question like this leads us to use tools from linear algebra and number theory, and serves as a model for more general questions about positioning objects in space.

A video of this lecture may be found here.

### Wednesday, September 19, 2012 5-7 PMFully Conservative Characteristic Methods for Transport by Professor Todd ArbogastRLM 7.124

A video of this lecture may be found here.

### Wednesday, September 12, 2012 5-7 PMThe Geometry of Numbers by Michael KellyRLM 12.166

We will introduce the basic objects and ideas of the geometry of numbers introduced over 100 years ago by Hermann Minkowski. Our goal will be to understand and prove Minkowski's convex body theorem.

A video of this lecture may be found here.

### Wednesday, September 5, 2012 5-7 PMSurprising Circles, Archimedes, Onions, and Calculus by Professor Michael StarbirdRLM 7.104

Cool insights about circles and basic geometry give magical proofs to some calculus results such as the derivative and integral of sin(x). In fact, some basically calculus insights preceded calculus itself by a couple thousand years. For example, Archimedes used ideas about levers and limits to deduce the equation for the volume of the sphere.

A video of this lecture may be found here.

### Wednesday, May 2, 2012 5-7 PMStudent Talks!!!RLM 12.104

Brandon Carter: Ramification and Reciprocity

Burns Healy: The First Fundamental Form

Jason Bowen: The Oddities of Perfect Numbers

A video of this lecture may be found here.

### Wednesday, April 25, 2012 5-7 PMReliable Communication and Linear Algebra by Felipe VolochRLM 12.104

Professor Voloch will speak about how linear algebra is used in the theory of error-correcting codes, which are extensively used in telecommunications. He will also cover the modern ideas that go into the new "low-density parity check" (LDPC) codes, as well as the mathematical challenges associated with them.

### Wednesday, April 18, 2012 5-7 PMThe Mathematics of Population Genetics by Rachel WardRLM 12.104

Why don't we all have brown eyes? This question remained unresolved for many years until in 1908, the mathematician G. Hardy (and, independently, W. Weinberg) derived the fundamental mathematical equilibrium equations of population genetics which Hardy himself referred to as "a little mathematics of the multiplication-table type." Today, scientific inference based on the Hardy-Weinberg Principle is standard in applications ranging from evolutionary biology to forensic science. We will give a short account of the history and mathematics of these equations, and discuss several mathematical implications of their discovery.

A video of this lecture may be found here.

### Wednesday, April 11, 2012 5-7 PMFiber Bundles and Symmetry Groups by Aaron RoyerRLM 12.104

Geometric and topological objects frequently come in families parameterized by other such objects. Depending on how intricate and/or symmetrical these objects are, the families can be "twisted.'' We will investigate the source of this behavior in concrete cases, and outline a program to understand such twisting in a large range of cases using topology.

A video of this lecture may be found here.

### Wednesday, April 4, 2012 5-7 PMA New Way to View Old Topics by Mark DanielsRLM 11.176

Abstract: "Hands-on" activities will be presented that emphasize creative, non-traditional, and, hopefully, enlightening ways to explore two topics: 1) Relations in the Polar Coordinate System and 2) The concept of complex roots of quadratics and higher degree polynomials. An article exploring the merits of these activities, authored by Mark Daniels and Efraim Armendariz, was recently published in the AMATYC Journal.

A video of this lecture may be found here.

### Wednesday, March 28, 2012 5-7 PMThe Top Ten Reasons Everyone Should Study Topology by Professor Jim VickRLM 11.176

Professor Vick explains why you should drop everything you're doing and start studying topology.

A video of this lecture may be found here.

### Wednesday, March 21, 2012 5-7 PMRecognizing the Maximum of a Sequence by John ChatlosRLM 12.104

In this talk, we will study what is variously known as the "Beauty Contest Problem," "Secretary Problem," or "Dowry Problem." That is, we will discuss optimum strategies for finding sequentially the maximum of a random sequence of fixed length. Our agenda for this talk will be: (1) Couch the problem in more modern terms, and (2) Find an asymptotic optimum strategy for our variant of the problem.

A video of this lecture may be found here.

### Wednesday, March 7, 2012 5-7 PMMath Club Movie Night!RLM 11.176

Next Wednesday, March 7, is going to be a movie night (no speaker). We're going to watch the Nova special on Andrew Wiles' proof of Fermat's Last Theorem, called The Proof. It will be in the 11th floor classroom from (roughly) 5 to 6, and then pizza at 6 like usual.

### Wednesday, February 29, 2012 5-7 PMYoung Tableaux, Symmetric Polynomials, and Representations by Iordan GanevRLM 12.104

Two common themes in mathematics are (1) the decomposition of objects into simple constituents and (2) the identification and analysis of the symmetries of an object. Young tableaux and symmetric polynomials provide powerful combinatorial tools to address questions of symmetry and decomposition in several mathematical fields, most notably representation theory and geometry. The aim of the talk is to introduce Young tableaux and symmetric polynomials, demonstrate some basic properties and manipulations, and give an idea of their applications to the representation theory of the symmetric group.

A video of this lecture may be found here.

### Wednesday, February 22, 2012 5-7 PMHomology that Persists, or How to Find Holes in Cellphone Reception by Nick ZufeltRLM 12.104

In this talk we'll discuss a topological tool called homology and hopefully get an understanding for what it's useful for and (roughly) how to compute it. Then we'll discuss ways to compute it when we're stuck with a less-than-optimal amount of information, arriving at a glimpse of a recently developed version of this tool called persistent homology.

A video of this lecture may be found here.

### Wednesday, February 15, 2012 5-7 PMThe Discriminant: A Classy Number by Bobby GrizzardRLM 12.104

We will investigate the 200-year-old problem of classifying integral binary quadratic forms with a given discriminant, first tackled by C. F. Gauss. This is a classical approach to the class number problem. No knowledge of number theory is necessary.

A video of this lecture may be found here.

### Wednesday, February 8, 2012 5-7 PMFourier Series: An Introduction by Chris WhiteRLM 12.104

Fourier analysis is one of the most powerful tools in modern mathematics. The (seemingly simple) idea of a Fourier series has surprisingly deep connections to almost every area of math, from representation theory to signal analysis, and can even be used to provide an elegant explanation of the Heisenberg Uncertainty Principle.

In this talk, I hope to provide a basic introduction to Fourier series. We will begin by reviewing some concepts, such as Hilbert spaces, which are needed to understand what Fourier series are all about. I will then state the Plancherel Theorem and use it to provide a simple solution to the classic "Basel problem," a problem which went unsolved for almost a century (we will solve it in about 5 minutes).

A video of this lecture may be found here.

### Wednesday, February 1, 2012 5-7 PMTopoisomerase: Knotenlöserin by Dr. Jennifer MannRLM 11.176

Close your eyes for a moment. Take a few slow, deep breaths. Begin to relax. Remember your Colorado vacation last summer. You are fishing a secluded, high-mountain lake. Tall Ponderosa pines surround you. The sky is a delightful blue with a few white, puffy clouds. The air is crisp and cool. You lure a 22 lb rainbow trout to bite your hook. The dance begins! Oh no! There is a knot in your line. The trout leaps out of the water, tries to spit out the hook, dives back into the water, pulls HARD on your line, and SNAP! Your line breaks. Now imagine you are sitting in math class in the big brown box of RLM. Chalk dust is flying at the front of the room. You are burning lead taking notes. A different dance has begun. Oh no! There is a knot in your DNA. How did this happen and what happens next?

A video of this lecture may be found here.

### Wednesday, January 25, 2012 5-7 PMComplex Tori and Elliptic Curves by Gil MossRLM 12.104

I will introduce doubly period functions on the complex plane and the Weierstrass-P function and its derivative. Using the Weierstrass-P, we can describe the space of elliptic functions for a given torus, and parametrize the torus itself with a curve in the projective plane: an elliptic curve. If time remains, I will point out how this unearths a modular form.

A video of this lecture may be found here.

### Wednesday, November 30, 2011 5-7 PMStudent talks!!!RLM 12.104

Austin Arlitt: Primes in Progressions

Brandon Carter: P-Adics: What are they and why do we care?

Jason Bowen: Topological Division Rings

### Wednesday, November 16, 2011 5-7 PMShapes of spaces: 2- and 3-manifolds by Marion CampisiRLM 12.104

If you took off in a faster than light rocket-ship and flew in a straight line forever, what would happen? Would you hit the edge of the universe? Would you keep going forever, always getting further away from home? Is there another possibility? Could the answer to these questions give us any clues about the shape of the Universe? While we do not know what the actual shape of the universe is, mathematicians have been able to determine possible shapes a 3-dimensional universe could have. Before we try to understand these shapes, called 3-manifolds, we will build our intuition by considering the perspective of beings living in 2-dimensional universes, or 2-manifolds. We will consider possible 2-manifolds and develop tools that a being living in such a space could use to distinguish them. Finally, we will develop a picture of several different 3-manifolds and consider if there is any way to know whether any of them might be the shape of our own universe.

A video of this lecture may be found here.

### Wednesday, November 9, 2011 5-7 PMDynamical Systems, Period Doubling, and Renormalization by Hans KochRLM 12.104

A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a lake.

### Wednesday, November 2, 2011 5-7 PMTriangles whose three angles are all zero! by Mark NorfleetRLM 12.104

If the sum of the three angles of a triangle where smaller than 180 degrees, what should we do? What hope can we have? Do not fear--hyperbolic geometry is here to save the day! We will discuss four different models of the hyperbolic plane (where these types of triangles live), illustrate the connection between the models, and conclude with how to "see" some actions on the hyperbolic plane.

A video of this lecture may be found here.

### Wednesday, October 26, 2011 5-7 PMColorful Knots by Dr. Brandy GuntelRLM 12.104

As a young knot theorist, you have the Reidemeister moves on your side when you're trying to tell if your two favorite knots are different. But you have been doing Reidemeister moves for the last two years, and you just don't know if that next Reidemeister move is the one that shows they're the same. One night while you're drawing knots, a big truck drives by and knocks over the three open cans of paint you keep on your desk. The paint seeps across your paper, and you realize that you need to color these knots, but you're just not sure how. In this talk, we will discuss a way for you to finally tell those knots are different.

A video of this lecture may be found here.

### Wednesday, October 19, 2011 5-7 PMStability of Soap Bubbles by Emanuel IndreiRLM 11.176

The formation of soap bubbles can be mathematically modeled with the classical isoperimetric inequality. We will show how one can apply optimal transport theory to give a short and elegant proof of the inequality. Moreover, we will discuss some recent isoperimetric-type stability results.

A video of this lecture may be found here.

### Wednesday, October 12, 2011 5-7 PMTensor Products: Universal Properties In Action! by John MethRLM 12.104

This week's talk will be by Dr. John Meth. John received his Ph.D. from UT Austin in 2009 and works in non-commutative algebra and algebraic geometry. His talk is titled Tensor Products: Universal Properties in Action! The concrete details of a given construction may be messy, but if the construction satisfies a universal property, one can forget all those details: all there is to know about the construct is already contained in the universal property. Proofs often become short and elegant if the universal property is used rather than the concrete details.

A video of this lecture may be found here.

### Wednesday, October 5, 2011 5-7 PMGeometric Gems by Dr. StarbirdRLM 11.176

Plain plane (and solid) geometry contains some of the most beautiful proofs ever--some dating from ancient times and some created by living mathematicians. Among my favorites are: the Dandelin Sphere argument that a plane intersects a cone in an ellipse; a method for computing areas under curves such as the tractrix developed by a living mathematician, Momikan Mnatsakanian; a modern proof by John Conway (refined by me) of Morley's Miracle; Archimedes' method of trisecting angles with only a little cheating; and many more. Geometry provides many treats!

A video of this lecture may be found here.

### Wednesday, September 28, 2011 5-7 PMGrad School Panel!RLM 11.176

If you are at all interested in applying to graduate school soon or at some point in the (not so) far-off future of your senior year, then you would be well-advised to attend our panel on applying to graduate school this week. Even if you are not applying to graduate school for mathematics but in one of the sciences, you will still get some valuable information.

There will be four panelists: the professors Ted Odell, Natasa Pavlovic, and Dan Knopf and first-year graduate student Cory Colbert. The faculty members are all involved in graduate admissions at UT (Knopf is the graduate advisor) and Cory successfully applied to graduate school: a feat many of us have not yet accomplished.

A video of this panel may be found here.

### Wednesday, September 21, 2011 5-7 PM"Let's Meet at the Euler Characteristic" by Professor Gary HamrickRLM 12.104

The Euler Characteristic of surfaces provides a wonderful opportunity to see the interplay of geometry, algebraic topology, and analysis that does not require sophisticated knowledge to grasp. It is actually a very special case of the celebrated Atiyah-Singer Index Theorem, a result that touches on virtually all areas of mathematics.

In the last few decades mathematics has progressed in such a way that its various branches such as algebra, analysis, geometry, and topology are becoming ever more inextricably linked together. An early such development occurred in the 17th century with Descartes' invention of algebraic geometry via his introduction of Cartesian coordinates. But the process has accelerated tremendously in relatively recent times.

A spectacular example in Perelman's proving Thurston's Conjecture relating the geometry and topology of 3-dimensional manifolds by the use of partial differential equations (analysis). A small corollary is the Poincare' Conjecture, one of the $1,000,000 Clay Institute Problems. Another such example is Andrew Wiles' proof of Fermat's Last Theorem the most famous problem in number theory. What Wiles actually proved was a result in algebraic geometry that had been known to imply Fermat's Last Theorem. A video of this lecture may be found here. ### Wednesday, September 14, 2011 5-7 PMTilings, Symmetry, and Dynamics by Professor Lorenzo SadunRLM 12.104 If you've every had a class with Prof. Sadun, then you know what a great speaker he is. If you haven't then come find out. This talk will more likely than not have a ton a nice pictures, which are always fun. A video of this lecture may be found here. ### Wednesday, September 7, 2011 5-7 PM Phil Monin on the Heat Equation and Optimal InvestingRLM 11.176 Our talk this week is at 5 pm in RLM 11.176 (or 11.166, the class room on the 11th floor). by UT grad student Phil Monin. Phil is a very good speaker and whether or not you think you like PDEs or mathematical finance, you will enjoy this talk. The title is The Heat Equation and Optimal Investing. The heat equation is the prototypical parabolic partial differential equation and describes "random walks," hence it's application to mathematical finance. It is also important in Riemannian geometry and thus topology: it was adapted by Richard Hamilton when he defined the Ricci flow that was later used by Grigori Perelman to solve the topological Poincaré conjecture (although this will not be the topic of Phil's talk). A video of this lecture may be found here. ### Wednesday, August 31, 2011 5-7 PMEric Staron with An Introduction to Rational TanglesRLM 12.104 Tangle theory can be considered analogous to knot theory except instead of closed loops we use strings whose ends are nailed down. An n-tangle can be defined as a proper embedding of a disjoint union of n arcs into a 3-ball. A rational tangle is a 2-tangle that is homeomorphic to the trivial 2-tangle as a map of pairs consisting of the 3-ball and two arcs. A video of this lecture may be found here. ### Wednesday, May 4, 2011 5-7 PMStudent Talks!RLM 12.104 Student talks this week! Come and hear some interesting talks from your peers! (Pizza as usual at 6.) Brandon Carter on Elliptic Curve Cryptography Josean Cortes on Numerical Integration Dustan Levenstein on Kuratowski's Theorem Joe Cunningham on The Poincare Recurrence Theorem and Corey Harris on Polytopes (probably) A video of this lecture may be found here. ### Wednesday, April 27, 2011 5-7 PMMusic and Mathematics by David BrownRLM 12.104 A video of this lecture may be found here. ### Wednesday, April 20, 2011 5-7 PMMathematics in Medical Imaging: Past and Present by Dr. RenRLM 12.104 A video of this lecture may be found here. ### Wednesday, April 13, 2011 5-7 PMHeights of Algebraic Numbers, a talk by UT grad student Zachary MinerRLM 12.104 An important characteristic of an algebraic number is its height, which is the analogue of the denominator of a rational fraction. The height of an algebraic number α is the greatest absolute value of the coefficients of the irreducible and primitive polynomial with integral rational coefficients that has α as a root. A video of this lecture may be found here. ### Wednesday, April 6, 2011 5-7 PMBrownian Motion, a talk by Dr. Maria GualdaniRLM 12.104 Brownian motion (named after Robert Brown, who first observed the motion in 1827, when he examined pollen grains in water), or pedesis (from the Greek "leaping") is the presumably random movement of particles suspended in a fluid (i.e., a liquid such as water or a gas such as air) or the mathematical model used to describe such random movements, often called a particle theory. The mathematical model of Brownian motion has se...veral real-world applications. An often quoted example is stock market fluctuations. However, movements in share prices may arise due to unforeseen events which do not repeat themselves. Brownian motion is among the simplest of the continuous-time stochastic (or probabilistic) processes, and it is a limit of both simpler and more complicated stochastic processes (see random walk and Donsker's theorem). This universality is closely related to the universality of the normal distribution. In both cases, it is often mathematical convenience rather than the accuracy of the models that motivates their use. This is because Brownian motion, whose time derivative is everywhere infinite, is an idealized approximation to actual random physical processes, which always have a finite time scale. A video of this lecture may be found here. ### Wednesday, March 30, 2011 5-7 PMSorting Shapes: a survey of some of the key ideas by which surfaces and 3-manifolds can be classified by geometric flows by Dan KnopfRLM 12.104 [poster] A survey of some of the key ideas by which surfaces and 3-manifolds can be classified by geometric flows. A video of this lecture may be found here. ### Wednesday, March 23, 2011 5-7 PMGroups, Representations and Particles by Andrew NeitzkeRLM 12.104 [notes] I will explain about group representations, with the idea that by the end we could understand what people mean by saying that the elementary particles are organized according to group theory, and what people mean when they talk about a "Grand Unified Theory". A video of this lecture may be found here. ### Wednesday, March 9, 2011 5-7 PMMath Club Movie Night!RLM 11.166 This week we will be watching N Is a Number: A Portrait of Paul Erdős. This film gave Erdős a Bacon number of 3. Trust me, you will be thoroughly and completely entertained. I believe will will watch it in the 11th floor classroom (RLM 11.166, I believe) at the usual 5 pm time. Pizza will be served in the lounge afterwards. ### Wednesday, March 2, 2011 5-7 PMThe Lehmer Conjecture by Bobby Grizzard RLM 12.104 [poster] ### Wednesday, February 23, 2011 5-7 PMChromatic Polynomials by Eric KatzRLM 12.104 [poster] [notes] The chromatic polynomial is a polynomial studied in algebraic graph theory. It counts the number of graph colorings as a function of the number of colors and was originally defined by George David Birkhoff to attack the four color problem. It was generalised to the Tutte polynomial by H. Whitney and W. T. Tutte, linking it to the Potts model of statistical physics. A video of this lecture may be found here. ### Wednesday, February 16, 2011 5-7 PMThe ABC Conjecture by Professor Jeffrey VaalerRLM 12.166 [poster] Our talk tomorrow is by Professor Jeffrey Vaaler (a man with an Erdős number of 1) on the ABC conjecture, a conjecture in number theory, first proposed by Joseph Oesterlé and David Masser in 1985. The conjecture is stated in terms of three positive integers, a, b and c (whence comes the name), which have no common factor and satisfy a + b = c. If d denotes the product of the distinct prime factors of abc, the conjecture essentially states that d is rarely much smaller than c. More precisely, the conjecture asks if there are, for every ε > 0, only finitely many triples of coprime positive integers a + b = c such that c > d1 + ε, where d denotes the product of the distinct prime factors of abc? This is a talk you do not want to miss. This talk will be in the 12th floor classroom (RLM 12.166), which is around the corner from our usual meeting place. A video of this lecture may be found here. ### Wednesday, February 9, 2011 5-7 PMModeling Electrical Activity in Neurons with Discrete Calculus by Andrew GilletteRLM 12.104 A video of this lecture may be found here. ### Wednesday, February 2, 2011 5-7 PMRamsey Theory by Professor Ted OdellRLM 12.104 This week's talk is going to be by Professor Ted Odell on the subject of Ramsey theory. Ramsey theory, named after Frank P. Ramsey, is a branch of mathematics that studies the conditions under which order must appear. Problems in Ramsey theory typically ask a question of the form: "how many elements of some structure must there be to guarantee that a particular property will hold?" If you want to know more, feel free to google "ramsey theory" and/or (preferably and) come to Prof. Odell's talk this week. As usual, there will be free pizza. A video of this lecture may be found here. ### Wednesday, January 26, 2011 5-7 PMWhat's so ideal about the ideal class group? by Keenan KidwellRLM 12.104 [poster] [notes] The ring Z of integers is just about as swell a ring as anybody could ever want. One of its most venerable qualities is that it possesses unique factorization into primes. This is the content of the Fundamental Theorem of Arithmetic. Unfortunately, some arithmetic statements that can be made "within Z" (like Fermat's Last Theorem) naturally lead one to consider rings like Z[z_n], where z_n is a primitive n-th root of unity; Z[z_n] is an example of a ring of algebraic integers. Such rings are similar to Z in some ways, but can be violently different in others. Most notably, they can fail to have unique factorization into primes. But all hope is not lost, and you can find out why at this talk. A video of this lecture may be found here. ### Wednesday, December 1, 2010 5-7 PMStudent Talks!RLM 12.104 [poster] 4 student talks!! Math Club this week will be 4 talks, 12-15 mins each, given by the following students: (in no particular order) • Shannon Hatfield - The Inca Quipu • Austin Arlitt - Incompleteness & Non-Contradiction • Dugan Hammock - Riemann Surfaces • Dustan Levenstein - Non-planar graphs A video of this lecture may be found here. ### Thursday, November 18, 2010 7-9 PMMath Club Movie Night!RLM 4th floor (ground floor) lecture hall Next Thursday, November 18, we'll be having Math Club Movie Night at 7 pm in Wheeler Hall (big lecture hall on RLM 4th Floor). If that room doesn't end up, working out, look for a sign on the door to tell you where to go. We'll be watching Proof unless anyone else has a suggestion! ### Wednesday, November 17, 2010 5-7 PMCassini Collage: You be the Astronomer by Sally Dodson-RobinsonRLM 12.104 [poster] We'll look at pictures of Saturn from the Cassini spacecraft, and use those pictures to figure out some key things about Saturn, its moons and its rings. ### Wednesday, November 10, 2010 5-7 PMContinued fractions and the ergodic theorem by Michael WilliamsRLM 12.104 [poster] Continued fractions, despite their unwieldy appearance, offer a representation of real numbers that is in some ways superior to our usual decimal notation. We will discuss a few basic facts about continued fractions, and derive a startling result with the help of the ergodic theorem from analysis. A video of this lecture may be found here. ### Wednesday, November 3, 2010 5-7 PMDonuts: Genus One Curves by Dr. Mirela Ciperiani RLM 12.104 [poster] Donuts: Genus One Curves No pizza! Lots of donuts! A video of this lecture may be found here. ### Wednesday, October 27, 2010 5-7 PMCreating the perfect calculus graphing problem by David RusinRLM 12.104 [poster] [notes] As we look for a function which can be analyzed by a freshman calculus student -- with all its "interesting" points being "nice" -- we find ourselves in the land of Arithmetic Algebraic Geometry! A video of this lecture may be found here. ### Wednesday, October 20, 2010 5-7 PMWater versus Ice by Charles RadinRLM 12.104 [poster] [notes] Dr. Radin will be speaking about phase transitions, specifically the difference between water and ice. Water and ice are both just large collections of H_2 O molecules, yet they seem so different. How does one understand that? A video of this lecture may be found here. ### Wednesday, October 13, 2010 5-7 PMComputers with Vision by Kristen GraumanRLM 12.166 [notes] Recognizing and Searching for Objects in Images 5:30 - 7:30 pm RLM 12.166 (around the corner from the regular room) Image and video data are rich with meaning, memories, or entertainment, and they can even facilitate communication or scientific discovery. However, our ability to capture and store massive amounts of interesting visual data has outpaced our ability to analyze it. Computer vision methods to search and organize images based on their visual cues are therefore necessary to make them fully accessible. In this talk I will discuss fundamental vision challenges, and then overview some of the UT-Austin Vision Group's work addressing scalable image search and object recognition. In particular, I will describe some recent work on randomized hashing algorithms for sub-linear time retrieval, and active or semi-supervised learning strategies to build category models using limited human supervision. This is an introductory high-level talk; no prior background on the subject is assumed. ### Wednesday, October 13, 2010 4-5 PMJoint Math and Physics Colloqium by Professor Jim YorkeRLM 4.102 Chaos 4:15 pm RLM 4.102 (ground floor auditorium) Professor Yorke coined the term "chaos" in 1975 and has been a pioneer in characterizing chaos (e.g., fractal basin boundaries, crises, chaotic scattering) and applying chaos to many problems, ranging from epidemics to optical communications to weather forecasting. ### Wednesday, October 6, 2010 5-7 PMCounting Solutions by Professor Dan FreedRLM 12.104 [poster] The problem of counting solutions to equations arises in algebra (polynomial equations) and geometric analysis (differential equations). Often topology can be brought to bear on such counting problems. Indeed, one can sometimes reverse the logic and count solutions to equations to shed light on topology. A video of this lecture may be found here. ### Wednesday, September 29, 2010 5-7 PMGraduate School Informational PanelRLM 12.104 [poster] Do you want to be a mathematician? Are you interested in going to graduate school? To address questions and concerns like these, we will be having a panel consisting of faculty members and graduate students from UT. We will go over things to think about as a math major, how to apply to graduate school, and we may even warn you about some signs that you may be turning into a mathematician. ### Wednesday, September 22, 2010 5-7 PMQuantum teleportation and quantum communication by Spencer Stirling RLM 12.104 [poster] Teleportation is real --at least in the quantum world. Although we can never know the exact state of a particle, nor can we copy it, we can transfer its state to an identical particle. Amazingly, this can be done instantaneously over large distances. We'll informally discuss this. No knowledge of quantum mechanics is necessary except some simple linear algebra. A video of this lecture may be found here. ### Wednesday, September 15, 2010 5-7 PMIdentity testing and the power of randomized algorithms by Andrew BlumbergRLM 12.104 [poster] [notes] We will talk about the sometimes surprising fact that incorporating coin flips into algorithms (and accepting bounded probability of failure) makes it possible to do things which seem intractable deterministically. A video of this lecture may be found here. ### Wednesday, September 8, 2010 5-7 PMFixed Points in a Changing Age by Professor StarbirdRLM 12.104 [poster] [notes] At this very instant, there must be two diametrically opposite points on Earth that have exactly the same temperature and also have exactly the same air pressure. This fact is a consequence of the Borsuk-Ulam Theorem, a proof of which involves ropes and winding. A similar proof involving ropes can prove the famous Brouwer Fixed Point Theorem. Perhaps these theorems illustrate how mathematics can bring stability to our shaky age. A video of this lecture may be found here. ### Wednesday, September 1, 2010 5-7 PMLinear Inequalities and Fourier-Motzkin Elimination, by Itamar GalRLM 12.104 [notes] In linear algebra we study systems of linear equations. A fundamental tool for answering questions about such systems is Gaussian elimination. If instead we consider systems of linear inequalities, then we are quickly led to many analogous questions which unfortunately cannot be resolved by the techniques of linear algebra. Luckily the theory of linear inequalities provides us with its own tools. We will introduce one such tool, namely Fourier-Motzkin elimination, and show how it can be used to answer questions about systems of linear inequalities. A video of this lecture may be found here. ### Wednesday, April 28, 2010 5-7 PMFour Student TalksRLM 12.104 [poster] [notes] Math Club members Dugan Hammock, Justin Hilburn, Kal Hourani, Lee McCuller, and Gideon McKee will each give short talks on several topics, including the Hairy Ball theorem, Fermat's Last Theorem, Galois Theory, the quasi-linear transport equation, and LU matrix decomposition. A video of this lecture may be found here. ### Wednesday, April 21, 2010 5-7 PMPeriod Three Implies Chaos (or, Nonlinearity is Insidious), by Sarah RichRLM 12.104 [poster] [notes] A simple quadratic equation seems innocent enough; after all, quadratic is almost linear. It's smooth, easy to describe---it's degree 2 for crying out loud! This is what the logistic function wants you to think. It lulls you into a false sense of security, and then it strikes. Dynamicist James Yorke had exactly this warning for the scientific community, which he elegantly captured and elucidated in his 1975 paper with Tien-Yien Li, "Period Three Implies Chaos." We will discuss the statement and proof of the main theorem of the paper, and then perhaps let things degenerate into a philosophical debate about whether we can ever grapple systematically yet effectively with the highly nonlinear world in which we live. ### Wednesday, April 14, 2010 5-7 PMKnotty Biology, by Dr. Jennifer MannRLM 12.104 [poster] What are instances of knotting in biological polymers? Are they advantageous or disadvantageous? When undesirable, how are they resolved? A video of this lecture may be found here. ### Wednesday, April 7, 2010 5-7 PMAn Introduction to Game Theory by Professor Max Stinchcombe RLM 12.104 [poster] [notes] A video of this lecture may be found here. ### Wednesday, March 31, 2010 5-7 PMThe Black-Scholes-Merton Formula and Risk-Neutral Pricing by Phil MoninRLM 12.104 [poster] [notes] The Black-Scholes-Merton formula, discovered in the early 1970s, led to a revolution in the world of pricing and hedging financial derivatives and helped spawn the fledgling academic field of mathematical finance. In this talk, I will introduce financial derivatives, discuss the rudiments of risk-neutral pricing, derive the Black-Scholes-Merton formula and discuss its limitations. If time permits, I will discuss other topics in mathematical finance. A video of this lecture may be found here. ### Wednesday, March 24, 2010 5-7 PMBlown Away: What Knot to Do When Sailing, by Sir Randolph Bacon III, cousin-in-law to Colin AdamsPAI 4.42 (NEW LOCATION!) [poster] Being a tale of adventure on the high seas involving great risk to the tale teller, and how an understanding of the mathematical theory of knots saved his bacon. No nautical or mathematical background assumed. A video of this lecture may be found here. ### Wednesday, March 10, 2010 5-7 PMDynamical Systems, Stochastic Processes, and Probabilistic Robotics, by David RosenRLM 12.104 [poster] [notes] Probabilistic robotics is the subfield of robotics concerned with perception and control in the presence of uncertainty. By explicitly modeling uncertainty and error in a mathematically sound way, probabilistic algorithms provide real-world autonomous systems with a level of robustness unattainable by purely deterministic methods. This talk will provide an introductory overview of the subject, focusing on the mathematical underpinnings of some of the more common classes of algorithms. A video of this lecture may be found here. ### Wednesday, February 24, 2010 5-7 PMThe infinitude of primes: the beauty of many proofs by Dr. Brad HenryRLM 12.104 [poster] Mathematicians derive immense pleasure from discovering elegant proofs of existing theorems. We are, after all, artists of reason. During this talk, we will investigate as many different proofs of the infinitude of primes as time allows. We will begin with Euclid's beautifully clever proof involving a few basic facts from arithmetic. From there we will venture into proofs involving calculus, infinite series, and topology. Each proof is truly elegant and each gives us new insight into the complexity of the prime numbers. A video of this lecture may be found here. ### Wednesday, February 17, 2010 5-7 PMAnalysis of PDEs and conformal geometry on the sphere, by Nestor Guillen RLM 12.104 [poster] [notes] The talk will be about the problem of conformal deformation of a Riemannian manifold (known as the Yamabe problem) and use it as an excuse to present several important ideas and tools from geometry and analysis of partial differentieal equations. The Yamabe problem arises when one tries to deform the geometry of a space in a way that preserves angles but not lengths. Usually one wants to deform a bad metric into a nicer one in a conformal fashion, and this problem is itself of great interest not only in geometry but in quantum field theory. It turns out that finding a conformal deformation that makes the geometry nice'' is equivalent to solving a nonlinear partial differential equation. Some of the tools needed to solve this particular equation will be presented. There will be particular emphasis on the direct method'' in the calculus of variations, inequalities of Sobolev-type, and the Laplace-Beltrami operator of a manifold. Knowledge of some differential geometry and topology would be helpful but is not required. A video of this lecture may be found here. ### Wednesday, February 10, 2010 5-7 PMFrom quantum to classical: On the emergence of physical laws and the underlying mathematics by Professor Thomas ChenRLM 12.104 [poster] [notes] While modern quantum theory describes matter as it appears in everyday life with incredible accuracy, and is "the correct theory", classical physical theories such as Newtonian mechanics, classical electrodynamics, or fluid dynamics are still as widely used and powerful as ever. As experience shows, it is not necessary to invoke quantum mechanics to determine the orbits of the planets, to design the wings of an airplane, or to build an electric engine. But why not? In this talk, it is described how certain examples of (semi)classical theories in physics are derived from first principles in quantum mechanics, in the process of which the former become independent of the latter. This typically involves a transition from small to large, from complex to average, or more generally, from one scale to a fundamentally different one. The examples addressed here allow for a fully mathematically rigorous analysis, based on a combination of techniques stemming from partial differential equations and quantum field theory. A video of this lecture may be found here. We will also have a representative from the NSA making a brief announcement at the beginning of the meeting. She will also be at the career fair on Friday. There are links to several handouts she's given us here and here. (This happened on the 10th instead of the original 3rd because of the gas leak in RLM on the 3rd.) ### Wednesday, January 27, 2010 5-7 PM"Curves, surfaces and their higher dimensional cousins" by Professor Bob GompfRLM 12.104 [poster] [notes] Curves and surfaces are examples of manifolds, one of the most important types of spaces arising in mathematics. These are frequently encountered in applications - the earth's surface, the universe and the space of configurations of a mechanical system are all examples. We will explore the theory of manifolds at the most fundamental level, namely their topology, from visualizing examples to investigating what is known regarding their classification. A video of this lecture may be found here. ### Saturday, December 5, 2009 12-12 AMPutnam Exam ### Wednesday, December 2, 2009 5-7 PMBouncing of the Balls: Indra's Pearls by Professor de la LlaveRLM 12.104 According to legend, the god Indra had a collar of pearls. Each of the pearls saw a reflection of all the other pearls, including the reflection of itself in the other pearls. A video of Professor de la Llave's lecture can be found here for part 1 and here for part 2. ### Wednesday, November 25, 2009 5-7 PMReminder: no meeting this week!RLM 12.104 Just a reminder, there won't be a meeting. Have a good Thanksgiving! ### Wednesday, November 18, 2009 5-7 PMThe enigma of the equations of fluid motion: a survey of existence and regularity results by Professor Natasa PavlovicRLM 12.104 [poster] The partial differential equations that describe the most crucial properties of the fluid motion are the Euler equations. They are derived for an incompressible, inviscid fluid with constant density. Some basic questions concerning Euler equations in 3 dimensions are still unanswered. For example, it is an outstanding problem to find out if solutions of the 3D Euler equations form singularities in finite time. The equations that describe the most fundamental properties of viscous fluids are the Navier-Stokes equations. As with the Euler equations the theory of the Navier-Stokes equations in 3D is far from being complete. The major open problems are global existence, uniqueness and regularity of smooth solutions of the Navier-Stokes equations in 3D. In this talk we will give a survey of some known results addressing existence and regularity of solutions to these equations. A video of Professor Pavlovic's lecture can be found here. ### Thursday, November 12, 2009 8-10 PMMath Club Movie NightRLM 12.104 • Thursday November 12th • 8:30pm (I know it says 8:00 at the top; it's really 8:30 • RLM 12.104 • Donald Duck in Mathmagicland... and more! ### Wednesday, November 11, 2009 5-7 PMKnot Theory - the algebro-geometric wayRLM 12.104 [poster] [notes] One way to gain insight into the topological structure of hyperbolic knot and link complements is through the existence of essential surfaces; surfaces which when embedded in the knot complement retain all of their topological information. How then does one detect essential surfaces in hyperbolic knot complements? It is through the algebro-geometric object called the character variety. Through discussing hyperbolic knot complements and describing their associated character varieties, the goal of my talk is to explain how topology and algebraic geometry come together. The opening act for this talk will be Danny Fast fingers preforming my theme song for my character Emie-Lou-the-Math-Guru after Bill Nye the Science Guy. A video of Emily's lecture can be found here. ### Wednesday, November 4, 2009 5-7 PMREU's and Summer Math Programs: an information session on mathematics programs for undergraduatesRLM 12.104 [poster] Have you ever wondered what research in math is like? Have you ever wanted to know what it is that graduate students spend all of their time doing? If so then a math REU (Research Experience for Undergrads) or math program is a great way to try to spend your summer. This Wednesday, come learn everything you want to know about REU's. We'll have experienced graduate and undergraduate students to relay their knowledge and answer all your questions. Here's a list of REU's. Some other summer programs include SPWM, PCMI, EDGE and a summer school at Carleton (ask Emily for more information). ### Sunday, November 1, 2009 1-3 PMVolunteers wanted for Math Adventure! Austin Math Circle is looking for volunteers to help run its Math Adventure -- a day of fun, outdoor math games for local area high school and middle school students. No experience is required; anyone who would enjoy playing games with children on a Sunday afternoon is welcome. The event will take place on Sunday, November 1st, from 1 PM to 3 PM. Contact Dave Jensen if you are interested. ### Wednesday, October 28, 2009 5-7 PMVisualizing the Fourth Dimension, from 'Flatland' to 'Sphereland' and Beyond by Professor Thomas Banchoff from Brown UniversityRLM 12.166 [notes] "How do mathematicians, artists, and philosophers try to comprehend geometry beyond our third dimension? Modern computer graphics approaches make it possible to see and interact with four-dimensional phenomena in ways never available before. With the aid of computer graphics images and animations I will discuss the 4th dimension. A video of Professor Banchoff's lecture can be found here. ### Monday, October 26, 2009 4-6 PM"Salvador Dali and the Fourth Dimension: A Mathematician's Personal Reflections" by Professor Thomas Banchoff of Brown UniversityART 1.120 [poster] Professor Thomas Banchoff of Brown University will be giving a lecture next Monday, Oct. 26 at 4:00 in the art building, ART 1.120. (The ART building is located at San Jacinto and 23rd, across from the stadium.) Salvador Dali was fascinated by science and mathematics, and geometric objects in various dimensions are central to many of his paintings, for example 'Corpus Hypercubicus'. Where did Dali get his mathematical inspirations and how did he incorporate them into his painting? This talk will recount meetings with the artist over a ten year period, and it will be illustrated by computer graphics images and animations. Professor Banchoff will also be giving a talk to the Math Club on October 28th. ### Wednesday, October 21, 2009 5-7 PMPhylogenetic Supertree Methods: tools for reconstructing the Tree of Life by Dr. Michelle SwensonRLM 12.104 [poster] Estimating the Tree of Life, an evolutionary tree describing how all life evolved from a common ancestor, is one of the major scientific objectives facing modern biologists. This estimation problem is extremely computationally intensive, given that the most accurate methods (e.g., maximum likelihood heuristics) are based upon attempts to solve NP-hard optimization problems. Most computational biologists assume that the only feasible strategy will involve a divide-and-conquer approach where the large taxon set is divided into subsets, trees are estimated on these subsets, and a supertree method is applied to assemble a tree on the entire set of taxa from the smaller "source" trees. Dr. Michelle Swenson will present supertree methods in a mathematical context, focusing on some theoretical properties of MRP (Matrix Representation with Parsimony), the most popular supertree method, and SuperFine, a new supertree method that outperforms MRP. A video of Dr. Swenson's lecture can be found here. ### Wednesday, October 14, 2009 5-7 PMMobius: The Original Invertebrate* by Nick RauhRLM 12.104 [poster] An introduction to number-theoretic functions and the principle of Mobius inversion. A little group theory, some identities/applications, and a generalization to posets. * Modern biology would likely classify Mobius as a vertebrate. Were he an invertebrate, even the claim to originality as such would be historically dubious, at best. A video of this lecture may be found here. ### Wednesday, October 7, 2009 5-7 PMEuler, Brouwer, Hopf: Topology by the Numbers, a talk by Dr. VickRLM 12.166 [poster] [notes] We will explore three ways in which integers arise in topology -- the Euler number, the Brouwer degree, and the index of a singularity of a vector field -- and see how the relationships among them tell us important facts about surfaces and manifolds. No prior knowledge of the subject will be assumed. A video of this lecture may be found here. ### Tuesday, October 6, 2009 5-7 PMPizza and Problem SolvingRLM 10.176 Starting Sept 29, the Putnam practice sessions (5PM on Tuesdays, 10.176 classroom) will include free food. Putnam registration is due by October 13th. Professor Sadun is organizing the Putnam exam for UT this year. The exam is on Saturday December 5th. So if you are interested in taking the exam send an email ASAP with subject: Putnam registration; your name, year and EID. If you have any questions please email Professor Sadun. ### Sunday, October 4, 2009 5-7 PMGoogle calendar (the date has no meaning)The Agenda tab has the information ### Friday, October 2, 2009 12-1 AM4-Dimensional Space with Jeff WeeksRLM 12.104 [poster] This will be an informal lunch session on 4-dimensional space. Jeff Weeks will start the discussion and then open the floor to questions. There will be pizza from 11:40-12pm and the talk will start at noon. Jeff Weeks is a great topologist and an amazing speaker. He has written the book "The Shape of Space", produced a SnapPea, a progam many topopogist use in their research, and programed torus games. As this is during lunch we will have some pizza. ### Thursday, October 1, 2009 7-RTG in Topology Public Lecture: The Shape of Space by Jeff WeeksThompson Conference Center 1.110 When we look out on a clear night, the universe seems infinite. Yet this infinity might be an illusion. During the first half of the presentation, computer games will introduce the concept of a “multiconnected universe”. Interactive 3D graphics will then take the viewer on a tour of several possible shapes for space. Finally, we'll see how recent satellite data provide tantalizing clues to the true shape of our universe. The only prerequisites for this talk are curiosity and imagination. For middle school and high school students, people interested in astronomy, and all members of the UT and surrounding communities. Reception to follow. ### Wednesday, September 30, 2009 5-7 PMMath Club meeting times updateRLM 12.104 There won't be a Wednesday Math Club this week, instead there will be Jeff Week's awesome talk on Thursday (Oct 1st)! He is also holding a session on Friday. ### Wednesday, September 23, 2009 5-7 PMEquivalence Classes in Higher Math by John MethRLM 12.104 [poster] [notes] We first encounter equivalence classes when we use fractions, but rarely question the fundamental objects we are dealing with. I want to explore a potpourri of examples of mathematical objects built out of equivalence classes, and the relationships of these objects to each other. We will see equivalence classes from most of the main branches of modern mathematics. A video of John Meth's talk can be found here. ### Wednesday, September 16, 2009 5-7 PMHow to Become a Mathematician in Just 5-7 YearsRLM 12.104 [poster] [notes] Do you want to be a mathematician? Are you interested in going to graduate school? To address questions and concerns like these, we will be having a panel consisting of faculty members and graduate students from UT. We will go over things to think about as a math major, how to apply to graduate school, and we may even warn you about some signs that you may be turning into a mathematician. ### Tuesday, September 15, 2009 5-7 PMWeekly Putnam Problem Solving sessions start this weekRLM 10.176 They will be every Tuesday. Professor Sadun will be running them. ### Tuesday, September 15, 2009 2-4 PMGRE study sessionRLM 12.166 Contact Lucia for more information. Tuesdays from 2-4 until Oct. 20!! Book(s) and study material will be provided. -Location may vary- ### Sunday, September 13, 2009 6-8 PM(Pi)cnicEastwoods/Harris Park Math Club Picnic! There will be awesome food and even more awesome people to meet! Everyone is welcome. ### Wednesday, September 9, 2009 5-7 PMCurvature, polyhedra, and modular origami by Professor SadunRLM 12.104 [poster] [notes] What does curvature mean when you have a polygon, or a polyhedron, with corners? In answering this question, we'll see why there are exactly 5 regular (Platonic) polyhedra. Later on, we'll see how to fold variants of these shapes out of square pieces of paper. A video of Professor Sadun's talk can be found here. ### Wednesday, September 2, 2009 5-7 PMCircles, Rings, and Tractors: Clever Cleaving for Finding FormulasRLM 12.104 [poster] [notes] How do we discover the formulas for the areas of objects such as circles and annuli and the volumes of solids such as cones, pyramids, and spheres? In each case, an effective strategy involves dividing the object into small pieces and seeing how the small pieces can be re-assembled to produce an object whose volume or area is easier to compute. Some of these methods were devised thousands of years ago and some of them seem to be relatively new. ### Wednesday, May 6, 2009 5-7 PM"Arithmetic Geometry: From Circles to Circular Counting" by Adriana SalernoRLM 12.104 In this talk, I will show you a glimpse of one of the most exciting and accessible facets of research in modern number theory: arithmetic geometry. We will start with a (gentle) introduction to this area of research through some familiar examples. Then we will move on to a not so familiar example where we count solutions of equations mod p. I will end by answering two of the oldest and most mystifying questions in mathematics: how does this work fit into the bigger picture, and who cares? ### Wednesday, April 29, 2009 5-7 PMStudent TalksRLM 12.104 We will have the following talks this week, all given by UT students: • Storm Search Modeling and the Constant Advection Equation by Nancy OkeudoIn This talk we will discuss the meteorological background, physics, and numerics the Con-stant Advection equation, which is a simplified version of the 1-dimensional Shallow Water equations. In particular, we will focus on how the 1900 Galveston hurricane led to further study of storm surge modeling, the derivation of the Constant Advection equation, and the basic properties of numerical methods for its approximation. • RSA: How It's Done by Christy Sheldon In this talk I will develop background on RSA. After discussing all the elements used to encode a message, I will describe how to then decode the message using Fermat's Little Theorem. • Folding Symmetries by Gilbert Bernstein In this talk I will discuss the old mathematical topic of planar symmetries from a fresh point of view. Since at least 1924 (if not as far back as the Egyptians) it has been known that there are only 17 possible types of planar symmetry, sometimes called the Wallpaper groups. We will look at what happens when we "fold" up these wallpaper patterns along their symmetries. • Erdos' Conjecture on Arithmetic Progressions by Michael Kelly Arithmetic progressions of natural numbers are sequences whose consecutive terms are equally spaced. Erdos conjectured (still an open problem) that if some very simple data about the "density" of a set in the natural numbers is given, then the set will necessarily contain arithmetic progressions of any finite length. We will introduce these concepts and give a few examples. • Multiplying Vectors and Determinants by Justin Hilburn Have you ever wondered why the cross product only works in dimension 3? Or where the formula for the determinant came from? It turns out that both of these questions can be answered by looking at the exterior algebra. ### Wednesday, April 22, 2009 5-7 PM" Expanding graphs" by Matthew StoverRLM 12.104 Expanders are families of graphs which are sparse and highly connected'. Conceptually, they represent very efficient communication networks, where vertices are well-connected by few edges. However, explicit constructions of expanders -- especially the best possible' expanders, called Ramanujan graphs -- were not found until the 80s. I will explain what expanders are, then explain the history behind the first constructions, giving an idea of the wide range of deep mathematical ideas required to build them. ### Wednesday, April 15, 2009 5-7 PM"'Am I knotted?' A conversation between Neil Hoffman and Knestor the knot about the Jones polynomial" by Neil HoffmanRLM 12.104 If you have a knot, one thing you might want to know is if that knot is actually knotted or if it is the unknot. In the event you have a knot that can talk back to you, it might just ask if it's knotted. We'll see how the Jones polynomial can help us give a good partial answer. ### Wednesday, April 8, 2009 5-7 PM"Lie groups, Lie algebras, and their "best" metrics: transforming transformation groups" by Dan KnopfRLM 12.104 TBA ### Thursday, April 2, 2009 5-7 PMTalk by Professor ReichlRLM 7.104 Part of the Math and Physics Lecture Series TBA ### Wednesday, April 1, 2009 5-7 PM"On billiards and time irreversibility... The birth of Statistical Mathematical Physics" by Professor GambaRLM 12.104 Part of the Math and Physics Lecture Series We will discuss the legacy of Ludwig Boltzmann in the connections between time irreversible stochastic processes and the theory of the Boltzmann Equation in the modeling of probabilistic dynamics of particle interactions modeled by elastic billiards, as well as connections to conservation laws and compressible fluid models. ### Thursday, March 26, 2009 5-7 PM"An Introduction to Anti-de-Sitter Space/Conformal Field Theory Correspondence" by Professor DistlerRLM 7.104 Part of the Math and Physics Lecture Series TBA ### Wednesday, March 25, 2009 5-7 PM"An application of quantum field theory to group theory" by Professor FreedRLM 12.104 Part of the Math and Physics Lecture Series TBA ### Wednesday, March 11, 2009 5-7 PM"Enumerative Geometry - Learning to count all over again" by Brian KatzRLM 12.104 Some of the most exciting mathematics is born from a connection between two seemingly disparate ideas. For example, enumerative geometry calls upon the tools of both combinatorics and algebraic geometry, and each sheds light on the other. In my opinion, the major player in this partnership is the idea of a moduli space. In this talk, I will flesh out these claims and use the geometry of a few moduli spaces to answer enumerative (counting) questions. If time allows, we will also parallel these ideas for tropical algebraic geometry. Most of my examples will be very familiar but viewed from a different perspective, so the majority of the talk will be accessible to any student with a basic understanding of linear algebra. ### Wednesday, March 4, 2009 5-7 PM"Data, and what to do with it" by Martin BlomRLM 12.104 Science tries to make models for what happens around us based on what we see. In this talk I will give an introduction to probability and Bayesian statistics, which is a mathematical framework for quantifying our belief in different models, and rules for how we should change our belief as new information becomes available. ### Wednesday, February 25, 2009 5-7 PM"Complex Numbers and the Beauty of Mandelbrot Set" by Prof. DanielsRLM 12.104 Some of the mathematics and properties behind one of the most intricate and interesting shapes in Complex Analytic Dynamics, The Mandelbrot Set, will be explored in this presentation. Background topics presented will include the Quadratic Map, Orbit Analysis, and properties of The Julia Set in addition to Mandelbrot Set properties. All topics will be introduced from the standpoint of discrete deterministic Chaos. Finally, computer graphics will be used to visually illustrate some of the mathematics discussed including how to “count” and “add” on the Mandelbrot Set. ### Wednesday, February 18, 2009 5-7 PM"The Jump to Light Speed" by Prof. AllcockRLM 12.104 Everyone knows that when your starship jumps to light speed: you see the stars suddenly rush past, so that it looks like an explosion of stars in front of you. But this isn't what actually happens: really, they appear to all rush together in front of you. So it looks like an *implosion* instead. Really this is all about the boundary of hyperbolic 3-space. Find out what this means! ### Wednesday, February 11, 2009 5-7 PM"Machines Reasoning about Machines" by Professor J. MooreRLM 12.166 Artificial intelligence,'' Lisp,'' theorem proving,'' program verification,'' formal methods,'' ... these are phrases that conjure up conflicting images in many people. The basic idea is this: Since (a) formal mathematical logic can be used to specify precisely what a computer program or piece of hardware is supposed to do, and (b) software can be written to manipulate formal mathematics to discover and check proofs, therefore (c) machines can check whether software and hardware designs do what they're supposed to. But how realistic is it to specify hardware and software precisely? Formally? What logic would you use? How realistic is it to apply AI-based theorem proving techniques to prove theorems about hardware and software? Is this a pipe-dream or a widely used industrial certification method or something in between? In this talk I'll describe 38 years of work on the subject of applying mechanized mathematics to hardware and software. The system I'll discuss is often called the Boyer-Moore theorem prover,'' which is actually a description of a series of theorem provers for pure Lisp dating back to 1971, written by Boyer, Moore, and Kaufmann, and used by such companies as AMD, IBM, Rockwell-Collins, and others. I'll describe how we got to this interesting point in history, where machines are sometimes able to reason about other machines -- and themselves. ### Wednesday, February 4, 2009 5-7 PM"Proof by Pictures: A Tour through Visual Mathematics" by Eric KatermanRLM 12.104 How do you evert a sphere? What's left once you remove a knot from space? Does every sphere bound a ball in space? We'll watch some amazing videos that will help us answer these questions. ### Wednesday, January 28, 2009 5-7 PM"Quadratic Reciprocity and Weil Reciprocity" by Dr. Jacob Lurie (MIT)RLM 12.104 I'll begin by reviewing the law of quadratic reciprocity over the ring of integers, and then explain how it can be generalized to the setting of a polynomial algebra over a finite field. ### Wednesday, December 3, 2008 5-7 PM"Spherical Geometry: Methods and Magic" by Cody PattersonRLM 12.104 If you passed high school geometry, you're familiar with how distances and angles "work" in \mathbb{E}^2, the Euclidean plane. But how do these concepts carry over to the unit sphere \mathbb{S}^2? Can we do trigonometry on the unit sphere as effortlessly as we can in the Euclidean plane? I'll show that the answer to this question is yes, and that in fact certain aspects of spherical trigonometry are even more elegant than their counterparts in Euclidean trigonometry. If you've ever wanted to know how to find the area of a spherical polygon, this is the talk for you. ### Wednesday, November 19, 2008 5-7 PM"The ABC Theorem and the ABC Conjecture" by Mark RothlisbergerRLM 12.116 In many ways, the ring of integers is similar to the ring of polynomials over the real numbers. For the purposes of this talk, the main similarity is that both are universal factorization domains. Over the integers, prime numbers play a key role, mirrored in many respects by irreducible polynomials over the integers. We will first prove ABC theorem for polynomials, the main consequence of which is that if two polynomials with zeros of large multiplicity are added together, their sum can not have any zeros of large multiplicity. It is easy enough to formulate a corresponding conjecture over the integers; however the presence of counterexamples forces modifications. Furthermore, the absence of a key tool used in the proof of the ABC theorem means that over the integers, nothing has yet been proved. However, we will discuss the far-reaching implications of the ABC conjecture on other areas of number theory. Finally, I will introduce to you a way that you can help progress towards the solution of this important, yet elusive result at home, even while you're asleep! ### Wednesday, November 12, 2008 5-7 PM"Down with sine and cosine!!! How these transcendental functions are holding you back from Geometry of a more Universal variety, and what you can do to stop them" by Sarah RichRLM 12.104 The talk will be based on a book called "Divine Proportions". The author of the book devises a method for doing "rational trigonometry" over any field, which leads to a concept of geometry over any arbitrary field of characteristic not equal to two. I will discuss both the foundations and some of the interesting results of his work, as well as possibly contemplate its merits and limitations. ### Wednesday, November 5, 2008 5-7 PM"Knots and How to Color Them" by BrandyRLM 12.104 How do we know when a knot is tricolorable? Namely, given a knot projection and three colors, can we color each arc of the knot so that at every crossing the arcs which meet there are either all the same color or all different colors? After discussing some properties of knots, we will talk about how to determine when knots are not just tricolorable but also n-colorable. ### Wednesday, October 29, 2008 5-7 PM"A Brief Introduction to the p-adic Numbers" by Keenan KidwellRLM 12.104 The set of real numbers is the completion of the set of rational numbers with respect to the familiar notion of distance furnished by the usual "Archimedean" absolute value; intuitively, this means that the reals consist of the rationals together with all the limits of sequences of rationals that ought to converge with respect to this absolute value but which fail to do so because of certain "holes." If p is any rational prime, then the set of p-adic rational numbers is an analogous object obtained by completing the rationals with respect to a different absolute value, the "non-Archimedean" p-adic absolute value, which measures the distance between two rationals based on the p-divisibility of their difference. Despite the fact that the sets of real and p-adic numbers both constitute completions of the rationals, the latter is a much more exotic object, both topologically and algebraically; for instance, every triangle in the p-adic world is isosceles, and any point of a ball is its center. We will encounter these and other peculiarities as we detail the construction of the p-adic numbers and explore some of their most interesting features. ### Wednesday, October 22, 2008 5-7 PM"The Mathematics of Cancer Biology" by NestorRLM 12.104 This talk has two goals: one is to show how mathematics helps us understand cancer and the other is to present some important ideas from analysis. In the last couple of decades we have seen the emergence of several mathematical models for cancer. These models have drawn inspiration from field is such as classical fluid dynamics. It is the mathematics behind these models that will very likely allow scientists to understand the behavior of tumors at a deeper level. In this way, mathematics will potentially play a role in the development of reliable and effective cancer treatments. After providing some background in biology, I will present a particular model for tumors and explain a few of the techniques behind its study. ### Wednesday, October 15, 2008 5-7 PM"Realizability of Polytopes" by Eric KatzRLM 12.104 Given a polytope (a higher dimensional generalization of a polygon), one can write down the data of the faces and how they fit together. Can such a process be reversed? Given abstract data of faces that satisfy certain axioms, can one reconstruct the polytope or even be sure that one can find a polytope with that face data? This is the realizability question. It turns out that realizability plays host to all sorts of bizarre phenomena. There is face data for which one can find no polytope. There is also face data for which one can only find polytopes whose vertices have irrational coordinates. We will explore such odd behavior and give a method for generating all sorts of weird examples. ### Wednesday, October 8, 2008 5-7 PM"Let's Meet at the Euler Characteristic" by Professor Gary HamrickRLM 12.104 The Euler Characteristic of surfaces provides a wonderful opportunity to see the interplay of geometry, algebraic topology, and analysis that does not require sophisticated knowledge to grasp. It is actually a very special case of the celebrated Atiyah-Singer Index Theorem, a result that touches on virtually all areas of mathematics. In the last few decades mathematics has progressed in such a way that its various branches such as algebra, analysis, geometry, and topology are becoming ever more inextricably linked together. An early such development occurred in the 17th century with Descartes' invention of algebraic geometry via his introduction of Cartesian coordinates. But the process has accelerated tremendously in relatively recent times. A spectacular example in Perelman's proving Thurston's Conjecture relating the geometry and topology of 3-dimensional manifolds by the use of partial differential equations (analysis). A small corollary is the Poincare' Conjecture, one of the$1,000,000 Clay Institute Problems.

Another such example is Andrew Wiles' proof of Fermat's Last Theorem the most famous problem in number theory. What Wiles actually proved was a result in algebraic geometry that had been known to imply Fermat's Last Theorem.

### Wednesday, October 1, 2008 5-7 PM"Surfaces, Tessellations and Hyperbolic Geometry" by Grant LakelandRLM 12.104

How does one find the shortest distance, and a path of that distance, between two points on a torus? How about on other surfaces? In this talk, I'll explain an answer to the first question involving tessellations of the Euclidean plane, and how it leads us to study tessellations of the hyperbolic plane to answer the second.

### Wednesday, September 24, 2008 5-7 PMHow to Become a Mathematician in Just 5-7 YearsRLM 12.104 [poster]

Do you want to be a mathematician? Are you interested in going to graduate school? To address questions and concerns like these, we will be having a panel consisting of faculty members and graduate students from UT. We will go over things to think about as a math major, how to apply to graduate school, and we may even warn you about some signs that you may be turning into a mathematician.

### Wednesday, September 17, 2008 5-7 PM"Mobius inversion and colorings of graphs" by Dr. Nicholas Proudfoot (University of Oregan)RLM 12.104 [poster]

A proper coloring of a graph is a way to label the vertices such that adjacent vertices get different labels. (The famous Four Color Theorem, proven in 1976, says that any graph that can be drawn on a blackboard without edge crossings admits at least one proper coloring with at most four different colors.) I'll discuss the beautiful theorem of Mobius inversion for functions on posets, and explain what it has to do with coloring graphs.

### Wednesday, September 10, 2008 5-7 PM"Discrete Mathematics for Molecular Models", by Andrew GilletteRLM 12.104 [poster]

A simplistic but useful model of a molecule treats its atoms as filled spheres of fixed radius with fixed relative position in space. In this talk, we will examine some of the discrete math concepts aiding such models, including Voronoi and Delaunay diagrams and their generalization known as power diagrams. If time permits, we will also discuss how such diagrams can be used to identify certain topological properties of the molecule. This talk assumes no background in biology and only a basic comfort with mathematical notions.

### Wednesday, September 3, 2008 5-7 PMThe Pythagorean Theorem, Euclid's Parallel Postulate, and non-Euclidean Geometry by Braxton CollierRLM 12.104 [poster]

In this talk I will explain a simple, visually intuitive proof of the Pythagorean theorem. Considerations of the ingredients that go into this deceptively simple proof lead to an examination of some basic questions concerning the foundations of geometry, and in particular the validity of Euclid's fifth, "parallel" postulate. Historically, attempts to prove this postulate from Euclid's other axioms led to the discovery of non-Euclidean geometry. Not only does non-Euclidean geometry play a vital role in modern mathematics, but it also features centrally in Einstein's description of gravity as a manifestation of space-time curvature.

### Wednesday, April 9, 2008 5-7 PMAbelian Sandpiles and My Favorite Open Math Problem, by Geir HelleloidRLM 12.104

The abelian sandpile model was invented by physicists to study physical phenomena like avalanches, but the idea was quickly co-opted by mathematicians who realized that they could do a lot of fun mathematics with the model. In fact, playing around with the model feels like playing a game, so it is sometimes called the chip-firing game. I'll show you a lot of the math behind the model, focusing on the group-theoretic aspects. You don't need to know anything about group theory though; in fact, coming to this talk is a good way to find out what a group is! Highlights will include crazy and beautiful fractal-like images and simulations, the entire audience standing up to physically compute sandpile addition, and my favorite open math problem.

### Wednesday, April 2, 2008 5-7 PMPublic Key Cryptography, by Brendan YoungerRLM 12.104

Public-key cryptography allows people to send encrypted messages to each other without ever having to get together to share a common secret. This makes it particularly attractive for performing secure transactions over the internet or sending super-secret spy messages. It's also rather intriguing in that it requires a "trap-door" operation which is very easy to perform in one direction and very difficult to perform the inverse of. In this talk, I will discuss the RSA cryptosystem and some of the attacks against it. I will then try to give an overview of elliptic curve cryptosystems and at least point out the difficulties in choosing appropriate parameters. If time permits, I will discuss cryptosystems based on the knapsack problem and why those have failed.

### Wednesday, March 26, 2008 5-7 PMThe Music of the Spheres, by Alex KahleRLM 12.104

Exotic spheres, infinite spheres, hairy spheres... who the humble sphere had so many surprises?

### Friday, March 21, 2008 3-4 PMDean's Scholars presents computer scientist Ron GrahamACES 2.302

There is no question that the recent advent of the modern computer has had a dramatic impact on what mathematicians do and on how they do it. However, there is increasing evidence that many apparently simple problems may in fact be forever beyond any conceivable computer attack. In this talk, Dr. Ron Graham will describe a variety of mathematical problems in which computers either have had, may have or will probably never have a significant role in their solutions.

Ron Graham was chief scientist at AT&T Bell Labs before taking a job at the University of California-San Diego in the Computer Science department. Dr. Graham is a former president of the American Mathematical Society. He has also been featured on Ripley's Believe It or Not as a talented mathematician, juggler, and trampolinist while also holding a spot in the Guinness Book of World Records for creating the worlds largest number used in a serious mathematical proof. He has produced over 300 papers, including several with his friend Paul Erdos, and won the annual Steele Prize for lifetime achievement from the American Mathematical Society in 2003. Without a doubt, Ron Graham is one of the world's foremost mathematicians in discrete mathematics.

### Wednesday, March 19, 2008 5-7 PMTilings: a mathematical model for crystals and quasicrystals, by Natalie FrankRLM 12.104 [notes]

Crystals are solids that have well-ordered, repeating atomic structures. Tilings of R^2, R^3, or even R^n are mathematical models of this structure. In the laboratory, scientists can measure the diffraction spectrum of a solid by shining x-rays through it. If the material is a crystal, the spectrum will have sharply defined brights spots known as Bragg peaks. Diffraction spectra can also be computed for a tiling, and if it is periodic, the spectrum will show Bragg peaks. Until the 1980's, it was thought that only crystals produce Bragg peaks. It was then that a new form of matter was discovered, one that had Bragg peaks in its spectrum, but could not have the well-ordered atomic structure of a crystal. This form of matter was named quasicrystal. We will discuss some of what is known about quasicrystals and their tiling counterparts.

No School!

TBA

### Wednesday, February 27, 2008 5-7 PMHuh? Mathematicians study knots just for the sake of it?, by Emily LandesRLM 12.104 [poster] [notes]

Take two ropes, loosely tie each into the same knot and fuse together the two free ends of each strand. Drop both knots on the ground. There, they each appear as a concoction of over and under crossings. Most likely, these identical knots will fall differently. Now work backwards. Start with two 2D concoctions of over and under crossings. When do they correspond to the same 3D knot? How can we be sure?

The classification of knots involves a search for knot invariants, properties that remain unchanged under three specific perturbations called Reidemeister moves. One such invariant is the Khovanov homology of a knot projection. As the machinery behind this invariant requires significant development, I will use my talk to present the intuitive picture.

### Wednesday, February 20, 2008 5-7 PMThe Million-Dollar Question: Is God a Geometer?, by Prof. Lorenzo SadunRLM 12.104 [notes]

Yang-Mills Theory is a way to cast fundamental physics in geometric terms. One of the million-dollar "millenium" problems posted by the Clay Institute is to rigorously construct a Yang-Mills theory in 4 dimensions and prove some properties about it. I'll go over the history of geometric constructs in physics, and explain what the Yang-Mills problem is all about.

### Wednesday, February 13, 2008 5-7 PMWhat are the possible shapes of space?, by Professor Dan KnopfRLM 12.104 [poster] [notes]

Manifolds are objects (like curves, surfaces, and our universe) that look like Euclidean space locally but whose global picture may be much different. We'll discuss some of what interests mathematicians when they study the topology and geometry of such objects. For 2-dimensional manifolds, a strong connection between their topology and geometry was known since the nineteenth century. For 3-dimensional manifolds, a similar connection has only recently been verified. We'll talk about this connection and some of the big ideas behind its proof. For 4-dimensional manifolds, we aren't even sure yet what the right questions are. Maybe you will study these some day.

### Wednesday, February 6, 2008 5-7 PMRamsey Theory and Distortion: Is Euclidean geometry inevitable?, by Professor Ted OdellRLM 12.104 [poster] [notes]

An example of a Ramsey theorem is that if we have 17 red and blue balls then there are at least 9 red balls or else 9 blue balls. We will discuss some more dramatic extensions of this theorem and then move on to different geometries in 2,3,4, or n- dimensional or even infinite dimensional space. As we shall explain the Ramsey problem in this setting is "Can you truly distort Euclidean space?"

### Wednesday, January 30, 2008 5-7 PMConfession of a Physicist to Mathematicians, by Professor Cecile DeWitt-MoretteRLM 12.104 [poster] [notes]

Listening to what mathematicians say is sometimes good and sometimes bad.

### Wednesday, January 23, 2008 5-7 PMKnot Theory and DNA, by Professor Jennifer MannRLM 12.104 [poster] [notes]

In our daily lives we encounter tangling and knotting in long, flexible objects such as extension cords and strings of Christmas tree lights. Often this knotting is an annoyance, and sometimes it compromises the function of the cord or string. Knotting also occurs in DNA. We will discuss the biological consequences of DNA knots and how DNA knots are resolved.

### Wednesday, December 5, 2007 5-7 PMBeyond Curves and Surfaces by Prof. Dan FreedRLM 12.104 [poster]

Curves are 1-dimensional and surfaces 2-dimensional. Geometers study shapes of arbitrary--even infinite--dimension. We will explore this idea, how such shapes (called manifolds) arise, and talk about some exciting recent work about 3-dimensional smooth manifolds.

### Wednesday, November 28, 2007 5-7 PMMovie Night!RLM 12.166 [poster]

Join us for another mathy movie: ENIGMA, starring Dougray Scott and Kate Winslet.

### Wednesday, November 14, 2007 5-7 PMGroup Action for Science Nerds by Brian KatzRLM 12.104 [poster]

In modern algebra, groups have become a very abstract idea, a structure worth investigating for its own sake. But this is not how groups came to be, and it's not how groups are used. Here's how you should think about groups, and all you need to know to understand this talk:

GROUPS DO THINGS.

The quintessential example of a group is the symmetries of some physical object, the ways to transform the object in space that make it look the same, like rotating a square 90 degrees. In particular, the symmetries of a molecule tell us how it will react to light, and conversely, we can use light to predict the symmetry and shape of molecules (which are way too small to look at). Hopefully this talk will be interesting to both mathematics and chemistry students.

### Wednesday, November 7, 2007 5-7 PMElliptic Curves: The Curves that keep on giving by Kim HopkinsRLM 12.104 [poster]

Elliptic curves have been a subject of great interest for mathematicians from the 18th century to present day. They combine algebra, number theory, and geometry in order to address problems such as the congruent number problem, Diophantine equations, and Fermat’s Last Theorem. They also provide a useful approach to public-key cryptography. In this talk we will explain the basics of elliptic curves and explain how they can be applied in the areas described above.

### Wednesday, October 31, 2007 5-7 PMShow and Tell!RLM 12.104

Today, a few members of the Math Club will show off some nifty mathematical tricks and treats, including mental divisibility tests and non-constructions using a straight-edge and compass. Spooky!

### Wednesday, October 24, 2007 5-7 PMInfo session: How to Use LaTeXRLM 12.104 [poster]

Eric and Mark will give a demonstration of how to use LaTeX to produce beautiful mathematical documents. We will also provide a style sheet to get you started.

### Wednesday, October 17, 2007 5-7 PMAnnual Women in Mathematics ReceptionRLM 12.104 [poster]

Pizza and chocolate of some sort will be served. A group of faculty members and graduate students will talk about career options and choices.

### Thursday, October 11, 2007 5-7 PMMovie Night: {proof}RLM 4.102 [poster]

Please note the special time/place of this week's Math Club meeting: Thursday night at 5pm in RLM 4.102, we will be watching {proof}, starring Gwyneth Paltrow, Anthony Hopkins, and Jake Gyllenhaal. Professor Vick will give a brief introduction to the movie, and refreshments will be served before the movie starts. Invite your friends! All undergraduates are invited!

### Wednesday, October 3, 2007 5-7 PMThe Mathematics of Juggling by Henry SegermanRLM 12.104 [poster]

How do you write down a juggling pattern? I'll talk about a system of notation that partially answers this question, and led to the discovery of many previously unknown patterns, as well as some interesting combinatorial problems. There will be many demonstrations. No prior juggling experience required.

### Wednesday, September 26, 2007 5-7 PMFixed points and stormy weather by Professor Michael StarbirdRLM 12.104 [poster]

"Somewhere on Earth at this very moment there are two antipodal points (that is, points directly opposite from one another through the Earth) where the temperatures are identical and the pressures are also identical. This meteorological fact follows immediately from the theorem in topology known as the Borsuk-Ulam Theorem. Also, at this moment there is a point on the Earth where the wind is not blowing. This other meteorological fact follows from the Hairy Ball Theorem. We'll see neat proofs of these and other facts whose primary tool is a wrapping rope."

Michael Starbird is Professor of Mathematics and a University Distinguished Teaching Professor at The University of Texas at Austin. He strives to present higher-level mathematics authentically to students and the general public and to teach thinking strategies that go beyond the mathematics. With those goals in mind, he wrote, with co-author Edward B. Burger, The Heart of Mathematics: An invitation to effective thinking, which won a 2001 Robert W. Hamilton Book Award and which is now used in hundreds of colleges and universities nationally each year. His promises to be an exciting talk!

### Wednesday, September 19, 2007 5-7 PMThe Evolution of a Mathematician: To Grad School, and Beyond!RLM 12.104 [poster]

This week we will have a panel consisting of faculty members and graduate students from UT discussing various topics concerning what it means to become a mathematician. We will go over things to think about as a math major, how to apply to graduate school, and we may even warn you about some signs that you may be turning into a mathematician.

### Wednesday, September 12, 2007 5-7 PMFilters and Social Choice, by David JensenRLM 12.104 [poster]

"When a large group of people have to make a decision together, bad things can happen. For example, in a strict plurality election system, it is possible for the majority of people to prefer any other candidate to the one that actually wins the election. It seems, then, that the plurality election system is unfair. What could we do to make it fair? Which election systems are the most fair? What does "fair" mean, anyway? We will consider these questions from a mathematical perspective, and we will discover a surprising answer."

David is a graduate student at UT, and in the past, he has given talks on the mathematics of juggling, nonstandard analysis, and Ramsey theory. This past summer, he was a counselor at Math Camp, where he taught algebraic geometry to high-school students from around the world.

### 5-7 PMTriangles whose three angles are all zero! by Mark NorfleetRLM 12.104

If the sum of the three angles of a triangle where smaller than 180 degrees, what should we do? What hope can we have? Do not fear--hyperbolic geometry is here to save the day! We will discuss four different models of the hyperbolic plane (where these types of triangles live), illustrate the connection between the models, and conclude with how to "see" some actions on the hyperbolic plane.