This is a Ph.D. Thesis Defense.

One aspect of quantum chaos are extreme values of high-energy eigenfunctions, which, on Riemannian manifolds of negative curvature, are not well understood and can depend heavily on the global geometry of the manifold. The principal result to be presented shows that there is a distinguished class of arithmetic hyperbolic 3-manifolds on which a sequence of L^2-normalized high-energy Hecke-Maass eigenforms achieve values as large as a power of the Laplacian eigenvalue. Power growth, first observed in this context by Rudnick and Sarnak, is (by far) not expected generically and stands in stark contrast with the statistical models suggested by the so-called random wave conjecture on eigenfunctions of classically ergodic systems. The arithmetic hyperbolic 3-manifolds on which the exceptional behavior is exhibited can be characterized in terms of their invariant quaternion algebras and are, up to commensurability, precisely those containing immersed totally geodesic surfaces, as described by Maclachlan and Reid. We will explain how number theory allows us to see some of the global geometry of the manifold and discuss the question of identifying the Hecke-Maass eigenforms achieving power growth through representation theory.

In this talk, I will discuss various obstructions for Seifert fibered surgery that arise from Heegaard Floer homology, including obstructions on the Alexander polynomial, the knot Floer homology, the surgery coefficient and the Seifert and four-ball genus of the knot.

There are three mountains (the preprojective variety, the quiver variety and the loop Grassmannian). These three mountains feed the spring at their base (the quantum group). At the spring live three nymphs, very similar, but different (the semicanonical basis, the canonical basis and the MV basis). Each of these is nourished by the special nutrients of the corresponding source (the preprojective variety, the quiver variety and the loop Grassmannian). At three particular moments of each day (dawn, noon, dusk) the shadow of the muses on the valley below is the same (the MV polytope). And, as we shall see, there is a Glass Bead Game in this story as well.

Let Q,M, and N be positive integers with MN and for each prime p less than Q let w(p) be a nonnegative integer with w(p) less than p. The large sieve is an inequality which provides an upper bound for the number of integers remaining in the interval [M+1,M+N] after w(p) residue classes modulo p are removed for each prime p less than Q+1. In this talk we will derive a form of large sieve inequality obtained by Vaaler and Graham which suggests an intriguing extremal problem.

I will describe a certain interacting one-dimensional system called the Calogero-Moser system. It turns out to be an integrable system and, moreover, a special case of the Hitchin system. I will then move on to the quantum case and explain what its quantization looks like using quantum Hamiltonian reduction. Along the way we will encounter a degeneration of a double affine Hecke algebra, which was used by Cherednik in his proof of Macdonald's conjectures.

For the two-dimensional Euler equation, we prove lower bounds for the growth of Sobolev norms. We also explain the merging of two centrally symmetric vortex patches and prove the double-exponential estimates for infinite time.

We construct an infinite family of knots in a genus 2 handlebody which have nontrivial handlebody surgeries. Several subfamilies of these knots have bridge number greater than one, disproving a conjecture of Wu. We'll discuss the techniques used in the proof, and look at some other interesting properties of the knots.

This is a thesis defense.

Join us for the last session of the Spring semester as Dr. Brian Conrey talks about strategies for estimating large numbers (gasp) without a calculator. Come prepared to think big!

We will discuss sufficient conditions for the Lyapunov stability (and instability) of time independent solutions to the Navier-Stokes equations. Our approach will be to view the linearized perturbation equations as an ODE on a suitable Banach space, and describe how spectral properties of the resulting operator are relevant to the stability of Navier-Stokes solutions. As an application we will provide several concrete examples of flows whose stability have been analyzed using this method.

One of the main topics in plasma physics concerns the energy production through thermonuclear fusion, which is achieved by magnetic confinement. From the mathematical point of view, the transport of charged particles under strong magnetic fields reduces to multiscale analysis. This paper deals with numerical schemes, based on macro-micro decomposition of the particle distribution function, which are uniformly consistent with the pde models for any order of the magnetic field. Numerical simulations, based on this approach, are presented for the resolution of the Vlasov equation. In collaboration with M. Lemou, N.Crouseilles and C. Caldini

In this talk I will introduce some key concepts of quantum mechanics. Most of what I will cover would typically be treated in a quantum mechanics course for physics majors. However, our main focus will be on developing a systematic mathematical framework.

Okay, be honest: do you really understand what a coequalizer is? If you don't, this talk is for you. I'll introduce category theory as a language for talking about the items that might be found and the things that might happen to them at your friendly neighborhood diner, physics lab, supercomputing facility, or crime scene. If you're lucky, you'll leave with a better intuition for things like monomorphisms, epimorphisms, (co)equalizers, (co)products, monoidal categories, and dagger categories. If time permits, I will discuss an application: explaining quantum mechanics to twelve-year-olds. Please note that this talk will consist almost entirely of definitions. The number of actual mathematical results mentioned will be between one and zero (inclusive), and nothing will be proven.