Large scale strucutre of singularities with
applications in enumerative geometry
Richárd Rimányi
2.10pm-3.00 pm April 26
First steps in Tropical Algebraic Geometry
Gregory Mikhalkin
3.10pm-4.00 pm April 26
939 Evans Hall, Berkeley
ABSTRACT OF FIRST TALK:
If $\eta$ is a singularity (defined in the lecture) and $f:N\to P$ is a
map between manifolds then we can consider the set of points in $N$ where
$f$ has singularity $\eta$. The cohomology class represented by this set
can be computed via the so called Thom polynomial of $\eta$. We'll give
an effective procedure to compute the Thom polynomials, obtaining numerous
enumerative topological and geometric results. Our procedure is based on
the ``local symmetry governs global topology'' principle.
We will also report on M. Kazarian's recent extension of this theory
to the case of multi-singularities---which gave numerous new
enumerative results, including confirmations of conjectures of
Gottsche, Kleiman and Piene.
We will also consider generalizations of this problem, computing Thom
polynomials of orbits of group actions, thus obtaining Chern class
formulas for quivers, reproving the Schubert calculus, etc. In addition
we will consider how this theory is related to the Atiyah-Bott-Kirwan
method of computing the cohomology ring structure of certain moduli spaces.
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ABSTRACT OF SECOND TALK: The set of real numbers equipped with two operations,
taking the maximum and the addition is a Tropical
Semiring. Such semirings appear, in particular, in
Computer Science as counterparts of Boolean Algebras.
(The name "tropical" is said to be given by French
computer scientists in honor of the Bazilian mathematician
Imre Simon.) It turns out that the geometric objects
associated to tropical polynomials are simpler and
more convenient than their classical counterparts.
In the same time tropical varieties model complex
and real algebraic varieties and can be approximated
by them. The talk will contain several examples.
