Mathematics

Fixed points of unitary decomposition complexes

Vesna Stojanoska (MIT)

Abstract: For a fixed integer n, consider the nerve $L_n$ of the topological poset of orthogonal decompositions of complex n-space into proper orthogonal subspaces. The space $L_n$ has an action by the unitary group U(n), and we study the fixed points for subgroups of U(n). Given a prime p, we determine the relatively small class of p-toral subgroups of U(n) which have potentially non-empty fixed points. Note that p-toral groups are a Lie analogue of finite p-groups, thus if we are interested in the U(n)-space $L_n$ at a fixed prime p, only the p-toral subgroups of U(n) play a significant role. The space $L_n$ is strongly related to the K-theory analogues of the symmetric powers of spheres and the Weiss tower for the functor that assigns to a vector space V the classifying space BU(V). Our results are a step toward a K-theory analogue of the Whitehead conjecture as part of the program of Arone-Dwyer-Lesh. This is joint work with J.Bergner, R.Joachimi, K.Lesh, K.Wickelgren.

Unicorns and Beyond

Sebastian Hensel

Abstract: In this talk, I will first present joint work with Piotr Przytycki and Richard Webb giving a new short proof of uniform hyperbolicity of curves and arc graphs. Namely, I will describe unicorn paths in arc and curve graphs and show that they form 1-slim triangles. Using this, one can deduce that arc graphs are 7-hyperbolic (and curve graphs are 17-hyperbolic) I will then overview some other results which, in a similar vein, give quick and purely topological-combinatorial proofs of curve graph results. If time permits, I will explain how such proofs can sometimes be adapted to work in the Out(F_n) setting.

Valued differential fields

Lou van den Dries (UIUC)

Abstract: We consider valued fields with a continuous derivation. This class of structures is rather diverse, including both monotone differential fields and asymptotic differential fields. (These terms will be defined.) Nevertheless, some results can be established uniformly for the entire class: algebraic extensions, construction of residue field extensions, the Equalizer Theorem, construction of immediate extensions, differential-henselianity. Next I will revisit Scanlon's thesis on the model theory of differential-henselian monotone differential fields with enough constants. Time permitting I will add some remarks on the case of asymptotic differential fields. The above is (a small part of) ongoing joint work with Matthias Aschenbrenner and Joris van der Hoeven focused on developing a model theory for differential fields of transseries.

Fourier-Deligne transform and Kronecker coefficients for the symmetric group

Galyna Dobrovolska (U. Chicago)

Abstract: I will derive a result of M. Brion on Kronecker coefficients from a computation of the Fourier-Deligne transform of certain (constructible) sheaves on a projective space corresponding to the irreducible representations of the symmetric group. I will also talk  about generalizations of this computation of the Fourier-Deligne transform which are related  to Bezrukavnikov's ideas for studying representation theory of the rational Cherednik algebra of type A in positive characteristic.

Hypergraph Turán Problems and Shadows

Alexandr Kostochka   [email] (UIUC Math)

Abstract: For positive integers $r,n$ and an $r$-uniform family ${\cal F}$, the Turán number is defined as follows: $ex_r(n, {\cal F})= \max|\{|H|: H \subset {[n] \choose r},\, F \not\subset H \quad \forall F \in {\cal F}\}|$. The goal of the talk is to describe recent results by D. Mubayi, J. Verstraete and the speaker on the Turán number of $r$-uniform paths, cycles and some trees. The talk is based on the slides by D. Mubayi.

Local cohomology with support in generic determinantal ideals

Claudiu Raicu (Princeton University)

Abstract: The space $Mat(m,n)$ of $m\times n$ matrices admits a natural action of the group $\textrm{GL}_m \times \textrm{GL}_n$ via row and column operations on the matrix entries. The invariant closed subsets are the closures of the orbits of constant rank matrices. I will explain how to describe the local cohomology modules of the ring $S$ of polynomial functions on $Mat(m,n)$ with support in these orbit closures, and mention some consequences of the methods employed to computing minimal free resolutions of invariant ideals in $S$. These ideals correspond to nilpotent scheme structures on the orbit closures, and their study goes back to the work of De Concini, Eisenbud and Procesi in the 80s. Joint work with Jerzy Weyman.

Equations of Parametric Curves and Surfaces via Syzygies

Eliana Duarte (UIUC Math)

Abstract: The problem of finding an implicit equation of a parametric curve or surface, known as the Implicitization Problem, dates back to 1862. The method of eliminating parameters and the use of resultants were the main tools to find implicit equations. In this talk I will explain Sederberg’s method(1997) of how to use syzygies to compute the implicit equation of a parametric curve or surface.

Understanding the Ocean Circulation

Carl Wunsch (Physical Oceanography, MIT)

Abstract: The ocean is a turbulent fluid in contact with another turbulent system (the atmosphere) and influenced by other so-far unpredictable external forces such as solar variations, ice sheet melt, runoff, Etc. Over the past approximately 20 years, the oceanographic and allied communities have developed observing systems that are quasi-global in scope, albeit still limited in coverage, subject to difficult calibration problems, and sometimes outright contradiction. Making best-use of these observations requires combining them with understanding of the presumed governing equations for fluid and thermo-dynamics on a grand scale. Mathematically, the combination problem is best-formulated as one of control theory in a stochastic environment, but of dimension vastly exceeding conventional systems (e.g., a control vector of 2x10^9 elements). Examples of what has been accomplished in practice raise numerous mathematical and physical problems, many of them related to the issues of uncertainty quantification.