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for events the day of Thursday, October 25, 2018.

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Thursday, October 25, 2018

12:30 pm in 464 Loomis,Thursday, October 25, 2018

A hierarchy of symplectic \sigma--models

Ryan Grady (Montana State)

Abstract: In this talk I will discuss a family of classical field theories in the Batalin--Vilkovisky formalism. These theories take as input n-symplectic Lie algebroids which correspond to symplectic manifolds, Poisson manifolds, and (higher) Courant algebroids. I will also discuss possible boundary conditions for these theories, e.g., Dirac structures and multiplectic structures. Finally, I will discuss aspects of the quantum theory in low dimensions.

2:00 pm in 241 Altgeld,Thursday, October 25, 2018

Ranks of elliptic curves

Siegfred Baluyot (Illinois Math)

Abstract: This will be a survey on ranks of elliptic curves over the field of rational numbers. We will review basic definitions about elliptic curves, and then discuss a few open problems and some progress towards them.

3:00 pm in 345 Altgeld Hall,Thursday, October 25, 2018

Crystals for Symmetric Grothendieck Polynomials

Travis Scrimshaw (University of Queensland)

Abstract: Kashiwara introduced crystals as a combinatorial framework for studying Lie algebra representations. In particular, crystals give a representation theoretic interpretation of a number of combinatorial operations on semistandard Young tableaux. Symmetric (aka Stable) Grothendieck polynomials arise from the K-theory of the Grassmannian and can be considered as a sum over set-valued tableaux. In this talk, we will show that set-valued tableaux have a crystal structure, giving a rule for decomposing symmetric Grothendieck polynomials into Schur functions. We will also discuss a potential K-theory extension of crystals to encode the multiplication of symmetric Grothendieck polynomials. No knowledge of crystals, representation theory, geometry, or tableaux combinatorics will be assumed.

4:00 pm in 245 Altgeld Hall,Thursday, October 25, 2018

Rigidity of uniform Roe algebras and coronas

Ilijas Farah (York University)

Abstract: Coarse geometry is the study of large-scale properties of metric spaces. Two metric spaces are coarsely equivalent if their "large-scale geometries" agree. The uniform Roe algebra $C^*_u(X)$ is a norm-closed algebra of bounded linear operators on the Hilbert space $\ell_2(X)$. It is the algebra of all bounded linear operators on $\ell_2(X)$ that can be uniformly approximated by operators of "finite propagation". The uniform Roe algebra is a coarse invariant of the space $X$. It includes $\ell_\infty(X)$ (as the algebra of all operators of zero propagation) and the algebra of compact operators. The uniform Roe corona $Q^*_u(X)$ is obtained by modding out the compact operators from $C^*_u(X)$. After introducing the basics of coarse spaces and uniform Roe algebras, we will study implications (or lack thereof) between the following three assertions and their variants:

IF.A.1. The spaces $X$ and $Y$ are (bijectively) coarsely equivalent.
IF.A.2. The uniform Roe algebras of $C^*_u(X)$ and $C^*_u(Y)$ are isomorphic.
IF.A.3. The uniform Roe coronas $Q^*_u(X)$ and $Q^*_u(Y)$ are isomorphic.

Under some additional assumptions on $X$ and $Y$ (the uniform local finiteness and a weakening of Yus property A — A stands for "amenability"), IF.A.2 implies IF.A.1. The implication from IF.A.3 to IF.A.2, even for uniformly locally finite spaces with property A, nontrivially (and possibly necessarily) involves set theory. This talk will be based on a joint work with B.M. Braga and A. Vignati.