Brane structures from the perspective of microlocal sheaf theory

Xin Jin
Northwestern Univ
March 13, 2017
Abstract: In this talk, I will present the following application of microlocal sheaf theory in
symplectic topology. For every closed exact Lagrangian L in the cotangent bundle of a manifold
M, we associate a locally constant sheaf of categories on L, which we call Brane_L, whose fiber is
the infinity-category of k-modules, for k any ring spectrum. I will discuss the relation of Brane_L
with the usual brane structures in Floer theory, and its connection to the J-homomorphism in

Calabi-Yau geometry and quantum B-model

Si Li
Tsinghua University
March 17, 2017
We discuss the Kadaira-Spencer gauge theory (or BCOV theory) on Calabi-Yau geometry. We explain Givental's loop space formalism at cochain level which leads to a degenerate BV theory on Calabi-Yau manifolds. Homotopic BV quantization together with a splitting of the Hodge filtration lead to higher genus B-model. We illustrate such quantization and higher genus mirror symmetry by the elliptic curve example.

"Small" representations of finite classical groups

Shamgar Gurevich
University of Wisconsin and Yale University
March 8, 2017

Suppose you have a finite group G and you want to study certain related structures (e.g., random walks, Cayley graphs, word maps, etc.). In many cases, this might be done using sums over the characters of G. A serious obstacle in applying these Fourier type formulas is lack of knowledge on the low dimensional representations of G. In fact, numerics shows that the "small" representations tend to contribute the largest terms to these sums, so a systematic knowledge on them might assist in the solution of important problems.

On small sums of roots of unity

Philipp Habegger
University of Basel
March 9, 2017

Let $k$ be a fixed positive integer. Myerson (and others) asked how small the modulus of a non-zero sum of $k$ roots of unity can be. If the roots of unity have order dividing $N$, then an elementary argument shows that the modulus decreases at most exponentially in $N$ (for fixed $k$). Moreover it is known that the decay is at worst polynomial if $k < 5$. But no general sub-exponential bound is known if $k \geq 5$.