Events

Uri Bader, The Weizmann Institute

"Cohomology of Arithmetic Groups, Higher Property T and Spectral Gap"

Groups of matrices with integer entries, aka arithmetic groups, are prominent objects of mathematics. From a geometric point of view, they appear as the fundamental groups of locally symmetric space. Topological invariants of such spaces could be seen as group invariants and vice versa. In my talk I will relate this useful link between topology and arithmetics with the theory of unitary representations. More precisely, I will focus on the cohomology of arithmetic groups with unitary coefficients, presenting a recent joint work with Roman Sauer which completely clarifies the theory in small degrees. By the end of the talk I will discuss the relation of the above with the phenomenon of spectral gap and state various related conjectures. I will make an effort to present the subject to a general audience.

MC 5501

Kiseok Yeon, Purdue University

"The Hasse principle for random homogeneous polynomials in thin sets"

In this talk, we introduce a framework via the circle method in order to confirm the Hasse principle for random homogeneous polynomials in thin sets. We first give a motivation for developing this framework by providing an overall history of the problems of confirming the Hasse principle for homogeneous polynomials. Next, we provide a sketch of the proof of our main result and show a part of the estimates used in the proof. Furthermore, by using our recent joint work with H. Lee and S. Lee, we discuss the global solubility for random homogeneous polynomials in thin sets.

Zoom link: https://uwaterloo.zoom.us/j/98937322498?pwd=a3RpZUhxTkd6LzFXTmcwdTBCMWs0QT09

Yash Singh, Department of Pure Mathematics, University of Waterloo

"Chern classes of vector bundles and applications"

We study Chern classes of vector bundles and their connection to grassmanians. We also study the problem of 27 lines on a cubic surface if time permits.

MC 5501

Rachael Alvir, Department of Pure Mathematics, University of Waterloo

"Computable Structure Theory II"

We will continue our discussion of forcing in computable structure theory.

MC 5479

Adi Glücksam, Northwestern University

"Multi-fractal spectrum of planar harmonic measure"

In this talk, I will define various notions of the multi-fractal spectrum of harmonic measures and discuss finer features of the relationship between them and properties of the corresponding conformal maps. Furthermore, I will describe the role of multifractal formalism and dynamics in the universal counterparts. This talk is based on a joint work with I. Binder.

MC 5501

AJ Fong, Department of Pure Mathematics, University of Waterloo

"Schemes in general"

We will introduce general schemes, and define important notions such as subschemes, local rings of schemes at a point, and morphisms. If time permits, we will also describe the gluing of arbitrary schemes by open subsets. This talk closely follows section I.2 of Eisenbud-Harris (with some necessary sheaf theory from I.1 which was omitted from the last talk for time).

MC 5417

Kieran Mastel, Department of Pure Mathematics, University of Waterloo

"Introduction to the PCP theorem"

This is the first meeting of a working seminar that will take place every Wednesday 1:30PM-3:00PM in QNC 1201. The PCP theorem from computational complexity theory was a important part of the groundbreaking MIP*=RE result, and has very strong connections to stability problems for representations of algebras and groups. We plan to study this theorem, the quantum PCP conjecture, and their connections to (approximate) representations and stability.  The first meeting will be organizational and will include an introductory talk by Kieran Mastel on the PCP theorem.

QNC 1201

Amy Huang, Auburn University

"Matrix Multiplication Complexity: Tensor Geometry and Commutative Algebra"

Tensors are just multi-dimensional arrays. Tensor decomposition also has a lot of applications in data analysis, physics, and other areas of science. I will survey my recent two results about matrix multiplication complexity and classification of special tensors. The first result computes the border rank of 3 X 3 permanent, which is important in the theory of matrix multiplication complexity. The second result classifies linear spaces of matrices of bounded rank 4, making progress on an old problem that has been open for decades in linear algebra society. I will also briefly discuss how the role of commutative algebra, algebraic geometry, and representation theory comes into the picture. 

Zoom link: https://uwaterloo.zoom.us/j/2433704471?pwd=aXJoSDh0NDF0aFREbkthSnFBOUI4UT09

Elliot Kaplan, McMaster University

"Generic derivations on o-minimal structures"

Let T be a model complete o-minimal theory that extends the theory of real closed ordered fields (RCF). We introduce T-derivations: derivations on models of T which cooperate with T-definable functions. The theory of models of T expanded by a T-derivation has a model completion, in which the derivation acts "generically." If T = RCF, then this model completion is the theory of closed ordered differential fields (CODF) as introduced by Singer. We can recover many of the known facts about CODF (open core, distality) in our setting. We can also describe thorn-rank for models of T with a generic T-derivation. This is joint work with Antongiulio Fornasiero.

MC 5479

Panagiotis Dimakis, Université du Québec à Montréal, CIRGET

"The moduli space of solutions to the dimensionally reduced Kapustin-Witten equations on $\Sigma\times\mathbb{R}_+$"

Since their introduction in 2006, the Kapustin-Witten (KW) equations have become the subject of a number of conjectures. Given a knot $K$ embedded in a closed $3$-manifold $Y$, the most prominent conjecture predicts that the number of solutions to the KW equations on $Y\times\mathbb{R}_+$ with boundary conditions determined by the embedding and with fixed topological charge, is a topological invariant of the knot. A major obstacle with this conjecture is the difficulty of constructing solutions satisfying these boundary conditions. In this talk we assume $Y\cong \Sigma\times\mathbb{R}_+$ and study solutions to the dimensionally reduced KW equations with the required boundary conditions. We prove that the moduli spaces are diffeomorphic to certain holomorphic lagrangian sub-manifolds inside the moduli of Higgs bundles associated to $\Sigma$. Time permitting, we explain how one could use this result to construct knot invariants.

MC 5417