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Jack Jia, University of Waterloo

Group Schemes: a Functor of Points Perspective

A group scheme is a group object in a category of schemes. This definition, much like other category theory mantras, is a great way to organize knowledge but falls short when one tries to work with it in a hands-on way. I will introduce a more hands-on classification for group schemes, which is aligned with how people work with them in practice. Time permitting, I will illustrate the advantage of this definition in the case of elliptic curves.

MC 5479

Erik Séguin, University of Waterloo

Selected Topics on Fourier-Stieltjes Algebras of Locally Compact Hausdorff Groups.

We discuss some selected topics on Fourier-Stieltjes algebras of locally compact Hausdorff groups.

MC 5403

Xinle Dai, Harvard University

Sectorial Decompositions of Symmetric Products and Homological Mirror Symmetry

Symmetric products of Riemann surfaces play a crucial role in symplectic geometry and low-dimensional topology. They are essential ingredients for defining Heegaard Floer homology and serve as important examples of Liouville manifolds when the surfaces are open. In this talk, I will discuss ongoing work on the symplectic topology of these spaces through Liouville sectorial methods, along with examples as applications of this decomposition construction to homological mirror symmetry.

MC 5417

Christine Eagles, University of Waterloo

Curve excluding fields IV

We continue to read Omar Leon Sanchez' paper.

MC 5403

Amanda Maria Petcu, University of Waterloo

A hypersymplectic structure on R^4 with an SO(4) action

Given a hypersymplectic manifold X^4, one can give a flow of hypersymplectic structures that evolve according to the equation

d_t w = d(Q d^*(Q^{-1} w), where w is the triple that gives the hypersymplectic structure and Q is a 3x3 symmetric matrix. In this talk we let X^4 be R^4 with an SO(4) action  The flow of the hypersymplectic triple then descends to a single flow of a function h. We will examine this flow, as well as solitons of the hypersymplectic flow in this set up. Furthermore, the triple w gives rise to a Riemannian metric g . We will conclude with a discussion about the Riemann and Ricci curvature tensors that are derived from this metric.

MC 5479

Kuntal Banerjee, University of Waterloo

Very stable and wobbly loci for elliptic curves

We explore very stable and wobbly bundles, twisted in a particular sense by a line bundle, over complex algebraic curves of genus 1. We verify that twisted stable bundles on an elliptic curve are not very stable for any positive twist. We utilize semistability of trivially twisted very stable bundles to prove that the wobbly locus is always a divisor in the moduli space of semistable bundles on a genus 1 curve. We prove, by extension, a conjecture regarding the closedness and dimension of the wobbly locus in this setting. This conjecture was originally formulated by Drinfeld in higher genus.

MC 5479

Spencer Unger, University of Toronto

Proofs of countable Ramsey theorems

We discuss the various proofs of Ramsey theorems involving colorings of countable sets with additional structure.  To illustrate a typical argument which proves an infinite Ramsey statement from a finite one, we sketch Baumgartner's proof of Hindman's theorem and report on some ongoing related projects.

MC 5479

Meenakshi McNamara, Perimeter Institute for Theoretical Physics

Exact quantum chromatic numbers of Hadamard graphs and products

Quantum chromatic numbers are defined in terms of non-local games on graphs. This talk gives a proof of the exact quantum chromatic number of Hadamard graphs using a conjugacy class graphs approach. This further allows us to consider graph products, and we compute the exact quantum chromatic number of the categorical product of Hadamard graphs. This work makes use of several results for the quantum chromatic numbers of quantum graphs, an operator algebraic generalizations of graphs. In particular, we also discuss results on products of quantum graphs from joint work with Rolando de Santiago.

MC 5417

Erik Seguin, University of Waterloo

Selected Topics on Fourier Algebras of Locally Compact Hausdorff Groups

We discuss some selected topics on Fourier algebras of locally compact Hausdorff groups.

MC 5403

Mark Rudelson, University of Michigan

When a system of real quadratic equations has a solution

The existence and the number of solutions of a system of polynomial equations in n variables over an algebraically closed field is a classical topic in algebraic geometry. Much less is known about the existence of solutions of a system of polynomial equations over reals. Any such problem can be reduced to a system of quadratic equations by introducing auxiliary variables. Due to the generality of the problem, a computationally efficient algorithm for determining whether a real solution of a system of quadratic equations exists is believed to be impossible. We will discuss a simple and efficient sufficient condition for the existence of a solution. While the problem and the condition are of algebraic nature, the proof relies on Fourier analysis and concentration of measure.

Joint work with Alexander Barvinok.

MC 5501