Events

Joey Lakerdas-Gayle, University of Waterloo

Infinitary Continuous Logic

We will introduce continuous analogues of infinitary logic following a survey of Christopher Eagle. We will also look at the Scott analysis for metric structures developed by Ben Yaacov, Doucha, Nies, and Tsankov.

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

Francisco Villacis, University of Waterloo

Computing the Quantum Cohomology

In this talk, I will compute the quantum cohomology ring of projective space and of the Grassmannian. If time permits, I will outline the computation of the quantum cohomology of generic quintic threefolds and their connections to the count of rational curves of a given degree on these.

MC 2017

Zahra Janbazi, University of Toronto

Extensions of Birch-Merriman and Related Finiteness Theorems

A classical theorem of Birch and Merriman states that, for fixed n, the set of integral binary n-ic forms with fixed nonzero discriminant breaks into finitely many GL(2, Z)-orbits. In this talk, I’ll present several extensions of this finiteness result.

In joint work with Arul Shankar, we study a representation-theoretic generalization to ternary n-ic forms and prove analogous finiteness theorems for GL(3,Z)-orbits with fixed nonzero discriminant. We also prove a similar result for a 27-dimensional representation associated with a family of K3 surfaces.

In joint work with Sajadi, we take a geometric perspective and prove a finiteness theorem for Galois-invariant point configurations on arbitrary smooth curves with controlled reduction. This result unifies classical finiteness theorems of Birch–Merriman, Siegel, and Faltings.

MC 5479

Kaleb D Ruscitti, University of Waterloo

Real Analytic Varieties and Singularities

Analytic varieties have the flavour of algebraic geometry, but are also foreign in many ways. Of course, over the complex numbers, Serre showed that analytic and algebraic varieties are strongly related. Over the real numbers however, things are more interesting.

In this talk I will review the definition of analytic completion, analytic spaces, and their relationship to algebraic varieties. Then I will focus on the real case, and talk about singularities of real analytic spaces and real normal crossings divisors.

MC 5479

Aleksa Vujicic, University of Waterloo

Fourier Algebras of Semi-Direct Product Groups of Local Fields

We look at Fourier Algebras of Semi-Direct Product Groups of Local Fields.

MC 5403

Yidi Wang, University of Waterloo

Local-global principles on stacky curves and its application in solving generalized Fermat equations. 

The primitive solutions of certain generalized Fermat equations, i.e., 
Diophantine equations of the form Ax^p+By^q = Cz^r, can be studied as 
integral points on certain stacky curves. In a recent paper by Bhargava and 
Poonen, an explicit example of such a curve of genus 1/2 violating 
local-global principle for integral points was given. However, a general 
description of stacky curves failing the local-global principle is 
unknown. In this talk, I will discuss our work on finding the primitive 
solutions to equation of the form when (p, q, r) = (2,2,n) by studying local-global principles for integral points on stacky curves constructed from such equations. 
The talk is based on a joint project with Juanita Duque-Rosero, 
Christopher Keyes, Andrew Kobin, Manami Roy and Soumya Sankar. 

MC 5417

Nicolas Banks, University of Waterloo

Non-Trivial Theorems with Trivial Proofs

One of the most fruitful things we can do as mathematicians is to think deeply about simple things. As students and researchers, perhaps we come across results with simple proofs and believe that not much can be learned from them. In this talk, I will challenge this misconception by diving into three important, non-trivial theorems with seemingly trivial proofs - Desargue's Theorem of planar geometry, the finite intersection property of compact sets, and Lagrange's Theorem from group theory. These will demonstrate three reasons that a profound truth need not be complicated.

MC 5501

(snacks at 17:00)

Ababacar Sadikhe Djité, Université Cheikh Anta Diop de Dankar & University of Waterloo

Shape Stability of a quadrature surface problem in infinite Riemannian manifolds

In this talk, we revisit a quadrature surface problem in shape optimization. With tools from infinite-dimensional Riemannian geometry, we give simple control over how an optimal shape can be characterized. The framework of the infinite-dimensional Riemannian manifold is essential in the control of optimal geometric shape. The covariant derivative plays a key role in calculating and analyzing the qualitative properties of the shape hessian. Control only depends on the mean curvature of the domain, which is a minimum or a critical point. In the two-dimensional case, Gauss-Bonnet's theorem gives a control within the framework of the algorithm for the minimum.

MC 5417