Events

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

Mark Poor, Cornell University

Some results about the pseudoarc and its homeomorphism group

It is known that the so called pseudoarc can be represented as a quotient of a zero-dimensional compact "prespace" under an appropriate equivalence relation (which is an inverse limit of linear graphs), and the automorphisms of this prespace densely embeds into the homeomorphism group of the pseudoarc. Although this embedding is only continuous, not a homeomorphic embedding, we can actually characterize the topology inherited from the homeomorphism group intrinsically, only in terms of the prespace. Using this characterization we show that not all homeomorphisms are conjugate to an automorphism, and we give a second proof to Kwiatkowska's conjecture, namely that there exists a homeomorphism with a dense conjugacy class.

This is joint work with S. Solecki.

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

Amanda Wilkens, Carnegie Mellon University

Poisson–Voronoi tessellations and fixed price in higher rank

We briefly define and motivate the Poisson point process, which is, informally, a "maximally random" scattering of points in space, and discuss the ideal Poisson–Voronoi tessellation (IPVT), a new random object with intriguing geometric properties when considered on a semisimple symmetric space (the hyperbolic plane, for example). In joint work with Mikolaj Fraczyk and Sam Mellick, we use the IPVT to prove a result on the relationship between the volume of a manifold and the number of generators of its fundamental group. We give some intuition for the proof. No prior knowledge on fixed price or higher rank will be assumed.

MC 5417

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)

Kira Bruschke, University of Watelroo, Centre for Career Development

Career Decision

The Career Talks seminar series aims to provide valuable advice and guidance for current graduate students. In this final seminar, a career advisor from the Centre for Career Development will provide a framework for career-decision making and resources for identifying career options.

MC 5417

Rahim Moosa, University of Waterloo

Model theory and the birational geometry of algebraic vector fields

I will try to illustrate how and why model theory (itself a branch of mathematical logic) can sometimes have something to say about algebraic geometry. I will focus on some results of mine (along with Jim Freitag and Remi Jaoui), from the last few years, on the birational geometry of algebraic vector fields.

MC 5479

Joey Lakerdas-Gayle, University of Waterloo

Infinitary Continuous Logic II

We will prove a continuous analog of Scott's Isomorphism Theorem using the Scott analysis for metric structures developed by Ben Yaacov, Doucha, Nies, and Tsankov.

MC 5479