Events

Gregory G. Smith, Queen's University

Sums of Squares and Projective Geometry

A multivariate real polynomial is nonnegative if its value at any real point is greater than or equal to zero.  These special polynomials play a central role in many branches of mathematics including algebraic geometry, optimization theory, and dynamical systems.  However, it is very difficult, in general, to decide whether a given polynomial is nonnegative.  In this talk, we will review some classical methods for certifying that a polynomial is nonnegative.  We will then present novel certificates in some important cases. This talk is based on joint work with Grigoriy Blekherman, Rainer Sinn, and Mauricio Velasco.

MC 5501

Agniva Dasgupta, University of Texas at Dallas

Second Moment of GL(3) L-functions

In this talk, I will discuss our recent result, joint with Wing Hong Leung and Matthew Young, on the second moment of GL(3) L-functions on the critical line. Moments of L-functions on the critical line have been studied for over a hundred years now, and still remains a very active field of research in number theory. The second moment of GL(3) L-functions has proved to be especially difficult, and only in the last couple of years have we seen some progress on this. Building on top of these works, we are able to obtain a strong upper bound for this moment. This allows us to deduce some nice corollaries including an improvement on the error term in the Rankin-Selberg problem, and on certain subconvexity bounds for GL(3) x GL(2) and GL(3) L-functions. As a byproduct of the method of proof, we also obtain an improved estimate for an average of shifted convolution sums of GL(3) Fourier coefficients. 

MC 5479

AJ Fong, University of Waterloo

Toric geometry without (most of) the geometry

Toric varieties are great because they lend themselves to combinatorial description and study. I will introduce some of the combinatorial objects involved, leaning heavily on and subsequently specialising to smooth toric surfaces as simple examples. If time permits, I will demonstrate the power of this approach by showing a result on Galois representations of toric surfaces.

MC 5403

Joey Lakerdas-Gayle, University of Waterloo

Fundamentals of Computability Theory 3

We will continue learning about priority constructions, now using the finite injury method, following Robert Soare's textbook.

MC 5403

Paul Cusson, University of Waterloo

The Kodaira embedding theorem and background material

The Kodaira embedding theorem is a crucial result in complex geometry that forms a nice bridge between differential and algebraic geometry, giving a necessary and sufficient condition for a compact complex manifold to be a smooth projective variety, that is, a complex submanifold of a complex projective space. The material and proof will follow the exposition in Griffiths & Harris's classic textbook.

MC 5479

Nikolay Bogachev, University of Toronto

Commensurability classes and quasi-arithmeticity of hyperbolic reflection groups

In the first part of the talk I will give an intro to the theory of hyperbolic reflection groups initiated by Vinberg in 1967. Namely, we will discuss the old remarkable and fundamental theorems and open problems from that time. The second part will be devoted to recent results regarding commensurability classes of finite-covolume reflection groups in the hyperbolic space H^n. We will also discuss the notion of quasi-arithmeticity (introduced by Vinberg in 1967) of hyperbolic lattices, which has recently become a subject of active research. The talk is partially based on a joint paper with S. Douba and J. Raimbault.

MC 5417

Matthew Harrison-Trainor, University of Illinois at Chicago

Back-and-forth games to characterize countable structures

Given two countable structures A and B of the same type, such as graphs, linear orders, or groups, two players Spoiler and Copier can play a back-and-forth games as follows. Spoiler begins by playing a tuple from A, to which Copier responds by playing a tuple of the same size from B. Spoiler then plays a tuple from B (adding it to the tuple from B already played by Copier), and Copier responds by playing a tuple from B (adding it to the tuple already played by Spoiler). They continue in this way, alternating between the two structures. Copier loses if at any point the tuples from A and B look different, e.g., if A and B are linear orders then the two tuples must be ordered in the same way. If Copier can keep copying forever, they win. A and B are isomorphic if and only if Copier has a winning strategy for this game.   Even if Copier does not have a winning strategy, they may be able to avoid losing for some (ordinal) amount of time. This gives a measure of similarity between A and B. A classical theorem of Scott says that for every structure A, there is an α such that if B is any countable structure, A is isomorphic to B if and only if Copier can avoid losing for α steps of the back-and-forth game, that is, when A is involved we only need to play the back-and-forth game for α many steps rather than the full infinite game. This gives a measure of complexity for A, called the Scott rank. I will introduce these ideas and talk about some recent results.

MC 5501

David McKinnon, University of Waterloo

How crowded can rational solutions be?

Say you've got the equation x^2-2y^2=z^4-1. Lots of rational solutions there, like (1,1,0). How are those solutions distributed in 3-space? In particular, how close can they get to (1,1,0)? This abstract has the questions, but the talk has the answers. Well, some of 'em.

MC 5479

Cynthia Dai, University of Waterloo

Height Modulis on Toric Stacks

In this talk we will go through Matt’s work on height modulis on weighted projective space, mainly its construction, and then its application. If time permits, I will talk about generalizations of this construction to toric stacks.

MC 5403

Matthew Harrison-Trainor, University of Illinois Chicago

Scott analysis of linear orders

The Scott analysis measures the complexity of describing a structure up to isomorphism, and equivalently the complexity of describing its automorphism orbits, and of computing isomorphisms between different copies. I will introduce the Scott analysis in general and talk about the Scott analysis of linear orders in particular. Linear orders have a few special properties which makes their behaviour quite interesting and sometimes different from structures in general.

MC 5479