Events

Lena Ji, University of Michigan

"Rationality of algebraic varieties over non-algebraically-closed fields"

The most basic algebraic varieties are projective spaces, and their closest relatives are rational varieties. These are varieties that agree with affine space on a dense open subset, and hence have a coordinate system on this open subset. Thus, rational varieties are the easiest varieties to understand. Historically, rationality problems have been of great importance in algebraic geometry: for example, Severi was interested in finding rational parametrizations for moduli spaces of Riemann surfaces (algebraic curves). Over the complex numbers, techniques from geometry and topology can be used to extract invariants useful for rationality questions. Over fields that are not algebraically closed (such as the rational numbers), the arithmetic of the field adds additional subtleties to the rationality problem. When the dimension of the variety is at most 2, there are effective criteria to determine rationality. However, in higher dimensions, there are no such known criteria. In this talk, I will first give a survey of some results on rationality of algebraic varieties. Then I will explain results on rationality obstructions for higher-dimensional varieties that involve the arithmetic of the field.

M3 3127

Eric Culf, Department of Applied Mathematics, University of Waterloo

"Approximation algorithms for noncommutative constraint satisfaction problems"

Constraint satisfaction problems (CSPs) are an important topic of investigation in computer science. For example, the problem of finding optimal k-colourings of graphs, Max-Cut(k), is NP-hard, but it is easy to approximate in the sense that it is possible to find a colouring that satisfies a large fraction of the constraints of an optimal one. We study a noncommutative variant of CSPs that is central in quantum information, where the variables are replaced by operators. In this context, even approximating general CSPs is known to be much harder than the classical case, in fact uncomputably hard. Nevertheless, Max-Cut(2) becomes efficiently solvable. We introduce a framework for designing approximation algorithms for noncommutative CSPs, which allows us to find classes of CSPs that are efficiently approximable but not efficiently solvable. To determine the quality of our approximation algorithm, we make use of results from free probability to characterise a distribution arising from random matrices. This talk is based on work with Hamoon Mousavi and Taro Spirig (arxiv.org/abs/2312.16765).

This seminar will be held both online and in person:

• Room: MC 5479
• Zoom link: https://uwaterloo.zoom.us/j/94186354814?pwd=NGpLM3B4eWNZckd1aTROcmRreW96QT09

Yvon Verberne, Western University

"Pseudo-Anosov Homeomorphisms"

The mapping class group is the group of orientation preserving homeomorphisms of a surface up to isotopy. In particular, the mapping class group encodes information about the symmetries of a surface. The Nielsen-Thurston classification states that elements of the mapping class group are of one of three types: periodic, reducible, and pseudo-Anosov. In this talk, we will focus our attention on the pseudo-Anosov elements, which are the elements of the mapping class group which mix the underlying surface in a complicated way. In this talk, we will discuss both classical and new results related to pseudo-Anosov mapping classes, as well as the connections to other areas of mathematics.

MC 5501

Kunjakanan Nath, University of Illinois, Urbana-Champaign

"Circle method and binary correlation problems"

One of the key problems in number theory is to understand the correlation between two arithmetic functions. In general, it is an extremely difficult question and often leads to famous open problems like the Twin Prime Conjecture, the Goldbach Conjecture, and the Chowla Conjecture, to name a few. In this talk, we will discuss a few binary correlation problems involving primes, square-free integers, and integers with restricted digits. The objective is to demonstrate the application of Fourier analysis (aka the circle method) in conjunction with the arithmetic structure of the given sequence and the bilinear form method to solve these problems.

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

Jiahui Huang, Department of Pure Mathematics, University of Waterloo

"Deformations and Lines on Cubics"

We continue studying the problem of lines on cubic surfaces. By considering first order deformations on the Hilbert scheme, we show that there are 27 distinct lines on any smooth surfaces.

MC 5501

Chatchai Noytaptim, Department of Pure Mathematics, University of Waterloo

"Potential function, transfinite diameter, and Fekete-Szego theorem"

We introduce the potential function attached to a probability measure and the transfinite diameter of any compact set in the Berkovich projective line. Time permitting, we briefly discuss a Berkovich version of the adelic Fekete-Szego theorem. The materials in this presentation cover sections 6.3-6.5 in Baker-Rumely’s monograph on “Potential Theory and Dynamics on the Berkovich Projective Line”.

MC 5417

Rachael Alvir, Department of Pure Mathematics, University of Waterloo

"Computable Structure Theory III"

We will continue our discussion on forcing and take a look at nontrivial structures. 

MC 5479

Jacques van Wyk, Department of Pure Mathematics, University of Waterloo

"(Some) Essentials of Schemes"

We continue with chapter 1 of Eisenbud and Harris, starting by introducing morphisms of schemes.

MC 5417

Faisal Romshoo, Department of Pure Mathematics, University of Waterloo

"Some computations with gauge transformations on a $G_2$ manifold"

Given a (torsion-free) $G_2$-manifold $(M, \varphi, g)$ and a gauge transformation $P: TM \rightarrow TM$, we want to look at the $G_2$ structures $\Tilde{\varphi} = P^*g$ and explore the conditions for it to be torsion-free. In this talk, we will start in a more general setting with a Riemannian manifold $(M, g)$ and obtain an expression for the tensor $B(X, Y) = \tilde{\nabla}_X Y -\nabla_X Y$ before moving on to computing the full torsion tensor $\tilde{T}_{pq}$ in the case when $M$ is a $G_2$ manifold.

MC 4058

Ross Willard, University of Waterloo

"Residually finite equational theories"

An equational theory T is said to be residually finite if every model of the theory can be embedded in a product of finite models of the theory.  Equivalently, T is residually finite if and only if its irreducible models (those that cannot be embedded in products of “simpler” models) are all finite.  In practice, it seems that whenever a theory is both “interesting” and residually finite, then there is a finite upper bound to the sizes of its irreducible models.  In other words, we see a sort of compactness principle for “interesting” equational theories: if such a theory has arbitrarily large finite irreducible models, then it must have an infinite irreducible model.  Whether or not this observation holds generally has been open for almost 50 years.  In this talk I will discuss some recent progress with collaborators Keith Kearnes and Agnes Szendrei.

MC 5479