Events

AJ Fong, Department of Pure Mathematics, University of Waterloo

"Schemes in general"

We will introduce general schemes, and define important notions such as subschemes, local rings of schemes at a point, and morphisms. If time permits, we will also describe the gluing of arbitrary schemes by open subsets. This talk closely follows section I.2 of Eisenbud-Harris (with some necessary sheaf theory from I.1 which was omitted from the last talk for time).

MC 5417

Kieran Mastel, Department of Pure Mathematics, University of Waterloo

"Introduction to the PCP theorem"

This is the first meeting of a working seminar that will take place every Wednesday 1:30PM-3:00PM in QNC 1201. The PCP theorem from computational complexity theory was a important part of the groundbreaking MIP*=RE result, and has very strong connections to stability problems for representations of algebras and groups. We plan to study this theorem, the quantum PCP conjecture, and their connections to (approximate) representations and stability.  The first meeting will be organizational and will include an introductory talk by Kieran Mastel on the PCP theorem.

QNC 1201

Amy Huang, Auburn University

"Matrix Multiplication Complexity: Tensor Geometry and Commutative Algebra"

Tensors are just multi-dimensional arrays. Tensor decomposition also has a lot of applications in data analysis, physics, and other areas of science. I will survey my recent two results about matrix multiplication complexity and classification of special tensors. The first result computes the border rank of 3 X 3 permanent, which is important in the theory of matrix multiplication complexity. The second result classifies linear spaces of matrices of bounded rank 4, making progress on an old problem that has been open for decades in linear algebra society. I will also briefly discuss how the role of commutative algebra, algebraic geometry, and representation theory comes into the picture. 

Zoom link: https://uwaterloo.zoom.us/j/2433704471?pwd=aXJoSDh0NDF0aFREbkthSnFBOUI4UT09

Elliot Kaplan, McMaster University

"Generic derivations on o-minimal structures"

Let T be a model complete o-minimal theory that extends the theory of real closed ordered fields (RCF). We introduce T-derivations: derivations on models of T which cooperate with T-definable functions. The theory of models of T expanded by a T-derivation has a model completion, in which the derivation acts "generically." If T = RCF, then this model completion is the theory of closed ordered differential fields (CODF) as introduced by Singer. We can recover many of the known facts about CODF (open core, distality) in our setting. We can also describe thorn-rank for models of T with a generic T-derivation. This is joint work with Antongiulio Fornasiero.

MC 5479

Panagiotis Dimakis, Université du Québec à Montréal, CIRGET

"The moduli space of solutions to the dimensionally reduced Kapustin-Witten equations on $\Sigma\times\mathbb{R}_+$"

Since their introduction in 2006, the Kapustin-Witten (KW) equations have become the subject of a number of conjectures. Given a knot $K$ embedded in a closed $3$-manifold $Y$, the most prominent conjecture predicts that the number of solutions to the KW equations on $Y\times\mathbb{R}_+$ with boundary conditions determined by the embedding and with fixed topological charge, is a topological invariant of the knot. A major obstacle with this conjecture is the difficulty of constructing solutions satisfying these boundary conditions. In this talk we assume $Y\cong \Sigma\times\mathbb{R}_+$ and study solutions to the dimensionally reduced KW equations with the required boundary conditions. We prove that the moduli spaces are diffeomorphic to certain holomorphic lagrangian sub-manifolds inside the moduli of Higgs bundles associated to $\Sigma$. Time permitting, we explain how one could use this result to construct knot invariants.

MC 5417

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