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Adam Jelinsky, University of Waterloo

"Properties of the Pseudo-randomness of a³ mod p²"

In this talk, I will be discussing methods and tactics used to quantify and understand the apparent pseudo-random distribution of a³ mod p², the evidence we have to fit the random model proposed, and the required steps in order to formally prove it.

MC 5403

Jiahui Huang, Department of Pure Mathematics, University of Waterloo

"Equivariant invariants for Quot schemes"

Deformation invariants on Quot schemes such as Donaldson-Thomas invariants are useful tools for studying the cohomology of moduli spaces. Equivariant versions of such invariants are obtained by integrating characteristic class of tautological bundles, over Quot schemes of quotients of a rank $N$ bundle on $\mathbb{C}^n$ for $n=2,3,4$. The $n=4$ case has been the subject of recent activity in relation to string theory and the DT/PT conjecture for Calabi-Yau 4-folds. This talk will demonstrate how integrations on Quot schemes are performed via equivariant localization and their connections to the usual invariants for compact manifolds.

MC 5417

Huixi Li, Nankai University

"On Covering Systems of Polynomial Rings Over Finite Fields"

In 1950, Erd\H{o}s posed a question known as the minimum modulus problem on covering systems for $\mathbb{Z}$, which asked whether the minimum modulus of a covering system with distinct moduli is bounded. This long-standing problem was finally resolved by Hough in 2015. In this presentation, we will discuss the analogous minimum modulus problem for $\mathbb{F}_q[x]$. We proof that the smallest degree of the moduli in any covering system for $\mathbb{F}_q[x]$ of multiplicity $s$ is bounded by a constant depending only on $s$ and $q$. This is a joint work with Shaoyun Yi, Biao Wang, and Chunlin Wang. 

Zoom: https://uwaterloo.zoom.us/j/98950813087?pwd=SEl1NlNqNHl0QzlYNGJzeDVla204QT09

Spiro Karigiannis, Department of Pure Mathematics, University of Waterloo

"Flows of G2 structures"

A G2-structure is a special type of 3-form on an oriented 7-manifold, which determines a Riemannian metric in a nonlinear way. The best class of such 3-forms are those which are parallel with respect to their induced Levi-Civita connections, which is a fully non-linear PDE. More generally, the torsion of a G2-structure is a 2-tensor which quantifies the failure of a G2-structure to be parallel. It is natural to consider geometric flows of G2-structures as a means of starting with a G2-structure with torsion and (hopefully) improving it in some way along the flow. I will begin with an introduction to all of these ideas, and  try to survey some of the results in the field. Then I will talk about recent joint work with Dwivedi and Gianniotis to study a large class of flows of G2-structures. In particular, we explicitly describe all possible second order differential invariants of a G2-structure which can be used to construct a quasi-linear second order flow. Then we find conditions on a subclass of these general flows which are amenable to the deTurck trick for establishing short-time existence and uniqueness.

MC 5403

Freid Tong, Harvard University

"On complete Calabi-Yau metrics and a free-boundary Monge-Ampere equation"

Calabi-Yau metrics are Ricci-flat, Kähler metrics, and they are central objects in Kähler geometry. The existence problem for Calabi-Yau metrics on compact manifolds was answered by Yau in his solution of the Calabi conjecture. The situation in the non-compact setting is much more delicate, and many questions related to the existence and uniqueness of non-compact Calabi-Yau metrics remain unanswered. A major difficulty lies in the lack of suitable model metrics that model the asymptotics of the Calabi-Yau metric at spatial infinity. In this talk, I will give an introduction to this subject and discuss some joint work with T. Collins and S.-T. Yau, on a new relationship between non-compact Calabi-Yau metrics and a free-boundary Monge-Ampere equation, which allows us to resolve this problem of the lack of model metrics.

MC 5501

Andy Zucker, Department of Pure Mathematics, University of Waterloo

"NIP"

We continue to read through Pierre Simon's a Guide to NIP Theories. 

MC 5403

Andy Zucker, Department of Pure Mathematics, University of Waterloo

"Effective Descriptive Set Theory 6"

We will continue to introduce effective descriptive set theory following Andrew Marks’s notes.

MC 5479

Akash Sengupta, Department of Pure Mathematics, University of Waterloo

"Furstenberg sets over finite fields"

A Kakeya set is a subset S of R^n that contains a unit line segment in every direction. The Kakeya conjecture in harmonic analysis states that a Kakeya set S in R^n has Hausdorff dimension n. The Kakeya conjecture is still open, however an analogous statement over finite fields is known due to a beautiful algebraic-geometric proof by Dvir. In this talk, we will consider a generalization of the Kakeya sets over finite fields, which are called Furstenberg sets. Furstenberg sets are subsets of F_q^n which have large intersection with linear spaces in every direction, where F_q is a finite field. We will discuss an algebraic geometric proof of lower bounds on the size of Furstenberg sets, due to Ellenberg-Erman.

MC 5417

Cynthia Dai, Department of Pure Mathematics, University of Waterloo

"Combinatorial aspects of Schubert calculus on Grassmannian"

We will explain how to take intersection products of two Schubert classes using combinatorics. If time permits, we will define Schubert polynomials.

This seminar will be held both online and in person:

• MC 5417
• Zoom link: https://uwaterloo.zoom.us/j/93801243240?pwd=blpUMVJ1dExUYVlEMkJVWjlscUNqZz09

Talk #1: Jason Hou, University of Waterloo

"Sieve methods in combinatorics"

In this talk, I will give a formulation for the Turán sieve and a 'simple sieve' in the context of bipartite graphs and apply it to a graph colouring problem.

Talk #2: Adam Jelinsky, University of Waterloo

"Properties of the Pseudo-randomness of a³ mod p²"

In this talk, I will be discussing methods and tactics used to quantify and understand the apparent pseudo-random distribution of a³ mod p², the evidence we have to fit the random model proposed, and the required steps in order to formally prove it.

MC 5403