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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

Leo Jimenez, Ohio State University

"Splitting differential equations using Galois theory"

An ordinary algebraic differential equation is said to be internal to the constants if its general solution is obtained as a rational function of finitely many of its solutions and finitely many constant terms. Any such equation has an algebraic group acting as its Galois group. In this talk, I will use decomposition theorems for algebraic groups to show that some internal equations (do not) split into a product of internal equations. The methods are model-theoretic and could be applied to other contexts. This is a joint work in progress with Christine Eagles.

MC 5479

Yuming Zhao, Department of Pure Mathematics, University of Waterloo

"Positivity and sum of squares in quantum information"

A multivariate polynomial is said to be positive if it takes only non-negative values over reals. Hilbert's 17th problem concerns whether every positive polynomial can be expressed as a sum of squares of other polynomials. Many problems in math and computer science are closely connected to deciding whether a given polynomial is positive and finding certificates (e.g., sum-of-squares) of positivity. In quantum information, we are interested in noncommutative polynomials in *-algebras. A well-known theorem of Helton states that an element of a free *-algebra is positive in all *-representations if and only if it is a sum of squares. The theorem provides an effective way to determine if a given element is positive, by searching through sums of squares decompositions. In this talk, I'll present joint work with Arthur Mehta and William Slofstra in which we show that no such procedure exists for the tensor product of two free *-algebras: determining whether an element of such an algebra is positive is coRE-hard. Consequently, tensor products of free *-algebras contain elements which are positive but not sums of squares. I will also discuss the connetions to quantum information theory.

This seminar will be held both online and in person:

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

Ian Charlesworth, Cardiff University

"Permutation matrices, graph independence over the diagonal, and consequences"

Graph products were first introduced by Green in the context of groups, giving a mixture of direct and free products. They have recently been studied in the contexts of operator algebras and of non-commutative probability theory by M\l{}otkowski, Caspers and Fima, Speicher and Wysocza\'nski, and others. It is interesting to ask how properties of a family of von Neumann algebras are witnessed in a graph product; while free products and tensor products are well understood, their interplay can be quite subtle in this more general setting. With Collins, I showed how conjugation by random unitary matrices in a tensor product of matrix algebras creates asymptotic graph independence, when the unitaries are independent and uniformly distributed but only on particular subalgebras. In this talk, after spending some time introducing the setting, I will discuss how techniques inspired by the work of Au, C\'ebron, Dahlqvist, Gabriel, and Male can be used to make a similar statement about random permutations leading to asymptotic graph independence over the diagonal subalgebra; the combinatorial techniques required involve some interesting subtleties which are not apparent at first glance. I will also discuss some consequences for von Neumann algebras. For example, suppose that $(M_v)_v$ is a collection of finite dimensional algebras. $M_v$ can be embedded into a larger matrix algebra in such a way that it is constant on the diagonal, and the standard matrix units of $M_v$ are embedded as elements whose entries are roots of unity. Then if $M$ is a graph product of the $M_v$, we can find matricial approximations of a generating set which enjoy the same properties, and this in turn allows us to show (using techniques of Shlyakhtenko) that if the if $M$ is diffuse and algebra generated by the $M_v$ within $M$ has vanishing first $L^2$ Betti number then $M$ is strongly 1-bounded in the sense of Jung. This is joint work with de Santiago, Hayes, Jekel, Kunnawalkam Elayavalli, and Nelson.

This seminar will be held both online and in person:

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