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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

Mikael De La Salle, Université de Lyon

"Variations around the Hilbert transform"

The Hilbert transform is the operator of convolution by the distribution 1/t. It is an emblematic example of a Fourier multiplier. I will present variants on other spaces than the real line, such as matrices, manifolds and groups, and applications to operator algebras, representation theory. I will present more or less recent results obtained in collaboration with Vincent Lafforgue, Tim de Laat, Javier Parcet and Eduardo Tablate.

MC 5501

Rachael Alvir, Department of Pure Mathematics, University of Waterloo

"Effective Descriptive Set Theory 5"

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

MC 5479

Laindon Burnett, Department of Pure Mathematics, University of Waterloo

"Honest Definitions in NIP"

We cover the idea of Honest Definitions from Pierre Simon's "A Guide to NIP Theories". In particular, we show that in an NIP theory, even if a particular type is not definable over a given set (and hence there is a bad formula which prevents it from being so), we may instead define this formula with parameters from an elementary extension, and this new defining formula is captured inside of the bad formula when we loosen ourselves and allow for object variables to come from the extension.

MC 5403

Brady Ali Medina, Department of Pure Mathematics, University of Waterloo

"Schubert Cycles"

I will introduce Schubert cycles and explain some facts about them. This corresponds to section 4.1 of 3264 & All That by Eisenbud and Harris.

This seminar will be held both online and in person:

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

Nathan Grieve, Acadia University

"On Schmidt's Subspace Theorem, Vojta's height inequalities and algebraic points in projective varieties: recent developments and progress"

I will report on a collection of recent results and ongoing work that surround extensions and applications of Schmidt's Subspace Theorem and Vojta's height inequalities.  As two examples: (i) It is of interest to understand the qualitative features of Diophantine exceptional sets; (ii) It is of interest to understand the extent to which algebraic points of a given bounded degree in a given general type projective variety are not-Zariski dense.  As I will explain, there are several logically equivalent points of departure for these results.  They build on a collection of my past contributions in addition to work of many others.

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

Félix Baril Boudreau, University of Lethbridge

"Arithmetic rank bounds for abelian varieties over function fields"

It is known since the works of Ogg (1962) and Shafarevich (1961) (under tameness assumptions), followed by Grothendieck (1968), that the rank of a given abelian variety over the function field of a curve is bounded by a quantity which depends on the genus of the base curve and on reduction data. This bound is "geometric" in nature. In particular, it holds if we replace the constant field by its algebraic closure.

Ulmer asked in 2004 if, for an elliptic curve, there was an arithmetic bound that could improve on the geometric one. This question recently obtained a positive answer (Gillibert and Levin, 2022).

In this talk, we present a similar arithmetic refinement of the geometric bound for higher-dimensional abelian varieties. When specialized to elliptic curves, we improve on Gillibert-Levin's bound. Time permitting, we will discuss some consequences of our result.

This is joint work with Jean Gillibert and Aaron Levin.

MC 5501

Junsheng Zhang, University of California Berkeley

"On complete Calabi-Yau manifolds asymptotic to cones"

We proved a "no semistability at infinity" result for complete Calabi-Yau metrics asymptotic to cones, by eliminating the possible appearance of an intermediate K-semistable cone in the 2-step degeneration theory developed by Donaldson-Sun. As a consequence, a classification result for complete Calabi-Yau manifolds with Euclidean volume growth and quadratic curvature decay is given. Moreover a byproduct of the proof is a polynomial convergence rate to the asymptotic cone for such manifolds. Joint work with Song Sun.

QNC 2501