Events

Christine Eagles, Department of Pure Mathematics, University of Waterloo

"Splitting the differential logarithm map 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. Such equations give rise to algebraic groups behaving as Galois groups. In this talk I give a characterisation of when the pullback of the differential logarithm of an equation is internal to the constants when the Galois group is unipotent or a torus. This is joint work in progress with Leo Jimenez.

MC 5479

Chris Schulz, Department of Pure Mathematics, University of Waterloo

"Automatic structure on Z[F]-modules"

The structure on the integers induced by the base-k representation has been well-studied using finite automata, by Büchi and others. Less well-explored are the extensions of these results to underlying groups other than Z. We will discuss a recent preprint of Francoise Point, in which the author uses the F-sets defined by Moosa and Scanlon in order to generalize Büchi's work. The end result is an expansion of a finitely generated Z[F]-module that has IP but maintains decidability.

MC 5479

Andy Royston, Penn State University

"Solitons and the Extended Bogomolny Equations with Jumping Data"

The extended Bogomolny equations are a system of PDE's for a connection and a triplet of Higgs fields on a three-dimensional space. They are a hybrid of the Bogomolny equations and the Nahm equations. After reviewing how these latter systems arise in the study of magnetic monopoles, I will present an energy functional for which solutions of the extended Bogomolny equations are minimizers in a fixed topological class. In this construction, the connection and Higgs triplet are defined on all of R^3 and couple to additional dynamical fields localized on a two-plane that are analogous to jumping data in the Nahm equations. Solutions can therefore be interpreted as finite-energy BPS solitons in a three-dimensional theory with a planar defect. This talk is based on work done in collaboration with Sophia Domokos.

MC 5417

Peter Pivovarov, University of Missouri

"A probabilistic approach to Lp affine isoperimetric inequalities"

In the class of convex sets, the isoperimetric inequality can be derived from several different affine inequalities. One example is the Blaschke-Santalo inequality on the product of volumes of a convex body and its polar dual. Another example is the Busemann--Petty inequality for centroid bodies. In the 1990s, Lutwak and Zhang introduced a related functional analytic framework with their notion of Lp centroid bodies, for p>1. Lutwak raised the question of encompassing the non-convex star-shaped range when p<1 (including negative values). I will discuss a probabilistic approach to establishing isoperimetric inequalities in this range. It uses a new representation of star-shaped sets as special averages of convex sets. Based on joint work with R. Adamczak, G. Paouris, and P. Simanjuntak.

This seminar will be held both online and in person:

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

Noah Snyder, Indiana University

"Tensor categories, string diagrams, and the Quantum Exceptional Series"

A representation of a group is a vector space on which the group acts linearly, and the collection of all finite dimensional representations of a group forms a structure called a tensor category. Unlike ordinary algebra which is written on a line (you can multiply on the left or on the right), tensor categories are better understood by doing calculations using diagrams in higher dimensions! In particular, "braided" tensor categories have 3-dimensional diagrams which are closely connected to knot polynomials like the Jones Polynomial, the Kauffman Polynomial, and the HOMFLY-PT polynomial. I will explain how the Kauffman polynomial is related to the family of orthogonal groups O(n), and at the end of the talk I will introduce a new conjectural knot polynomial related to the Exceptional Lie groups (from work joint with Thurston and joint in part with Morrison arxiv:2402.03637).

MC 5501

Amir Akbary, University of Lethbridge

"eta-Quotients whose Derivatives are eta-Quotients"

The Dedekind eta function is defined by the infinite product
\[
\eta(z) = e^{\pi i z/12}\prod_{n=1}^\infty (1 - e^{2 \pi i z}) = q^{1/24}\prod_{n=1}^\infty (1 - q^n).
\]
and
\[
f(z) = \prod_{t\mid N} \eta^{r_t}(tz),
\]
where the exponent r_t are integers. Let k be an even positive integer, p be a prime, and m be a nonnegative integer. We find an upper bound for orders of zeros (at cusps) of a linear combination of classical Eisenstein series of weight k and level p^m. As an immediate consequence, we find the set of all eta quotients that are linear combinations of these Eisenstein series and, hence, the set of all eta quotients of level p^m whose derivatives are also eta quotients.

This is joint work with Zafer Selcuk Aygin (Northwestern Polytechnic).

MC 5417

Chatchai Noytaptim, Department of Pure Mathematics, University of Waterloo

"Adelic equidistribution theorem for points of small height"

Bilu’s celebrated equidistribution theorem asserts that if there is an infinite sequence of distinct algebraic numbers with low  arithmetic complexity, then its Galois orbit is equidistributed with respect to the uniform probability measure on the complex unit circle. We present the proof of an adelic version of Bilu-type equidistribution theorem in dynamical setting. The material in this presentation covers section 7.9 in Baker-Rumely’s monograph on “Potential Theory and Dynamics on the Berkovich Projective Line”.

MC 5417

Gian Cordana Sanjaya, Department of Pure Mathematics, University of Waterloo

"Even More Examples of Schemes"

Last time, we looked at reduced schemes over algebraically closed fields. Now we remove the algebraically closed condition, and look at even more interesting schemes.

MC 5417

Joey Lakerdas-Gayle, Department of Pure Mathematics, University of Waterloo

"Isomorphism Spectra and Computably Composite Structures"

If $\mathcal{A}$ and $\mathcal{B}$ are two computable copies of a structure, their isomorphism spectrum is the set of Turing degrees that compute an isomorphism from $\mathcal{A}$ to $\mathcal{B}$. We introduce a framework for constructing computable structures with the property that the isomorphisms between arbitrary computable copies of these structures are constructed from isomorphisms between computable copies of their component structures. We call these \emph{computably composite structures}. We show that given any uniformly computable collection of isomorphism spectra, there exists a pair of computably composite structures whose isomorphism spectrum is the union of the original isomorphism spectra. We use this to construct examples of isomorphism spectra that are not equal to the upward closure of any finite set of Turing degrees.

MC 5479

Charles Cifarelli, CIRGET & Stony Brook

"Steady gradient Kähler-Ricci solitons and Calabi-Yau metrics on C^n"

I will present recent joint work with V. Apostolov on a new construction of complete steady gradient Kähler-Ricci solitons on C^n, using the theory of hamiltonian 2 forms, introduced by Apostolov-Calderbank-Gauduchon-Tønnesen-Friedman, as an Ansatz. The metrics come in families of two types with distinct geometric behavior, which we call Cao type and Taub-NUT type. In particular, the Cao type and Taub-NUT type families have a volume growth rate of r^n and r^{2n-1}, respectively. Moreover, each Taub-NUT type family contains a codimension 1 subfamily of complete Ricci-flat metrics.

MC 5417