Events

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

Roberto Hernandez Palomares, Department of Pure Mathematics, University of Waterloo

"C* Quantum Dynamics"

A subfactor is a unital inclusion of simple von Neumann algebras, which can be presented as a non-commutative dynamical system governed by a tensor category. Popa established that in ideal scenarios, dynamical data is a strong invariant for hyperfinite subfactors. These reconstruction results in a way give an equivariant version of Connes' classification for amenable factors. On the topological side, after the recent culmination of the classification program for amenable C*-algebras, whether there is an analogue of Popa's Reconstruction results is not clear. In this talk, I will describe the transfer of subfactor techniques to C*-algebras, introducing the largest class of inclusions of C*-algebras admitting a quantum dynamical invariant akin to subfactors. Examples include the cores of Cuntz algebras, certain semicircular systems, and crossed products by actions of tensor categories. Time allowing, we will discuss some interactions with the C* classification program. This is based on joint work with Brent Nelson.

This seminar will be held both online and in person:

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

Akash Sengupta, Department of Pure Mathematics, University of Waterloo

"Approximation of rational points and a characterization of projective space"

Given a real number x, how well can we approximate it using rational numbers? This question has been classically studied by Dirichlet, Liouville, Roth et al, and the approximation exponent of a real number x measures how well we can approximate x. Similarly, given an algebraic variety X over a number field k and a point x in X, we can ask how well can we approximate x using k-rational points? McKinnon and Roth generalized the approximation exponent to this setting and showed that several classical results also generalize to rational points algebraic varieties.

In this talk, we will define a new variant of the approximation constant which also captures the geometric properties of the variety X. We will see that this geometric approximation constant is closely related to the behavior of rational curves on X. In particular, I’ll talk about a result showing that if the approximation constant is larger than the dimension of X, then X must be isomorphic to projective space. This talk is based on joint work with David McKinnon.

MC 5417