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Konstantin Tikhomirov, Carnegie Mellon University

"On the width of random polyhedra"

We consider the problem of estimating the width of a polyhedron defined as the intersection of m i.i.d random affine subspaces of n-dimensional space. Such polyhedra naturally appear in probabilistic analysis of linear programs, as well as in convex geometric analysis as extremizers of various quantities associated with convex sets. For a wide range of parameters m, n, we obtain sharp estimates of the width of the polyhedron in any given direction.

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

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

"More Examples of Schemes"

We describe a number of examples of schemes and how to make sense of them.

MC 5417

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

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

Peter Oberly, University of Rochester

"Some Bounds on the Arakelov-Zhang Pairing"

The Arakelov-Zhang pairing (also called the dynamical height pairing) is a kind of dynamical distance between two rational maps defined over a number field. This pairing has applications in arithmetic dynamics, especially as a tool to study the preperiodic points common to two rational maps. We will discuss some bounds on the Arakelov-Zhang pairing of f and g in terms of the coefficients of f and investigate some simple consequences of this result.   

MC 5417

Kieran Mastel, Department of Pure Mathematics, University of Waterloo

"An Aperiodic Monotile"

Last year, David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss found the first example of an aperiodic monotile (or ‘einstein’), solving a longstanding open problem. We will look at the ‘hat’ tile they define and try to visually understand why it tiles the plane and why none of its tilings are periodic.

MC 5501

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

"Computable Structure Theory VII"

We will discuss degree spectra of structures, following Antonio Montalbán's monograph.

MC 5479

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

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