Events

Stephen Melczer University of Waterloo,

Analytic Combinatorics in One and Several Variables

The field of analytic combinatorics develops effective methods to compute the asymptotic behaviour of combinatorial sequences from analytic properties of their generating functions. This talk surveys the classical methods of analytic combinatorics and details the newer area of analytic combinatorics in several variables (ACSV), which handles multivariate sequences and their multivariate generating functions. Applications to several areas of mathematics and computer science, including number theory, will be discussed. This talk will be complemented by a presentation of Erica Liu on January 27 describing some recent progress on new approaches to ACSV.

MC 5479

Jashan Bal University of Waterloo,

Strong convergence of random permutations

We will start proving that i.i.d random permutations strongly converge to Haar unitaries.

MC 5479

Spencer Kelly, University of Waterloo

Proper Group Actions and the Slice Theorem in Finite Dimensions

In this talk we will begin by reviewing important properties of group actions on manifolds, and characteristics of proper actions. We then define isotropy and orbit types, discuss the slice theorem (on finite dimensional manifolds), and go over non-trivial examples of slice bundles. This will set us up to conclude with the principal orbit theorem and the stratification of the orbit space.

MC 5403

Zhihao Zhang, University of Waterloo

Spectra of Beurling Algebras of Discrete Abelian Groups

We will discuss a variant of the group algebra, called the Beurling algebra. These algebras differ from their classical counterpart through the addition of a weight function modifying the norm. The Gelfand spectrum of the group algebra of absolutely integrable functions on an abelian group, G, is well known to be the Pontryagin dual of G. In the case of a Beurling algebra, the Gelfand spectrum can be much larger for suitable weights. We will focus on the Beurling algebra of a discrete abelian group, G, and give a description of its Gelfand spectrum in terms of a seminorm constructed from a symmetric weight.

MC 5417 or Join on Zoom

Erica Liu, University of Waterloo

Toric Compactifications and Critical Points at Infinity in Analytic Combinatorics

The field of Analytic Combinatorics in Several Variables (ACSV) provides powerful tools for deriving asymptotic information from multivariate generating functions. A key challenge arises when standard saddle-point techniques fail due to the presence of critical points at infinity (CPAI), obstructing local analyses near singularities. Recent work has shown that Morse-theoretic decompositions remain valid under the absence of CPAI, traditionally verified using projective compactifications. In this talk, I will present a toric approach to compactification that leverages the Newton polytope of a generating function to construct a toric variety tailored to the function’s combinatorial structure. This refinement not only tightens classification of CPAI but also enhances computational efficiency. Through concrete examples and an introduction to tropical and toric techniques, I will demonstrate how these methods clarify the asymptotic landscape of ACSV problems, especially in combinatorially meaningful settings. This talk draws on joint work studying toric compactifications as a bridge between algebraic geometry and analytic combinatorics.

MC 5479

Leigh Foster, University of Waterloo

Learning Seminar on lozenge tilings and the single dimer model

The study of lozenge tilings and of the dimer model is a well-established area of research, going back to the 1960's and

still subject to active research at present. (Some references, also showing connections to other directions of research,

are listed below.) We will start the learning seminar on this topic with a series of three meetings giving an introduction

to the dimer model in its single-dimer version, and considered on a finite hexagonal grid.

--In the first meeting, on January 27, we will introduce the single dimer model and we will discuss its connection to other

combinatorial objects, with particular emphasis on the connection between the dimer model and the notion of "stack of

boxes".

--In the second meeting, on February 3, we will discuss enumeration questions related to the configurations introduced

in the preceding week.

--The third meeting, on February 10, will be open to requests from the audience. One potential direction for this meeting

will be to look at some probabilistic results on random dimer configurations of a given shape, as well as various limit shape

phenomena, pointing to the study od some intriguing objects called "amoebas".

The learning seminar is intended to continue after the reading week, covering the more recent research topic of double-dimer

and more generally multiple-dimer models.

The seminar is addressed to all interested audiences, with very few assumptions regarding background knowledge.

Come, listen and ask questions - everyone is welcome, and interruptions are hoped for!

MC 5403

Viktor Majewski, University of Waterloo

Dirac Operators on Orbifold Resolutions

In this talk we discuss Dirac operators along degenerating families of Riemannian manifolds that converge, in the Gromov-Hausdorff sense, to a Riemannian orbifold. Such degenerations arise naturally when analysing the boundary of Teichmüller spaces of special Riemannian metrics as well as moduli spaces appearing in gauge theory and calibrated geometry. Here sequences of smooth geometric structures on Riemannian manifolds may converge to an orbifold limit. To understand and control these degenerations, we introduce smooth Gromov-Hausdorff resolutions of orbifolds, that are, smooth families (X_t,g_t), which collapse to the orbifold (X_0,g_0) as t goes to 0.

The central analytic problem addressed in this paper is to understand the behaviour of Dirac operators along such resolutions, in particular in collapsing regimes where classical elliptic estimates fail. We develop a uniform Fredholm theory for the family of Dirac operators on the Gromov-Hausdorff resolution. Using weighted function spaces, adiabatic analysis, and a decomposition of X_t into asymptotically conical fibred (ACF), conically fibred (CF) and conically fibred singular (CFS), we obtain uniform realisations of the model operators and prove a linear gluing exact sequence relating global and local (co)kernels. As a consequence, we construct uniformly bounded right inverses for D_t, and derive an index additivity formula.

The theory developed here provides the analytic foundation for nonlinear gluing problems in gauge theory and special holonomy geometry, including torsion-free G-structures, instantons, and calibrated submanifolds of Riemannian manifolds close to an orbifold limit.

MC 5403

Beining Mu, University of Waterloo

Algorithmic randomness and Turing degrees 3

In this seminar we talk about coding strategies to encode an arbitrary set into a 1-random set in a sense that every set is wtt-reducible to a 1-random set. We will also have a review of the jump operator and lowness of Turing degrees to explore the distribution of 1-random sets in terms of Turing degrees.

MC 5403

Luke Postle, University of Waterloo

A New Proof of the Existence Conjecture and its Applications to Extremal and Probabilistic Design Theory

We discuss the recently developed method of refined absorption and how it is used to provide a new proof of the Existence Conjecture for combinatorial designs. This method can also be applied to resolve open problems in extremal and probabilistic design theory while providing a unified framework for these problems. Crucially, the main absorption theorem can be used as a ``black-box'' in these applications obviating the need to reprove the absorption step for each different setup.

MC 5403

Carlo Pagano, Concordia University

Reconstructing curves from their Galois set of points

Mazur—Rubin asked to what extent one can reconstruct a curve C over a number field K from its set of points over bar(K), viewed as a Galois set. They asked the same question specifically about the set of fields where C acquires new points and gave evidence for a positive answer for curves of genus 0. In this talk we will present upcoming work with Zev Klagsbrun where we provide a positive answer for a generic pair of elliptic curves with full 2-torsion over a number field. The method of proof uses the combination of additive combinatorics and descent introduced in joint work of the speaker and Koymans in 2024. I will overview several other recent results obtained, by a number of authors, with that method.

MC 5479