Welcome to Pure Mathematics

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

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