Geometry and Combinatorics

Speaker: Taylor Brysiewicz (Western)

"A ((numerical) algebraic) geometer's guide to algebraic matroids"

Time: 15:30 - 16:30

Room: MC 108

A finite set of elements in a field extension L/K defines a matroid via algebraic independence. From the perspective of algebraic geometry, every irreducible variety X in C^N therefore determines an algebraic matroid of rank dim(X) through the field extension C(X)/C. The bases of this matroid correspond precisely to coordinate projections that are branched covers.

This geometric perspective enriches the structure of algebraic matroids: bases and circuits carry natural degrees, and additionally, bases come equipped with a Galois (or monodromy) group. We describe these connections between algebraic matroids and algebraic geometry and make the case that numerical algebraic geometry is a powerful computational framework for handling such objects.

We end with two (open) examples of families of algebraic varieties whose matroids remain unknown.

Colloquium

Speaker: Frank Sottile (Texas A&M University)

"Webs and Welschinger signs"

Time: 15:30 - 15:30

Room: MC 107

A 3-dimensional subspace f of real polynomials defines a map f : P^1 -> P^2 whose image is a rational plane curve. It is maximally inflected when all of its flexes are real, equivalently, when its Wronski determinant has only real roots. We associate two a priori distinct signs (\pm 1) to f: the Welschinger invariant of the rational curve and the degree of the Wronski map at f. Extensive computation suggests that these signs coincide. While studying this conjecture we were led to a deeper conjecture: From f, we define a a function W : CP^1 -> CP^1 which encodes some real geometry of f and conjecturally gives an object called a web. We conjecture that known bijections between webs and standard Young tableaux and between tableaux and maximally inflected curves recovers the curve.

This talk will explain this picture with compelling evidence and beautiful pictures. It is joint work with Brazelton, Karp, Le, Levinson, McKean, Peltola, and Speyer.

Algebra Seminar

Speaker: Stefan Gille (University of Alberta)

"Stronger versions of Rost nilpotence"

Time: 14:30 - 15:30

Room: MC 108

Given two Chow motives M and N satisfying Rost nilpotence, a natural question is whether their direct sum has the same property. Rather obviously this question is closely related to the famous and old Kothe conjecture. If this conjecture is true the answer to above question is yes. However it seems that most ring theorists believe Kothe's conjecture does not hold. This leads to studying stronger versions of Rost nilpotence, which (surprisingly?) hold in (almost?) all cases where usual Rost nilpotence is known, as I will explain in my talk.

Graduate Seminar

Speaker: Theofanis Chatzidiamantis (Western)

"Fixed point properties in synthetic homotopy theory"

Time: 16:30 - 17:30

Room: MC 108

There are many results in topology showing that certain continuous maps from a space to itself have fixed points (most famously, Brouwer's fixed-point theorem). These results are often not accessible from the homotopy-theoretic point of view, since they usually depend on more than just the homotopy type of a given space, and many are also not constructive, making use of the law of excluded middle. After introducing the language of synthetic homotopy theory, we will see that studying fixed point properties in that setting provides a different, homotopy-invariant notion, and we will present (counter-)examples that can be obtained using constructive methods. For this talk, we also aim to avoid type-theoretic terminology, instead working from the topologist's perspective.

Colloquium

Speaker: Andy Zucker (University of Waterloo)

"Ramsey theory for binary relational structures"

Time: 15:30 - 16:30

Room: MC 107

Ramsey's theorem states that given a partition of the n-element subsets of a countably infinite set S into finitely many pieces, there is an infinite subset H of S so that all of the n-element subsets of H belong to the same piece. There are multiple ways one can attempt to generalize this result. In one direction, one can ask about coloring the infinite subsets of S. Here one needs to put some definability constraints on the partition (for instance, demanding that each piece is Borel), but upon doing so, Ellentuck's theorem gives a very satisfactory positive result. In another direction, one can add more structure to the infinite set S and demand that the witness H share this structure. For instance, S might be the rationals, and H would then be a subset of S which is order-isomorphic to the rationals. Here we can no longer demand that the n-element subsets of H belong to one piece of the partition, but we can put an absolute bound on how many pieces are needed. It turns out that these two ways of generalizing Ramsey's theorem can be combined, and this is the subject of joint work with Natasha Dobrinen.

Colloquium

Speaker: Janusz Adamus (Western)

"Introduction to Arc-analytic Geometry."

Time: 15:30 - 16:30

Room: MC 107

Abstract: Arc-analytic geometry is a fairly new branch of real analytic and algebraic geometry, inspired by John Nash's questions about the role of arcs in geometry. It studies the so-called arc-analytic functions and arc symmetric sets. In the algebraic setting, it provides the most satisfactory real counterpart of Zariski topology over an algebraically closed field. In the analytic setting, in contrast, it consists at present mostly of tough open problems. I will give a brief history of the subject with an overview of classical results from the late 80's and the 90's, and discuss both the recent progress and questions. The talk should be accessible to senior math undergrads.

Colloquium

Speaker: Junping Shi (College of William & Mary, VG, USA)

"Reaction-diffusion models for animal movement with spatial memory and nonlocal advection"

Time: 15:30 - 16:30

Room: MC 107

Animal populations often self-organize into territorial structure from movements and interactions of individual animals. Spatial memory is one of cognitive processes that may affect the movement and navigation of the animals. We will review several mathematical approaches of animal spatial movements: (i) reaction-diffusion-advection model with time-delayed memory-based movement; and (ii) reaction-diffusion-advection model with a non-local advection term driven by a cognitive map representing memory of past animal locations embedded in the environment. The well-posedness of models and bifurcation of spatiotemporal patterns will be discussed.

Mathematics Departmental Presentation 2026

Speaker: Adrian Chitan, Junqi Liu, Rujing Zhao (Western)

"TBA"

Time: 15:30 - 17:00

Room: MC 107