Applied dynamical system seminar

Speaker: Xiangyuan Liang (Western)

"TBA"

Time: 11:30 - 12:30

Room: MC 204

Introduction: Topics of these seminars include differential equations (ODEs, PDEs, DDEs, FDEs, etc.), dynamical systems theory, and their applications (often in mathematical biology). To get the brain gears turning, each session will kick off with a fun trivia!

Western Quantiers

Speaker: Asghar Ghorbanpour (Western)

"Mathematical Principles of Stabilizer Codes-II"

Time: 15:30 - 16:30

Room: MC 107

Stabilizer codes offer a robust and efficient framework for encoding quantum information and detecting errors. This family includes a large class of codes such as CSS codes, surface codes and Toric codes. In this talk, we will focus on the fundamental mathematical principles of stabilizer codes. Using the aspects of subgroups of Pauli groups, this family of the codes offers a unified scheme for detecting and correcting errors in quantum world. This unification simplifies both error detection and error correction for these family of codes.

Department Meeting

Speaker: (Western)

"N/A"

Time: 15:30 - 16:30

Room:

Transformation Groups Seminar

Speaker: Tao Gong (Western)

"Contractibility of quotients of real toric varieties from Weyl groups II"

Time: 09:30 - 11:00

Room: MC 108

Given a reduced crystallographic root system $R$ with the associated Weyl group $W$, the Weyl chambers from a fan and then give out a complex toric variety and its real part $X_{\mathbb{R}}$. We will see that the underlying topological space $X_{\mathbb{R}}/W$ is contractible.

This is a continuation of last week's talk.

Graduate Seminar

Speaker: Nathan Kershaw (Western)

"Efficient computations of discrete cubical homology"

Time: 15:30 - 16:30

Room: MC 108

We will present the fastest known algorithm for computing discrete cubical homology, a valuable graph invariant with a wide range of applications, including matroid theory, hyperplane arrangements, and topological data analysis. This invariant is capable of detecting certain types of "holes" within a graph, providing insight into its structure.

We will begin by defining discrete cubical homology and outlining the standard approach to its computation. We will then present an algorithm designed to improve efficiency by using techniques such as faster generation of singular cubes, reducing chain complex dimensions through quotients over automorphisms, and preprocessing graphs using results from discrete homotopy theory. These advancements aim to make the invariant more accessible computationally for applications. We are now able to compute examples that were previously considered out of reach by experts.

This talk is based on the paper: Kapulkin, Kershaw, Efficient computations of discrete cubical homology, arXiv:2410.09939.

Flower Hour

Speaker: TBA (Western)

"TBA"

Time: 11:00 - 12:00

Room: WSC 187

Applied dynamical system seminar

Speaker: Ruchita Amin (Western)

"TBA"

Time: 11:30 - 12:30

Room: MC 204

Introduction: Topics of these seminars include differential equations (ODEs, PDEs, DDEs, FDEs, etc.), dynamical systems theory, and their applications (often in mathematical biology). To get the brain gears turning, each session will kick off with a fun trivia!

Final Presentation

Speaker: David Johnson (Western)

"Finding configurations of 9 points lying on circles"

Time: 11:30 - 12:30

Room: MC 107

We will discuss the problem of finding the maximum number of 4-point circles given 9 points, where a 4-point circle is a circle on which lie exactly 4 of the 9 given points. This problem is solved for points in $\mathbb{R}^2$, we present a generalization to $\mathbb{C}^2$. Our approach uses techniques from polyhedral geometry and group theory. In particular, we will construct a polytope whose lattice points represent candidate point-circle incidences of 9-point configurations. This method works until the number of circles is greater than 4, at which point solving for the lattice points becomes computationally infeasible. We present then another method which instead recursively constructs these configurations. We reduce computations by leveraging symmetries of $S_9$. Finally, we discuss the original motivation for this work, coming from an interpolation problem in mechanical engineering involving four-bar linkages.

This is a final presentation for the course AM 4999Z.

Geometry and Topology

Speaker: Dan Isaksen (Wayne State University)

"Unstable homotopy groups of spheres"

Time: 15:30 - 16:30

Room: MC 107

A basic and naive problem in homotopy theory is to compute the sets $[S^m, S^n]$ of homotopy classes of maps between spheres of different dimensions. I will describe the preliminary results of a machine-based approach to these computations. Historically, there are two separate paradigms for such computations: the EHP sequence, and the unstable Adams spectral sequence. Our approach exploits both, and the interaction between them.

I will not assume any familiarity with either the EHP sequence or the unstable Adams spectral sequence.

Colloquium

Speaker: Kasra Rafi (Toronto)

"What does a random surface look like?"

Time: 15:30 - 16:30

Room: WSC 240

Building on her seminal work regarding moduli space volumes for Riemann surfaces, Mirzakhani also calculated expected values for various geometric functions on moduli space. Notably, she examined the expected Cheeger constant, the injectivity radius at a random point, and the statistical distribution of different types of curves on surfaces of large genus. We will review several of Mirzakhani’s key results, which collectively offer insights into the geometry of random surfaces in high genus. Following this, we will explore some extensions of her findings in the context of translation surfaces.

Pizza Seminar

Speaker: Geoff Wild (Western)

"Implications of vertical transmission for pathogen-host co-evolution"

Time: 17:30 - 18:30

Room: MC 107

(Based on work by recent MSc Thesis student, George Shillcock)

Understanding the capacity of pathogens to cause severe disease is of fundamental importance to human health and preserving biodiversity. Many of those pathogens are not only transmitted horizontally between unrelated hosts but also vertically between parents and their progeny. It is widely accepted that vertical transmission leads to the evolution of less virulent pathogens, but this idea stems from research that neglects the evolutionary response of hosts. Here, we use a game-theory model of coevolution between pathogen and host to show that vertical transmission does not always lead to more benign pathogens. We highlight scenarios in which vertical transmission results in pathogens exhibiting more virulence. However, we also predict that more benign outcomes are still possible (a) when generating new horizontal infections inflicts too much damage on hosts, (b) when clearing an infection is too costly for the host, and (c) when vertical transmission is promoted by a greater growth rate of the host population. Though our work offers a new perspective on the role of vertical transmission in pathogen–host systems, it does agree with previous experimental work.

Graduate Seminar

Speaker: Mieke Fink (Western)

"Solving problems in matroid theory"

Time: 15:30 - 16:30

Room: MC 108

Many matroid invariants take values in polynomial rings. The study of the coefficients and roots of matroid invariants is an important topic in algebraic combinatorics. Examples are the characteristic polynomial, the Tutte polynomial or the Kazhdan-Lusztig polynomial of a matroid. Connections between matroid theory to the intersection theory of toric varieties has played an important role in the proof of several long-standing conjectures in recent years. In this talk I will explain techniques that have been successfully used to investigate properties of such polynomials.

Flower Hour

Speaker: TBA (Western)

"TBA"

Time: 11:00 - 12:00

Room: WSC 187

Applied dynamical system seminar

Speaker: Haifeng Wang (Western)

"TBA"

Time: 11:30 - 12:30

Room: MC 204

Introduction: Topics of these seminars include differential equations (ODEs, PDEs, DDEs, FDEs, etc.), dynamical systems theory, and their applications (often in mathematical biology). To get the brain gears turning, each session will kick off with a fun trivia!

Western Quantiers

Speaker: Asghar Ghorbanpour (Western)

"Mathematical Principles of Stabilizer Codes-III"

Time: 15:30 - 16:30

Room: MC 108

Stabilizer codes offer a robust and efficient framework for encoding quantum information and detecting errors. This family includes a large class of codes such as CSS codes, surface codes and Toric codes. In this talk, we will focus on the fundamental mathematical principles of stabilizer codes. Using the aspects of subgroups of Pauli groups, this family of the codes offers a unified scheme for detecting and correcting errors in quantum world. This unification simplifies both error detection and error correction for these family of codes.

Colloquium

Speaker: (Basic Notions) Taylor Brysiewicz (Western)

"Sparse Polynomial Systems"

Time: 15:30 - 16:30

Room: MC 107

As implied by Bezout's theorem, n generic polynomials of degrees d1,...,dn in C^n have exactly d1*d2*...*dk common roots. Here, the degrees of each polynomial are specified, but they are otherwise generic. Adding constraints, one may impose which *monomials* are involved in each polynomial, resulting in a 'sparse polynomial system'. The analogue of Bezout for sparse systems is the celebrated Bernstein-Kouchnirenko-Khovanskii (BKK) theorem.

The BKK theorem relates a solution count to the polyhedral geometry of the monomial support of a sparse system. Relating other algebro-geometric features to combinatorial data of sparse systems is an active area of research. I will give a survey of what is known about some of these connections, how the associated theorems are used in practice, and what has yet to be discovered.

Transformation Groups Seminar

Speaker: Kumar Shukla (Western)

"Syzygies in equivariant cohomology of toric varieties with respect to subtori"

Time: 09:30 - 11:00

Room: MC 108

Let $X$ be a smooth toric variety with dense torus $T$. It is known that the syzygy order of equivariant cohomology $H_T^*(X)$ as an $H^*(BT)$-module can be computed from the combinatorics of the underlying fan. Suppose $S\subset T$ is a subtorus, we will compute the syzygy order of equivariant cohomology $H_S^*(X)$ with respect to $S$ as an $H^*(BS)$-module in some cases.

Algebra Seminar

Speaker: Daniel Litt (University of Toronto)

"On the converse to Eisenstein's last theorem"

Time: 14:30 - 15:30

Room: MC 108

I'll explain a conjectural characterization of algebraic solutions to (possibly non-linear) algebraic differential equations, in terms of the arithmetic of the coefficients of their Taylor expansions, strengthening the Grothendieck-Katz p-curvature conjecture. I'll give some evidence for the conjecture coming from algebraic geometry: in joint work with Josh Lam, we verify the conjecture for algebraic differential equations (both linear and non-linear) and initial conditions of algebro-geometric origin. In this case the conjecture turns out to be closely related to basic conjectures on algebraic cycles, motives, and so on.

Graduate Seminar

Speaker: Shubhankar (Western)

"Geometry of polar convex sets"

Time: 15:30 - 16:30

Room: MC 108

Polar convex sets can be seen as images of convex sets under some M\"obius transform. The notion of polar convex sets in the complex plane has been used to analyze the behavior of critical points of polynomials. In this talk, we go over the not too long history of polar convex sets in the plane. Following which, we extend the notion to finite dimensional Euclidean spaces. The goal of the talk is to build a theory of polar convexity and to show that the introduction of a pole creates a richer geometry compared to classical convex sets. Polar convexity enjoys a beautiful duality that does not exist in classical convexity. We will also formulate polar analogues of several classical results in convex optimization. Finally, we give a full description of the convex hull of finitely many points with respect to finitely many poles.

Flower Hour

Speaker: TBA (Western)

"TBA"

Time: 11:00 - 12:00

Room: WSC 187

Applied dynamical system seminar

Speaker: Dr. Terry Moschandreou (TVDSB)

"TBA"

Time: 11:30 - 12:30

Room: MC 204

Introduction: Topics of these seminars include differential equations (ODEs, PDEs, DDEs, FDEs, etc.), dynamical systems theory, and their applications (often in mathematical biology). To get the brain gears turning, each session will kick off with a fun trivia!

Geometry and Combinatorics

Speaker: Chris Kapulkin (Western)

"McCord's theorem via abstract homotopy theory"

Time: 15:30 - 16:30

Room: MC 108

Here is a fun question: what has four points and the weak homotopy type of the circle? If you enjoyed this one, here is a generalization: what has 2n+2 points and the weak homotopy type of the n-sphere? If you solved this one too, then it might come as no surprise to you that finite spaces present all finite homotopy types, which is to say that for every finite simplicial complex there is a finite topological space weakly equivalent to it, and for every finite space, there is a finite simplicial complex weakly equivalent to it. This arguably surprising result was proven by McCord in 1966.

In joint work with Daniel Carranza (Johns Hopkins University), we revisit McCord's theorem through the lenses of abstract homotopy theory and give a new proof of his result. The key fact used in our proof is the fact that a pushout of two open embeddings is a homotopy pushout.

In the talk, I will present the new proof of McCord's theorem. No background in abstract homotopy theory will be assumed.

Colloquium

Speaker: Jon Merzel (Soka)

"$G,\sigma$-circulant rings and generalizations of join rings"

Time: 14:30 - 15:30

Room: MC 108

Given a finite group $G$ and action $\sigma$ of $G$ on a finite set $S$, the $G-\sigma$ circulant ring is the ring of $S \times S$ matrices invariant under the induced action of $G$ on $S \times S$ and hence on $M_{S,S}$. We will see that this provides a generalization not only of circulant and $G$ circulant matrices but also the recently defined notion of a join of group rings. This provides a new approach to investigating an important family of matrices which have found many multidisciplinary applications. Several examples of generalizations to this context of properties of join rings will be given.

Geometry and Topology

Speaker: Diego Manco Berrio (Western)

"Witt vectors and K-theory"

Time: 15:30 - 16:30

Room: MC 107

By work of Almkvist and Campbell, for a commutative ring R, the ring of big Witt vectors on R, W(R), is a completion of the Grothendieck ring of the category of endomorphisms of finitely generated projectives over R. On the other hand, the usual operations on Witt vectors, like Verschiebung and Frobenius can be defined at a categorical level. We extend the categorical operations of Verschiebung and Frobenius to the case where R is a non-commutative ring and show that these satisfy some expected properties. We also show that these play well with the iterated traces of endomorphisms, or ghost maps. Our work is inspired by the definition of Witt vectors in the non-commutative setting by Dotto, Krause, Nikolaus, and Patchkoria. This is Joint work with Agarwal, Campbell, Ponto, and Sun.

Colloquium

Speaker: Siyuan Yu, Tao Gong, Shubankar (PhD Candidates) (Western)

"Mathematics Departmental Presentations 2025"

Time: 15:30 - 16:30

Room: MC 107

Transformation Groups Seminar

Speaker: Kumar Shukla (Western)

"Syzygies in equivariant cohomology of toric varieties with respect to subtori"

Time: 09:30 - 11:00

Room: MC 108

continuation from previous week

Flower Hour

Speaker: TBA (Western)

"TBA"

Time: 11:00 - 12:00

Room: WSC 187

Applied dynamical system seminar

Speaker: Qian Qin (Western)

"TBA"

Time: 11:30 - 12:30

Room: MC 204

Introduction: Topics of these seminars include differential equations (ODEs, PDEs, DDEs, FDEs, etc.), dynamical systems theory, and their applications (often in mathematical biology). To get the brain gears turning, each session will kick off with a fun trivia!

Geometry and Combinatorics

Speaker: Leo Jiang (University of Toronto)

"Topology of real matroid Schubert varieties"

Time: 15:30 - 16:30

Room: MC 108

Every linear representation of a matroid determines a matroid Schubert variety whose geometry encodes combinatorics of the matroid. When the representation is over the real numbers, we study the topology of the real points of the variety. Our main tool is an explicit cell decomposition, which depends only on the oriented matroid structure and can be extended to define a combinatorially interesting topological space for any oriented matroid. This is joint work with Yu Li.