Geometry and Combinatorics

Speaker: Graham Denham (Western)

"matroids and toric varieties I"

Time: 15:30 - 16:30

Room: MC 108

I will give the first of two(?) expository talks on how toric and tropical geometry can be useful to study matroids.

Transformation Groups Seminar

Speaker: Vladimir Gorchakov (Western)

"Cohomology of Free Loop Spaces"

Time: 09:30 - 11:00

Room: MC 108

In this talk, we will discuss the cohomology groups of the free loop space of a topological space $X$, which is the space of all continuous maps from $S^1$ to $X$. Using the Eilenberg–Moore spectral sequence, we will connect these cohomology groups and Hochschild homology and compute them in specific cases. We will mostly follow the article "On the Characteristic Zero Cohomology of the Free Loop Space" by L. Smith.

Geometry and Topology

Speaker: Matthias Franz (Western)

"The homology of fibre bundles"

Time: 15:30 - 16:30

Room: MC 107

We review several known, but unfortunately not well-known results about the (singular) homology of fibre bundles. We use simplicial sets and in particular twisted Cartesian products, which are the simplicial analogues of fibre bundles. The central result is the twisted Eilenberg-Zilber theorem, which relates the chains on a bundle to a twisted tensor product of the chains on base and fibre. The Eilenberg-Moore theorems are easy consequences of it.

Graduate Seminar

Speaker: Alexander Zwart (Western)

"Rationality of Algebraic Tori"

Time: 15:30 - 16:30

Room: MC 107

Given a variety X a natural question to ask is whether it is birational to projective space. In general, this is quite a very hard question to answer. We restrict ourselves to a subclass of objects known as algebraic tori. It turns out that a slightly weaker notion related to rationality can be "cleanly" understood in terms of the character lattice for a given torus. I will give some background to state this result and then discuss the work that has been done on the rationality question for algebraic tori.

PhD Thesis Defence

Speaker: Doli Rani Pal (Western)

"Modeling and Analysis of Mosquito population dynamics and Mosquito-borne disease dynamics"

Time: 13:30 - 14:30

Room: Zoom

This presentation summarizes the findings from three interrelated projects conducted throughout my PhD research. By deploying differential equation models, this research explores issues concerning the dynamics of the mosquito population and diseases transmitted by mosquitoes. In the first project, we introduce a non-autonomous model for the transmission of a mosquito-borne disease, specifically Dengue fever, between two distinct regions: (A) rural areas with a fraction of the human population and a higher mosquito presence, and (B) urban areas where most workplaces are located, where the majority of people reside, and where mosquito numbers are relatively low. This model integrates periodic mosquito-biting rates and periodic shifts in the workforce. This project calculates the basic reproduction number, $\mathcal{R}_0$, and establishes the epidemic threshold. In particular, we illustrate how changes in the workforce's shift patterns impact the disease's spread. In the second project, we present a mathematical model that characterizes the interactions between wild mosquitoes and genetically modified mosquitoes that carry the {\it Serratia AS1} bacteria. The primary concern is determining whether AS1 can establish itself in the mosquito population, and if so, in what manner: either replacing or co-existing with wild mosquitoes. After confirming the well-posedness of the model, we investigate two sub-models: one disregards environmental AS1 infection, and the other assumes no cross-vertical transmission of AS1 within mosquitoes. We perform an in-depth analysis of each sub-model to identify the conditions under which AS1-carrying mosquitoes either replace or suppress wild mosquitoes or fail to establish. In addition, we performed numerical simulations to support our theoretical results. The third project builds on the second one by examining the influence of AS1 in malaria control. Using the model presented in the second project, we construct an extensive model that categorizes mosquitoes into three groups: wild, AS1-carrying, and malaria-carrying. By analyzing the dynamics of the model and comparing the results with those of a sub-model that omits \textit{Serratia AS1} (termed the malaria-only model), we investigate the potential role of the AS1 bacterium, introduced as a malaria control measure, in mitigating or eradicating malaria.

Geometry and Combinatorics

Speaker: Taylor Brysiewicz (Western)

"Computing with matroids"

Time: 15:30 - 16:30

Room: MC 108

TBA

Transformation Groups Seminar

Speaker: Fedor Vylegzhanin (NRU Higher School of Economics)

"Moment-angle complexes in the flag case (and beyond)"

Time: 09:30 - 10:30

Room: Zoom Meeting ID: 990 6584 3212

Loop homology of a moment-angle complex is a subalgebra in the loop homology of Davis-Januszkiewicz space, which is isomorphic to the Yoneda algebra $Ext_{k[K]}(k,k)$. (Here $k[K]$ is the Stanley-Reisner ring for the simplicial complex $K$). If $K$ is a flag complex, this Yoneda algebra is known; this allows to give a presentation for loop homology for the moment-angle complex and to describe homotopy groups of these spaces in terms of homotopy groups of spheres (using recent results of L. Stanton). If time permits, we will also consider the case of "almost flag" simplicial complexes.

Dept Oral Exam

Speaker: Prakash Singh (Western)

"Maximal torus in Hofer geometry and Embeddings in S^2 \times S^2"

Time: 09:30 - 10:30

Room: TBA

In the first part, we will discuss some geometric properties of the group of hamiltonian diffeomorphisms on M, Ham(M), associated to a closed symplectic manifold (M,\om) with respect to the Hofer metric. This group, although infinite dimensional, exhibits properties similar to compact Lie groups. Pushing this philosophy, it has been observed, classically, that when the symplectic manifold is endowed with a toric action, the centralizer of this action plays the role of a maximal torus in Ham(M). In this talk, we present results that support the Hofer geometric arguments supporting this philosophy. We also present some results w.r.t the intrinsic hofer geometry on the centraliser.

In the second part of the talk, we will study the embedding space of two disjoint standard symplectic balls of capacities (sizes) c1 and c2 in $S^2\times S^2$ with respect to any symplectic form. The set of admissible capacities for such embeddings is subdivided into polygonal regions in which the homotopy type of the embedding space is constant. We present the set of all stability chambers and also present the homotopy type of the relevant embedding spaces in some of these chambers.

Geometry and Topology

Speaker: Tao Gong (Western)

"Homotopy Types of Quotients of Toric varieties from Weyl Groups"

Time: 15:30 - 16:30

Room: MC 107

Given a reduced crystallographic root system with a fixed simple system, it is associated to a Weyl group $W$ and a polytope $P$ which is the convex hull of the $W$-orbit of a dominant weight. $P$ and $P/W$ are polytopes, associated with complex toric varieties $X_P$ and $X_{P/W}$ respectively. We will see a homotopy equivalence between $X_P/W$ and $X_{P/W}$, and contractibility of the real real points $X_{P}^{\mathbb{R}}/W$ for small ranks.

Colloquium

Speaker: Kelvin Chan (Western)

"TBA"

Time: 15:30 - 16:30

Room: MC 108

Graduate Seminar

Speaker: Zack Dooley (Western)

"An Introduction to Proof Assistants"

Time: 15:30 - 16:30

Room: MC 107

Proof assistants are a variety of software tools which can be used to assist in proof writing and verifying mathematical statements. Recently, proof assistants have been gaining popularity not just for their ability to verify the correctness of complicated proofs, but also as a tool of collaboration for mathematicians. In this talk I will introduce the basics of what proof assistants are and how to use them, in particular, focusing on the proof assistant Coq. I will show how to write basic definitions and proofs in Coq and show how libraries can help us with collaboration and proof writing.

Transformation Groups Seminar

Speaker: Vladimir Gorchakov (Western)

"TBA"

Time: 09:30 - 11:00

Room: MC 108

Geometry and Topology

Speaker: Nathan Kershaw (Western)

"Efficient computations of discrete cubical homology"

Time: 15:30 - 16:30

Room: MC 107

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. Part of the motivation for this work was a joint project with the research group of R. Laubenbacher (Dept. of Medicine, University of Florida) on analyzing gene regulatory networks.

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

Pizza Seminar

Speaker: Masoud Khalkhali (Western)

"What is not random about random matrices?"

Time: 17:30 - 19:00

Room: MC 107

Imagine you don't have perfect knowledge of the entries of a matrix (which is a what happens usually in applications of matrices). What can be said about the eigenvalues of such a matrix?  Is there a pattern to the eigenvalues at all? Can we say anything about them? In this talk I shall start with very simple examples and gradually  examine the question, using some computer calculations and some basic undergraduate mathematics.