Geometry and Combinatorics

Speaker: Taylor Brysiewicz (Western)

"Computing with matroids"

Time: 15:30

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

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

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

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)

"Symmetric functions and 1, 2, 5, 14, 42, ..."

Time: 15:30

Room: MC 108

In this episode of Basic Notions, we introduce symmetric functions, build quotient rings and retrace the history of a beautiful (still) open problem in coinvariants theory — the q,t-Catalan joint symmetry.

Graduate Seminar

Speaker: Zack Dooley (Western)

"An Introduction to Proof Assistants"

Time: 15:30

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.

Dept Oral Exam

Speaker: Siyuan Deng (Western)

"Symbolic-numeric algorithms for simplifying differential systems and their application to the determination of approximate Lie symmetry algebras"

Time: 15:30

Room: MC 327

An important and challenging computational problem is to identify and include the missing compatibility (integrability) conditions for general systems of partial differential equations. The inclusion of such missing conditions is executed by the application of differential-elimination algorithms. Differential equations arising during modeling generally contain both exactly known coefficients and coefficients known approximately from data. Very little research has been done on this case. This talk focuses on our recent work on approximate differential-elimination methods. Those methods are applied to approximate Lie symmetry algebras defining systems of differential equations. We illustrate this with applications to a class of Schrodinger equations, and other systems. Methods for computing the reliability of the methods are given as well.