Transformation Groups Seminar

Speaker: Tao Gong (Western)

"Introduction to root systems and Weyl groups"

Time: 10:30

Room: MC 108

In a Euclidean space, a (crystallographic) root system is special set of nonzero vectors, called roots. Each root determines a reflection in the Euclidean space, and these reflections generate a finite group, called the Weyl group. I will show some interesting facts about root systems and Weyl groups. In particular, I will talk about classification of root systems, and group actions of (extended) Weyl groups on the Euclidean space.

Geometry and Topology

Speaker: Yvon Verberne (Western)

"The grand arc graph"

Time: 15:30

Room: MC 107

One of the key tools to study surfaces of finite-type is the curve graph. Masur and Minsky showed that the curve graph is both infinite diameter and Gromov hyperbolic. Additionally, Masur and Minsky showed the curve graph's utility by using it to study the geometry of the mapping class group for surfaces of finite-type. Unfortunately, for surfaces of infinite-type the curve graph has diameter 2. In this talk, we introduce the grand arc graph and show that for large collections of infinite-type surfaces, the grand arc graph has infinite diameter and is Gromov hyperbolic. This work is joint with Assaf Bar-Natan.

Pizza Seminar

Speaker: Olga Trichtchenko (Western)

"Solitons and Machine Learning: Capturing Nonlinear Waves with Physics-Informed Neural Networks"

Time: 17:30

Room: MC 108

> Solitons (or solitary waves) are waves that travel at a constant speed while maintaining their shape, due to the perfect balance of nonlinear and dispersive effects. These were first described in 1834 by J.S. Russell as he travelled on horseback beside a canal, pursuing a wave for almost two miles. However, solitons are more ubiquitous in nature from rogue waves that can overturn boats, to being responsible for transmitting signals in the brain. They are well studied in different models like the Korteweg-de Vries equation first introduced in 1877 using various methods such as inverse scattering. On the other hand machine learning techniques, in particular physics informed neural networks (PINNs) are much newer methods, only formally introduced in the last few years. They rely on the governing physical laws and employ modern optimisation techniques to find solutions to the underlying equations. In this talk, we discuss the preliminary work bridging the two fields together. We examine how well machine learning methods can capture solitons as well as their nonlinear interactions and where the new methods may come up short.

Ph.D. Presentation

Speaker: Sinan Nurlu (Western)

"TBA"

Time: 15:30

Room: MC 107

Ph.D. Presentation

Speaker: Priya Bucha Jain (Western)

"TBA"

Time: 16:00

Room: MC 107

Ph.D. Presentation

Speaker: Siyuan Deng (Western)

"TBA"

Time: 16:30

Room: MC 108

Transformation Groups Seminar

Speaker: Tao Gong (Western)

"Introduction to root systems and Weyl groups (continued)"

Time: 10:30

Room: MC 108

In a Euclidean space, a (crystallographic) root system is special set of nonzero vectors, called roots. Each root determines a reflection in the Euclidean space, and these reflections generate a finite group, called the Weyl group. I will show some interesting facts about root systems and Weyl groups. In particular, I will talk about classification of root systems, and group actions of (extended) Weyl groups on the Euclidean space.

Topology and geometry seminar

Speaker: Michael Francis (Western)

"Detecting the orientation class of a singular foliation"

Time: 15:30

Room: MC 107

Given a foliated smooth manifold, you'd like to do analysis/topology on the space of leaves. Bad news: this quotient is poorly-behaved! Instead, you can work with a smooth proxy called the holonomy groupoid (which defines the same stack). Using this gadget, Connes defined things like the orientation class and fundamental class of the leaf space. Trying to do similar things with singular foliations, one encounters weird and interesting phenomena such as "continuous holonomy". We'll explore these ideas by concentrating on a special class of examples.

Colloquium

Speaker: Michael Albanese (Waterloo)

"The Yamabe Invariant of Complex Surfaces"

Time: 15:30

Room: MC 108

To any suitable geometric space (closed smooth manifold), one can associate a real number called the Yamabe invariant which arises from considerations in Riemannian geometry. For surfaces, this number is a familiar quantity, but in higher dimensions, it is less understood. However, as we will see, more can be said if one restricts to those spaces which admit a complex structure, e.g., orientable surfaces. This talk is partly based on joint work with Claude LeBrun.

Ph.D. Public Lecture

Speaker: Tianyu Cheng (Western)

"Study of Behaviour Change and Impact on Infectious Disease Dynamics by Mathematical Models"

Time: 09:00

Room: MC 107

This work uses mathematical models to study human behavior changes' effects on infectious disease transmission dynamics. It centers on two main topics. The first concerns how behavior response evolves during epidemics and the effects of adaptive precaution behavior on epidemics. The second topic is how to build general framework models incorporating human behavior response in epidemiological modelling.

Graduate Seminar

Speaker: Nathan Kershaw (Western)

"Closed symmetric monoidal structures on the category of graphs"

Time: 15:30

Room: MC 107

Discrete homotopy theory is a relatively new area of mathematics, concerned with applying methods from homotopy theory in topology to the category of graphs. In order to do this, a notion of a product between graphs is required. Classically two products have been considered, the box product and the categorical product. These products lead to two different homotopy theories, namely A-theory and X-theory, respectively. This leads us to the question of why these two products are considered, and if one can define other products to study discrete homotopy theory with instead. In this talk, we will answer this question by fully characterizing all closed symmetric monoidal products on the category of graphs. 

This talk will be based on joint work with C. Kapulkin (arxiv:2310.00493).

Transformation Groups Seminar

Speaker: Kumar Sannidhya Shukla (Western)

"Complexity 0 torus actions on manifolds"

Time: 10:30

Room: MC 108

Let T be an n-dimensional torus acting on a ‘nice’ 2n-manifold M effectively, with nonempty set of fixed points and suppose that all the isotropy groups are connected. If the action satisfies another hypothesis (equivariant formality) then the quotient space M/T has the structure of a homology cell complex and is in fact a homology disk. In this talk we will give first discuss some general facts about orbits of torus actions on manifolds and about locally standard actions. Then using the Atiyah-Bredon-Franz-Puppe sequence we give a detailed proof of the fact that under the assumptions stated above, M/T is a homology disk.

Analysis Seminar

Speaker: Blake Boudreaux (Western)

"Generalizations of Rational Convexity"

Time: 14:30

Room: WSC 187

A compact $K\subset\mathbb{C}^n$ is called rationally convex if for every point $p\not\in K$ there is a polynomial $P$ with $P(z)=0$ but $P^{-1}(0)\cap K=\varnothing$. Rational convexity is important in view of the Oka--Weil theorem, which states that a holomorphic function defined in a neighbourhood of a rationally convex compact $K$ is the uniform limit on $K$ of a sequence of rational functions. $$ $$ It is not obvious how to generalize rational convexity to the setting of a Stein manifold $X$. For instance, what should play the role of the polynomial? A first guess would be to replace the polynomial with a holomorphic function on $X$, but a second guess would be to replace the zero set of the polynomial with a general analytic hypersurface in $X$. In Euclidean space, it is well known that every analytic hypersurface has a global representation as the zero set of an entire function, so these notions coincide. $$ $$ In this talk, we will compare and contrast these different notions of rational convexity and flesh out their relevant properties. In particular, each generalization has its own version of an Oka--Weil theorem. We will also explore connections to weak and strong meromorphic functions. This is joint work with Rasul Shafikov.

Geometry and Topology

Speaker: Udit Mavinkurve (Western)

"Seifert-van Kampen theorems in discrete homotopy theory"

Time: 15:30

Room: MC 107

Discrete homotopy theory is a homotopy theory designed for studying simple graphs, detecting combinatorial, rather than topological, "holes." Central to this theory are the discrete homotopy groups which, just like their continuous counterparts, are easy to define but generally hard to compute. A discrete analogue of the Seifert-van Kampen theorem is thus a crucial tool to have in our computational toolbox. However, the version found in literature turns out to be too restrictive and is not applicable to several examples of interest. In this talk, we will state and sketch a proof of a new version that applies to a wider range of examples, and along the way, introduce some techniques with broader applicability. This talk is based on joint work with C. Kapulkin (arxiv:2303.06029).

Algebra Seminar

Speaker: Hyun Jong Kim (University of Wisconsin)

"An integral big monodromy theorem"

Time: 14:30

Room: MC 108

Associated to a family of curves $C\to S$ are $\ell$-adic monodromy representations, which generalize Galois representations. I will discuss part my ongoing thesis work demonstrating a big monodromy result for the moduli space of superelliptic curves. This result uses an arithmeticity result of reduced Burau representations of Venkataramana and clutching methods of Achter and Pries. Time permitting, I will also describe applications of this big monodromy result in other parts of my thesis --- it can be used to prove a Cohen-Lenstra result for function fields and to prove a result on the vanishing of zeta functions for Kummer curves over the projective line over finite fields.

Transformation Groups Seminar

Speaker: Kumar Sannidhya Shukla (Western)

"Complexity 0 torus action on manifolds (Part 2)"

Time: 10:30

Room: MC 108

Let T be an n-dimensional torus acting on a ‘nice’ 2n-manifold M effectively, with nonempty set of fixed points and suppose that all the isotropy groups are connected. If the action satisfies another hypothesis (equivariant formality) then the quotient space M/T has the structure of a homology cell complex and is in fact a homology disk. In this talk we will give first discuss some general facts about orbits of torus actions on manifolds and about locally standard actions. Then using the Atiyah-Bredon-Franz-Puppe sequence we give a detailed proof of the fact that under the assumptions stated above, M/T is a homology disk.

Ph.D. Public Lecture

Speaker: Shubhankar (Western)

"On Polar Convexity and Refinements of the Gauss-Lucas Theorem"

Time: 13:30

Room: MC 260

This report is based on the paper by Prof. Hristo Sendov and a joint work in preparation. The aim is to discuss the results presented in the paper in the light of the notions of polar convexity and convince the reader of the usefulness of this approach in studying polynomials. We start by recalling some relevant concepts and basic Definitions from polar convexity. We will then set up the stage for the main theorem with some supporting results and then finally prove it. Then we go on to discuss my own work in extending the notion of polar convexity to any finite dimension Euclidean space. The problem holds interesting consequences regarding non-linear optimization and somehow polar convex sets turn up very naturally when considering zeroes or critical points of polynomials.

Analysis Seminar

Speaker: Blake Boudreaux (Western)

"Generalizations of Rational Convexity II"

Time: 14:30

Room: MC 108

There are two generalizations of the notion of rational convexity on $\mathbb{C}^n$ to a general Stein manifold $X$. For a given compact $K\subset X$, the two associated hulls are $$ h(K)=\left\{z\in X\,:\,f^{-1}(0)\cap K\neq\varnothing\text{ for every }f\in\mathcal{O}(X)\text{ satisfying }f(z)=0\right\} $$ and $$ H(K)=\left\{z\in X\,:\,\text{every $\mathbb{C}$-hypersurface in $X$ passing through $z$ intersects $K$}\right\}. $$ In the last talk, it was shown these hulls are equal to the hulls with respect to meromorphic and "strong" meromorphic functions on $X$, respectively. In this talk we continue this comparison by showing a generalization of Runge's theorem and a theorem of Duval-Sibony for strong meromorphically convex compacts.

Geometry and Topology

Speaker: Abdul Zalloum (University of Toronto)

"Injective metric spaces"

Time: 15:30

Room: MC 107

A metric space X is said to be injective if any collection of pairwise intersecting balls admits a total intersection. Equivalently, injective metric spaces are exactly the injective objects in the category of metric spaces with respect to 1-Lipshitz map. In the talk I will:

1) Define injective spaces and investigate some of their properties, 2) Discuss what can be learnt about a group G that admits a ``nice" action on an injective metric space, and 3) Give a criterion for building actions on injective metric spaces with some applications.

The talk is based on two joint works, one is with Sisto and the other is with Petyt and Spriano.

Pizza Seminar

Speaker: Taylor Brysiewicz (Western)

"Computational Enumerative Geometry"

Time: 17:30

Room: MC 107

In this talk, I will tell the story of 'computational enumerative geometry', the area of research which uses computers to study enumerative problems. An enumerative problem asks how many geometric figures have a prescribed position to a given set of fixed, generic, geometric figures. For example: "how many conics are tangent to five conics in the plane". The answer to this question is famously 3264, but actually computing 3264 conics is another story. This task, and several others, is a job for the emerging field of computational enumerative geometry!

Ph.D. Public Lecture

Speaker: Siyuan Yu (Western)

"Symplectic Embeddings of Closed Balls in the Complex Projective Plane"

Time: 10:00

Room: WSC 248

Department Meeting

Speaker: (Western)

"Department Meeting"

Time: 15:30

Room: MC 108

Colloquium

Speaker: Alexandre Odesskii (Brock University)

"When the Fourier transform is one loop exact?"

Time: 15:30

Room: MC 107

We investigate the question: for which functions $f(x_1,...,x_n),~g(x_1,...,x_n)$ the asymptotic expansion of the integral $\int g(x_1,...,x_n) e^{\frac{f(x_1,...,x_n)+x_1y_1+...+x_ny_n}{\hbar}}dx_1...dx_n$ consists only of the first term. We reveal a hidden projective invariance of the problem which establishes its relation with geometry of projective hypersurfaces of the form $\{(1:x_1:...:x_n:f)\}$. We also construct various examples, in particular we prove that Kummer surface in $\P^3$ gives a solution to our problem. This is a joint work with Maxim Kontsevich.

Graduate Seminar

Speaker: Nathan Pagliaroli (Western)

"The Gaussian Unitary Ensemble and the Enumeration of Maps"

Time: 16:30

Room: MC 1078

In this talk I will introduce the notation of a matrix ensemble with a focus on the Gaussian Unitary Ensemble (GUE) as an example. I will introduce its basic properties in connection with map enumeration. In particular, I will outline a proof of the Genus Expansion Formula for moments of the GUE. Time permitting we will discuss the famous Harer-Zagier formula.

Transformation Groups Seminar

Speaker: Kumar Sannidhya Shukla (Western)

"Complexity 0 torus action on manifolds (Part 3)"

Time: 10:30

Room: MC 108

Let T be an n-dimensional torus acting on a ‘nice’ 2n-manifold M effectively, with nonempty set of fixed points and suppose that all the isotropy groups are connected. If the action satisfies another hypothesis (equivariant formality) then the quotient space M/T has the structure of a homology cell complex and is in fact a homology disk. In this talk we will give first discuss some general facts about orbits of torus actions on manifolds and about locally standard actions. Then using the Atiyah-Bredon-Franz-Puppe sequence we give a detailed proof of the fact that under the assumptions stated above, M/T is a homology disk.