Flower Hour

Speaker: (Western)

"Mathematical Biology Seminar"

Time: 11:00

Room: WSC 187

Geometry and Combinatorics

Speaker: Mieke Fink (University of Bonn)

"Schubert matroids and valuative invariants"

Time: 08:30

Room: Zoom

contact Graham for zoom link, thanks.

Transformation Groups Seminar

Speaker: Tao Gong (Western)

"On the quotient of a real toric variety from a Weyl group"

Time: 09:30

Room: MC 108

For a Weyl polytope $P$, there is an associated real toric variety $X_P^{\mathbb{R}}$. The quotient of $X_P^{\mathbb{R}}$ by the Weyl group action is obtained by gluing copies of cubes together. In this lecture, we will see the sufficient condition for the glued subspace of the cube to be contractible, and hence the corresponding gluing operation is a homotopy equivalence.

Pizza Seminar

Speaker: Nicole Lemire (Western)

"Triangulations of regular polygons and associated stories."

Time: 17:30

Room: MC 107

In 1751, Euler wrote a letter to Goldbach in which he conjectured a formula for the number of triangulations of a regular polygon with n sides. It turns out that the triangulations of a regular polygon are in bijection with many other geometric and combinatorial sets of objects. There is a mythical polytope, called the associahedron, whose vertices correspond to the triangulations of a regular polygon. The associahedron itself has a long mathematical history, starting with work of Tamari and Stasheff. The associahedron today has connections to many diverse areas of mathematics, including moduli spaces and topology, quiver representation theory, cluster algebras and toric varieties. We will discuss the beginnings of this story, starting with cutting small polygons into triangles using non-crossing diagonals.

Colloquium

Speaker: Kasra Rafi (Toronto)

"What does a random surface look like? CANCELLED"

Time: 15:30

Room: PAB 148

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.

Professional Development

Speaker: Aaron Crighton (Fields Institute)

"Fields Quantitative Information Security Specialist Program"

Time: 12:00

Room: https://zoom.us/j/99334167838

Representatives from Fields Information Security will present on their Information Security Specialist Program. Email Cassandra Schultz at for more information and program brochure.

Colloquium

Speaker: Hamid Hezari (UC Irvine)

"The inverse spectral problem for ellipses"

Time: 15:30

Room: MC 108

This talk is about Kac's famous inverse problem from 1966: "can one hear the shape of a drum?" The question asks whether the frequencies of vibration of a bounded domain determine the shape of the domain. First we present a quick survey on the known results. Then we discuss the key connection between eigenvalues of the Laplacian and the dynamics of the billiard, which is governed by the so called "Poisson Summation Formula". Finally we discuss our main theorem that "one can hear the shape of nearly circular ellipses". This is a joint work with Steve Zelditch (1953-2022).

Graduate Seminar

Speaker: Thomas Thorbjornsen (Western)

"Constructive Homological Algebra"

Time: 15:30

Room: MC 107

Homological algebra is a powerful tool to differentiate between structures and study obstructions. For instance, homology of spaces is an invariant that is classically simple to compute. For these reasons, it is desirable to develop these tools constructively, that means without using the law of excluded middle and the axiom of choice. We will discuss the relevance of these axioms, what breaks, and different approaches to fix or side-step the problems all together.

Flower Hour

Speaker: (Western)

"Mathematical Biology Seminar"

Time: 11:00

Room: WSC 187

PhD Thesis Defence

Speaker: Marwa Tuffaha (Western)

"Mutational Bias Shifts and Severe Environmental Stress Promote Mutator Emergence"

Time: 10:00

Room: MC 204

This thesis explores how elevations in mutation rates, also known as the rise of mutators, can be affected by two main factors: mutational biases and harsh environmental challenges. We show here that shifts in mutational biases -especially reductions or reversals- increase an organism's access to previously under-sampled mutations, resulting in higher frequencies of beneficial de novo mutations. Through a discrete-time mathematical model and simulations, we demonstrate that this enhanced access facilitates the rise of mutator strains with larger fitness effects. We also consider how evolutionary rescue can promote mutator lineages under abrupt or gradual environmental stress. Using branching processes and deterministic models, supported by simulations, we show that de novo mutators are likely to hitchhike due to evolutionary rescue events when the wildtype mutation rate is intermediate, while pre-existing mutators in the populations have a significant advantage when mutation costs are minimal due to low wildtype mutation rates. Unsurprisingly, the stronger a mutator is, the more effective it is if the wildtype mutation rate is low, while its relative advantage decreases in populations where the wildtype itself is a mutator. Finally, by analyzing cancer mutational data, we show that our theoretical predictions apply to human cancer. We find that non-hypermutated tumors exhibit a reversal of germline mutation biases such that a similar mutation spectrum across tissues shows signs of positive selection in cancer genes, whereas hypermutated tumors potentially access cancer-driver mutations through their high mutation rates without the need for bias shifts. Altogether, these findings underscore the important role that mutational biases and severe environmental stresses have on mutator emergence in asexual organisms, point to mechanisms of adaptive evolution and drug resistance development, and suggest possible therapeutical implications for the treatment of cancer.

PhD Thesis Defence

Speaker: Prakash Singh (Western)

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

Time: 09:00

Room: MC 204

This talk consists of two parts: In the first part, we will discuss geometric properties of the group of Hamiltonian diffeomorphisms (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 this centraliser. In the second part of the talk, we will discuss 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 these sets of all stability chambers. We also present the homotopy type of the relevant embedding spaces in some of these chambers.

Colloquium

Speaker: Olguta Buse (Indianapolis)

"Homotopic stability chambers in irrational blow up ruled surfaces"

Time: 15:30

Room: MC 107

We will give a gentle introduction to questions about the homotopy type of symplectomorphism groups of ruled symplectic 4-manifolds. Expanding results from the nineties on minimal rational ruled surfaces, several strides have been made in more recent years in the cases of blow-ups of such manifolds. We focus on understanding at large how such groups behave as we deform the symplectic forms within the cohomology cone in the case of irrational ruled surfaces with arbitrarily many blow ups. Using improved inflation techniques and a better understanding of the spaces of $J$ holomorphic curves, we will introduce a chamber structure on the reduced symplectic cone of such manifolds, so that the symplectomorphisms groups remain homotopically the same within the chambers. We will then discuss how an instance of such extremal ray limiting behaviour yields nontrivial symplectic isotopies, contrasting to the minimal cases. This is joint work with Jun Li.

Flower Hour

Speaker: (Western)

"Mathematical Biology Seminar"

Time: 11:00

Room: WSC 187