Ph.D and M.Sc Defences

Friday, May 17th, 2024

Time: 1:00pm

Student: Sonja Ruzic (PhD)

Supervisor: Ivan Dimitrov and David Wehlau

Title: Certain Weyl Modules of Infinite Dimensional Lie Superalgebras

PDEs & Applications Seminar

Tuesday, April 9th, 2024

Time: 9:30 a.m.  Place: Jeffery Hall, Room 319 (Via Zoom)

Speaker: Mark Veraar (Delft University of Technology)

Title: Stochastic partial differential equations in critical spaces

Abstract: In this talk I will give an overview of several recent developments on quasi- and semi-linear stochastic PDEs in critical spaces. I will present a new method to prove local and global well-posedness results, and new bootstrap method to show higher order regularity of the solution. In the talk several applications to reaction diffusion equations will be discussed in details. In particular, the new setting allows to prove global well-posedness for several systems which do not satisfy classical coercivity estimates. The talk is based on joint work with Antonio Agresti.

Curves Seminar

Thursday, April 4th, 2024

Time: 4:00 p.m.  Place: Jeffery Hall, Room 319

Speaker: Luke Steverango

Title: Defining Cluster Algebras by Generators and Relations

Abstract: A traditional way to describe a commutative algebra over a ring is as a quotient of a polynomial ring. One would hope to get an analogous way of describing cluster algebras in a similar way. Unfortunately, that runs into two issues. These issues are that the set of cluster variables is typically infinite and the exchange relations do not, in general, generate the ideal of all relations among cluster variables. We explore each of these issues, with particular emphasis on the second issue.

Math & Stats Department Colloquium (3MT Competition)

Friday, April 5th, 2024

Time: 2:30 p.m.  Place: Jeffery Hall, Room 234

Speaker(s): Jerin Farin (Giusy Mazzone)
Nic Fellini (Ram Murty)
Annika Fuernsinn (Bahman Gharesifard)
Skye Griffith (Glen Takahara and Wes Burr (Trent))
Neil MacVicar (Francesco Cellarosi and Jamie Mingo)
Richard Zhao (Felicia Magpantay)
Sasha Zotine (Mike Roth and Greg Smith)

Details: The Department of Mathematics and Statistics will once again host its own 3 Minute Thesis Competition. Seven graduate students will explain some aspect of their research in just three minutes following the official 3MT rules copied below. The winners will be selected by secret ballot following the judging criteria also copied below. All department members –graduate students, faculty, postdocs and staff - who attend the entire event and listen to all 7 presentations will be able to cast one ballot selecting their top 3 choices, and the winners will be announced in the 5th floor lounge immediately after the competition.

PDEs & Applications Seminar

Tuesday, April 2nd, 2024

Time: 9:30 a.m.  Place: Jeffery Hall, Room 319 (Via Zoom)

Speaker: Adilbek Kairzhan (Nazarbayev University)

Title: A Hamiltonian Dysthe equation for deep-water gravity waves with constant vorticity

Abstract: In this talk I present a study of the water wave problem in a two-dimensional domain of infinite depth in the presence of nonzero constant vorticity. A goal is to describe the effects of uniform shear flow on the modulation of weakly nonlinear quasi-monochromatic surface gravity waves. Starting from the Hamiltonian formulation of this problem and using techniques from Hamiltonian transformation theory, we derive a Hamiltonian Dysthe equation for the time evolution of the wave envelope. Consistent with previous studies, we observe that the uniform shear flow tends to enhance or weaken the modulational instability of Stokes waves depending on its direction and strength. Our method also provides a non-perturbative procedure to reconstruct the surface elevation from the wave envelope, based on the Birkhoff normal form transformation to eliminate all non-resonant triads. This model is tested against direct numerical simulations of the full Euler equations and against a related Dysthe equation derived in previous studies. This is a joint work with P. Guyenne and C. Sulem.

Math Club

Thursday, March 28th, 2024

Time: 5:30 p.m.  Place: Jeffery Hall, Room 118

Speaker: Francesco Cellarosi (Queen's University)

Title: Concentration of measure on the sphere

Abstract: If we pick a point uniformly at random on a unit sphere, what is the probability that we are in close to the equator?

It turns out that for high-dimensional spheres this probability is very close to 1, showing that the "surface area" is concentrated near the equator. This already counterintuitive statement turns paradoxical if we consider the fact that it must be true for every equator…

We will discuss these facts and their connection with the isoperimetric inequality.

Number Theory Seminar

Monday, March 25th, 2024

Time: 2:30 p.m.  Place: Jeffery Hall, Room 202

Speaker: Mike Roth (Queen's University)

Title: Galois groups as monodromy groups in étale cohomology

Abstract: This talk is a companion to the talk of David Nguyen earlier in the term. That talk concerned estimating the size of certain trigonometric sums, and the method was to interpret those sums as coming from étale sheaves on an open subset of P^1, and then use the weight machinery of étale cohomology.

In the talk I will try and give a simple introduction to the idea of a sheaf of locally constant sections over a curve, and related ideas in the purely topological case, and then say how those notions can be expressed in terms of representations of Galois groups in the characteristic p case. Hopefully there will be time to explain the idea of the ‘weights’ of a sheaf, and the weights of the action on cohomology. Finally, I hope to briefly discuss Grothendieck’s viewpoint of ’sheaves as functions’, and so return to the problem of estimating trigonometric sums.

None of these interpretations or constructions are new. They are all part of the beautiful synthesis of number theory and geometry that is étale cohomology, as envisioned by Grothendieck, and as developed by Grothendieck, Artin, Deligne, and collaborators in the 1960’s and 70’s.

PDEs & Applications Seminar

Tuesday, March 26th, 2024

Time: 9:30 a.m.  Place: Jeffery Hall, Room 319 (Via Zoom)

Speaker: Afroditi Talidou (University of Ottawa & Krembil Brain Institute)

Title: Influence of heterogeneous myelination patterns on axonal conduction and vulnerability to demyelination

Abstract: Axons of the mammalian brain display significant variations in their myelination motifs. Far from being regular and uniform, the distribution patterns of myelin sheaths vary significantly between axons, and across brain areas. To explore the influence of such variability on axonal conduction, we developed an axon model based on a system of PDEs, exhibiting myelin distributions mirroring those observed experimentally in different regions of the central nervous system of mice. We also examined how varying myelination patterns predispose axons to failure. Our study shows such variability significantly impacts axonal conduction timing and reliability. Action potential propagation was found to be highly sensitive to the specific arrangement and ordering of myelinated and/or exposed segments along axons, indicating that axonal conduction is non-linear and path-dependent. Furthermore, properties of axonal conduction were found to differ between cortical and callosal axons, influencing their vulnerability to demyelination, while shaping both conduction time and predisposition to failure. Our analysis indicates that callosal axons are particularly sensitive to myelin changes, especially after damage. These findings highlight the crucial role of myelination profiles in brain function and disease.