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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.

Curves Seminar

Thursday, March 21st, 2024

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

Speaker: Calvin Fletcher

Title: Further examples of finite type

Abstract: Last week we explored examples of cluster algebras of finite type, specifically type A_n and B_n. This week, we will see some settings in which type C_n and D_n arise. To conclude, we will introduce the Starfish lemma.

Math & Stats Department Colloquium

Friday, March 22, 2024

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

Speaker: Xiao-Li Meng (Harvard)

Title: Multi-resolution Meandering: Personalized Treatments, Individual Privacy, Machine Unlearning, and aWorld without Randomness

Abstract: Data science revolutionizes the granularity of human inquiries and even offers the promise of personalized assessments. However, how can we assess individual treatment effect before treating the individual? Transitional Inference addresses this dilemma through the concept of “transfer to the similar,” a notion that has been pondered by philosophers since Galen of the Roman Empire. This talk presents a Multi-Resolution Framework (Li and Meng, 2021, JASA) for transitional inference, where similarity is prescribed probabilistically by concomitantly specifying the sameness — the shared distributional form — and the differences — the individual realizations. This framework avoids the concept of randomness and defines “individual probability” as a deterministic limit with infinite resolution. These conceptualizations help us operationalize the meaning of personalized treatments, clarify what individual privacy is protected by differential privacy, and anticipate the challenges of preserving an individual’s right to be forgotten through machine unlearning. Furthermore, it reveals a world that is resistant to overfitting when the resolutions of our data and (deep) learning far exceed the resolution necessary for pattern recognition.

Bio: Xiao-Li Meng, the Founding Editor-in-Chief of Harvard Data Science Review and the Whipple V. N. Jones Professor of Statistics at Harvard University, is well known for his depth and breadth in research, his innovation and passion in pedagogy, his vision and effectiveness in administration, as well as for his engaging and entertaining style as a speaker and writer. Meng was named the best statistician under the age of 40 by the Committee of Presidents of Statistical Societies (COPSS) in 2001, and he is the recipient of numerous awards and honours for his more than 150 publications. In 2020, he was elected to the American Academy of Arts and Sciences. Meng received his BS in mathematics from Fudan University in 1982 and his PhD in statistics from Harvard in 1990. He was on the faculty of the University of Chicago from 1991 to 2001 before returning to Harvard, where he served as the Chair of the Department of Statistics (2004–2012) and the Dean of the Graduate School of Arts and Sciences (2012–2017).

Curves Seminar

Thursday, March 14th, 2024

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

Speaker: Julia McClellan

Title: Cluster Algebras and Coordinate Rings

Abstract: In this talk we will revisit our previous result that under certain conditions, the coordinate ring of an algebraic variety can be naturally identified with a cluster algebra. We will then use the familiar example of the Plücker ring to see that it can carry two non-isomorphic cluster structures of classical types – A_m and B_m.

Math Club

Thursday, March 14th, 2024

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

Speaker: James Mingo (Queen's University)

Title: Up and down and down and up

Abstract: We look at the number of ways of arranging 1, 2, 3, $\ldots$, $n$, so that the numbers go up, then down, then back up, and so on. While there isn’t a simple closed formula for this number as a function of $n$, there is a very simple way of analyzing the number and a simple connection to the Taylor series of some very familiar functions.

Number Theory Seminar

Monday, March 11th, 2024

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

Speaker: Nic Fellini (Queen's University)

Title: Diophantine equations, linear recurrences, and p-adic analysis

Abstract: In some instances, problems from Diophantine equations can be translated into problems concerning linear recurrences. A question that arises through this connection is the following: If (u_n) is a sequence solving an r-th order linear recurrence relation with integer coefficients, what does the zero set {u_n = 0} "look" like? In a surprising twist, the only known solutions to this question are p-adic in nature. The goal for this talk is to give an example driven explanation of how we can translate problems from Diophantine equations to studying the sets {u_n = a} using p-adic analysis.