Organizers: Ioakeim Ampatzoglou, Dan Ginsberg, Weilin Li, Vincent Martinez, and Azita Mayeli
Time: Friday, 2:00-4:00pm EST
Location (in-person and hybrid): GC 9116
Zoom: link Passcode: HAPDE2024
Current Schedule (Fall 2025)
August 29
(First Week of classes) No Seminar
September 5
No Seminar
September 12
Vincent Martinez, Hunter College & GC (Department of Mathematics)
Title: The Secret Life of Mathematical Fluids
Abstract: Fluids influence our lives in a multitude of ways, ranging from the mundane (when we stir milk into our coffee) to the spectacular (the formation of galaxies). It is a great achievement of the human intellect that we are able to study such phenomenon abstractly through mathematics. This talk will present some ways for how fluids can be studied mathematically and introduce a few interesting problems, both theoretical and practical, of ongoing scientific relevance, for prospective PhD students.
September 19 2:30 pm
Zoe Wyatt, University of Cambridge (Department of Pure Mathematics and Mathematical Statistics)
Title: A new phase transition in cosmological fluid dynamics
Abstract: On a background Minkowski spacetime, the Euler equations (both relativistic and not) are known to admit unstable homogeneous solutions with finite-time shock formation. Such shock formation can be suppressed on cosmological spacetimes whose spatial slices expand at an accelerated rate. However, situations with decelerated expansion, which are relevant in our early universe, are not as well understood. I will present some recent joint work in this direction, based on collaborations with David Fajman, Maciej Maliborski, Todd Oliynyk and Max Ofner.
(Note the change in time)
September 26
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October 3
Ely Sandine, UC Berkeley (Mathematics)
Title: TBD
Abstract: TBD
October 10
Luke Evans, Flatiron Institute (Center for Computational Mathematics)
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October 17
Allison Byars, University of Wisconsin–Madison (Mathematics)
Title: Global dynamics for the derivative nonlinear Schrödinger equation
Abstract: We will discuss the long-time dynamics of the derivative nonlinear Schrödinger equation. For small, localized initial data, where no solitons arise, we prove dispersive estimates globally in time. Under the same assumptions, we further prove modified scattering and asymptotic completeness. To the best of our knowledge, this is the first result to achieve an asymptotic completeness theory in a quasilinear setting. Our approach combines the method of testing by wave packets of Ifrim and Tataru, a bootstrap argument, and the Klainerman–Sobolev vector field method.
October 24
(Monday Schedule) No Seminar
October 31
Hung-Hsu Chou, University of Pittsburgh (Mathematics)
Title: Neural Networks’ Implicit Bias Towards Sparse Solutions
Abstract: Despite their recent successes, most modern machine learning algorithms lack theoretical guarantees, which are crucial to further development towards delicate tasks. One mysterious phenomenon is that, among infinitely many possible ways to fit data, the algorithms often find the "good" ones, even when the definition of "good" is not specified by the designers. In this talk I will approach this from both the microscopic view and the macroscopic view, with empirical and theoretical study of the connection between the good solutions in neural networks and the sparse solutions in compressed sensing. The key concepts are the implicit bias/regularization in Bregman divergence of ODEs, and the neural collapse phenomenon justified by the block structure of neural tangent kernel, which can be used for out-of-distribution detection.
November 7
Bob Strain, University of Pennsylvania (Mathematics)
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November 14
Leonardo Abbrescia, Georgia Institute of Technology (Mathematics)
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November 21
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November 28
(Thanksgiving Break) No Seminar
December 5
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December 12
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