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Xiaoyu Huang, Temple University
We will begin by discussing local fields and valuations, followed by an exploration of inertia groups and Frobenius elements. Next, we will cover unramified extensions and then connect these concepts to the global field setting.
Susanna Haziot, Princeton University
The Peskin problem describes the flow of a Stokes fluid through the heart valves. We begin by presenting the simpler 2D model and investigate its small data critical regularity theory, with initial data possibly containing small corners. We then present the 3D problem and describe the challenges that arise to proving global well-posedness.
The first part is joint work with Eduardo Garcia-Juarez, and the second is on-going work with Eduardo Garcia-Juarez and Yoichiro Mori.
Ari Shnidman, IAS
An algebraic curve C of genus g > 1 can be embedded in its Jacobian variety J in two different ways: {x - e : x in C} and {e - x : x in C}. The Ceresa cycle k(C) is the formal difference of these two curves-- it vanishes if and only if these two curves can be algebraically deformed to each other within J. To date, the only known cases of k(C) = 0 are hyperelliptic curves. However, many examples of k(C) being torsion have recently been found, among curves/Jacobians with symmetries. I'll survey these recent examples, with an emphasis on what number theory has to say about this (a priori) purely algebro-geometric question.
Narek Hovsepyan, Rutgers University
Second Harmonic Generation (SHG) is a process inwhich the input wave (e.g. laser beam) interacts with a nonlinearmedium and generates a new wave, called the second harmonic, atdouble the frequency of the original input wave. Weinvestigate whether there are situations in which the generatedsecond harmonic wave does not scatter and is localizedinside the medium, i.e., the nonlinearinteraction of the medium with the probing wave isinvisible to an outside observer. This leadsto the analysis of a semilinear elliptic system formulatedinside the medium with non-standard boundary conditions. Moregenerally, we set up a mathematical framework needed to investigate amultitude of questions related to the nonlinear scatteringproblem associated with SHG (or other similar multi-frequency opticalphenomena). This is based on a joint work with Fioralba Cakoni, MattiLassas and Michael Vogelius.
There are no conferences next week.