Department of

Mathematics


Seminar Calendar
for events the day of Thursday, January 30, 2020.

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Thursday, January 30, 2020

11:00 am in 241 Altgeld Hall,Thursday, January 30, 2020

Modularity of some $\mathrm{PGL}_2(\mathbb{F}_5)$ representations

Patrick Allen (Illinois)

Abstract: Serre's conjecture, proved by Khare and Wintenberger, states that every odd two dimensional mod p representation of the absolute Galois group of the rationals comes from a modular form. This admits a natural generalization to totally real fields, but even the real quadratic case seems completely out of reach. I'll discuss some of the difficulties one encounters and then discuss some new cases that can be proved when p = 5. This is joint work with Chandrashekhar Khare and Jack Thorne.

2:00 pm in 347 Altgeld Hall,Thursday, January 30, 2020

The Semicircle Law for Wigner Matrices

Kesav Krishnan (UIUC Math)

Abstract: I will introduce Wigner Matrices and their universal properties. I will then state the semi-circle law and sketch out three district proofs, in analogy to the proof of the usual central limit theorem. Talk 1 will sketch out the proof via the Stieltjes transform and via the energy entropy balance.

4:00 pm in 245 Altgeld Hall,Thursday, January 30, 2020

Local and global boundary rigidity

Plamen Stefanov   [email] (Purdue University)

Abstract: Abstract: The boundary rigidity problem consist of recovering a Riemannian metric in a domain, up to an isometry, from the distance between boundary points. We show that in dimensions three and higher, knowing the distance near a fixed strictly convex boundary point allows us to reconstruct the metric inside the domain near that point, and that this reconstruction is stable. We also prove semi-global and global results under certain an assumption of the existence of a strictly convex foliation. The problem can be reformulated as a recovery of the metric from the arrival times of waves between boundary points; which is known as travel-time tomography. The interest in this problem is motivated by imaging problems in seismology: to recover the sub-surface structure of the Earth given travel-times from the propagation of seismic waves. In oil exploration, the seismic signals are man-made and the problem is local in nature. In particular, we can recover locally the compressional and the shear wave speeds for the elastic Earth model, given local information. The talk is based on a joint work with G.Uhlmann (UW) and A.Vasy (Stanford). We will also present results for a recovery of a Lorentzian metric from red shifts motivated by the problem of observing cosmic strings. The methods are based on Melrose’s scattering calculus in particular but we will try to make the exposition accessible to a wider audience without going deep into the technicalities.