Graduate Student Math Conference

Brown University

Tuesday, February 23rd 2016

Titles & Abstracts

10:00 - 10:50

Chiara Gallarati

TU Delft

Maximal regularity for systems of non-autonomous evolution equations

In this talk I will explain a new approach to maximal $L^p$-regularity for parabolic PDEs with time dependent generator $A(t)$. The novelty is that I merely assume a measurable dependence on time. I will firstly show that there is an abstract operator theoretic condition on $A(t)$ which is sufficient to obtain maximal $L^p$-regularity. As an application I will obtain optimal $L^p(L^q)$ estimates, for every $p,q\in\(1,\infty)$, for systems of non-autonomous differential equations of order 2m, with coefficients that depends only measurable on time. This is a joint work with Mark Veraar.

11:00 - 11:50

Yang-Mills Replacement

Harmonic replacement is a technique for reducing the energy of a map $u\colon\Sigma^2\to M$ by replacing it on a small ball $B^2$ with a harmonic map $v\colon B^2\to M$ with the same values on the boundary as $u$. Harmonic replacement has proven to have a wide range of applications, such as the Perron method for constructing global harmonic functions, and, more recently, Colding and Minicozzi's proof of the finite extinction of Ricci flow on homotopy $3$-spheres. We develop an analogous technique for Yang-Mills connections on $4$-manifolds, where we replace a connection $A$ on a small $4$-ball $B^4$ with a Yang-Mills connection $B$ that has the same restriction to the boundary as $A$, thereby decreasing the energy. In both settings, the maps $u$ and connections $A$ are assumed to be $L^2_1$, not necessarily continuous, leading to subtleties involving borderline Sobolev spaces. It is hoped that this Yang-Mills replacement technique could be used to simplify the proofs in Taubes's work on the stable topology of the moduli spaces of anti-self-dual connections, as well as to provide a simpler alternative to Yang-Mills gradient flow in certain applications.

12:00 - 1:00


Lunch will be provided for those who attend.

1:00- 1:50

log-log blow up solutions of the NLS at exactly m points

We will first introduce the log-log blow up solutions to NLS which have been studied by Merle and Raphael and many other authors. Then, we will illustrate our construction about certain solutions to NLS, which blow up at $m$ prescribed points according to log-log law. The idea is to use bootstrap to show the $m$ bubbles can be decoupled, and then use soft topological argument to balance different bubbles, making them blow up simultaneously at the prescribed points.

2:00 - 2:50

Xin Sun


Random planar maps and Liouville quantum gravity

We begin with a well known bijection between planar trees and walks on $\Z_{\ge 0}$. Then we present a bijection between spanning tree decorated planar maps and walks on $\Z^2_{\ge 0}$. Both bijections can be understood as quotient spaces. Thanks to that random walk converges to Brownian motion, both bijections have continuum counterpart. In particular, the 2D bijection amounts to gluing two continuum trees into a topological sphere with a space filling curve. This construction gives us access to the fascinating world of Liouville quantum gravity and 2D random geometry. No background is required other than undergraduate probability.

3:00 - 3:30


3:30 - 4:20

Haitian Yue


Nondeterministic well-posedness for the periodic 4D cubic NLS


4:30 - 5:20

Simone Cecchini


Spectral theory of von Neumann algebra valued differential operators over non-compact manifolds

We provide criteria for self­adjointness and τ­-Fredholmness of first and second order differential operators acting on sections of infinite dimensional bundles, whose fibers are modules of finite type over a von Neumann algebra A endowed with a trace τ. We extend the Callias­-type index to operators acting on sections of such bundles and show that this index is stable under compact perturbations. (Joint work with Maxim Braverman).

Organized by Ken Ascher, Dori Bejleri, Mamikon Gulian, and Laura Walton at the Brown Math Department

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