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The colloquium typically meets Mondays at 4:00 PM in Room 617 on the sixth floor of Wachman Hall.
The colloquium is preceded by tea starting at 3:30 in the Faculty Lounge, adjacent to Room 617. Click on title for abstract.
Andre Guerra, Institute for Advanced Study
Quasiconvexity is a fundamental notion in the vectorial Calculus of Variations and is essentially equivalent to the applicability of the Direct Method. A fundamental problem, considered by Morrey in the 50s and 60s, is whether quasiconvexity is equivalent to ellipticity (in the sense of Legendre-Hadamard). In 1992 Vladimir Sverak showed that in 3 or higher dimensions they are not equivalent, but the two-dimensional case remains open. In this case one can expect a "complex analysis miracle", and we will discuss deep connections of Morrey's problem to old questions in Quasiconformal Analysis.
Elliptic curves are fundamental and well-studied objects in arithmetic geometry. However, much is still not known about many basic properties, such as the number of rational points on a "random" elliptic curve. We will discuss some conjectures and theorems about this "arithmetic statistics" problem, and then show how they can be applied to answer a related question about the number of integral points on elliptic curves over Q. In particular, we show that the second moment (and the average) for the number of integral points on elliptic curves over any number field is bounded (joint work with Levent Alpöge).
In this talk, I will survey what is known or conjectured about maps between configuration spaces and moduli spaces of surfaces. Among them, we consider two categories: continuous maps and also holomorphic maps. We will also talk about some results and conjectures about maps between finite covers of those spaces. Unlike the case of the whole spaces, much less is known about their finite covers.
In the middle of the nineteenth century, Kummer observed striking congruences between certain values of the Riemann zeta function, which have important consequences in number theory. In spite of its potential, this topic lay mostly dormant for nearly a century until breakthroughs by Iwasawa in the middle of the twentieth century. Since then, advances in arithmetic geometry and number theory (in particular, for modular forms, certain analytic functions that play a central role in number theory) have revealed similarly consequential congruences in the context of other arithmetic data. This remains an active area of research. In this talk, I will survey old and new tools for studying such congruences. I will conclude with some unexpected challenges that arise when one tries to take what would seem like immediate next steps beyond the current state of the art.
We use relaxation as in Nash’s work, but replace his iteration (in low codimension) or continuous flow (in high codimension) with a stochastic flow. The main issue in the derivation of our flow is a principled resolution of a semidefinite program. The same fundamental structure applies to several hard constraint systems and nonlinear PDE.
2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022