Informal Geometric Analysis Archives for Academic Year 2019

Multiple solutions for the van der Waals-Allen-Cahn-Hilliard equation

When: Fri, September 7, 2018 - 2:00pm
Where: MATH 2300
Speaker: Paolo Piccione (Sao Paolo) -
Abstract: The classical Allen-Cahn equation gives a bridge between the
theory of phase transition and the theory of minimal surfaces. In this talk
I will discuss the existence of multiple solutions for a suitable variant
of this equation, satisfying a volume constraint. This aims naturally at an
existence theory for constant mean curvature hypersurfaces. Joint work
with Vieri Benci (Pisa) and Stefano Nardulli (UFABC).

Pluripotential theory and convex bodies

When: Tue, October 23, 2018 - 4:30pm
Where: Kirwan Hall 3206
Speaker: Norman Levenberg (Indiana University) -
Abstract: Given a convex body P one can associate a natural class of plurisubharmonic functions: those that grow like the logarithmic indicator function. These generalizations of Lelong classes in standard pluripotential theory arise in the theory of random sparse polynomials and in problems involving polynomial approximation. We give some examples of extremal plurisubharmonic functions in this setting and discuss other results in the general theory as well as connections with complex geometry.

A Dirichlet problem for flat hermitian metrics

When: Tue, December 4, 2018 - 4:30pm
Where: Math 2300
Speaker: Kuang-Ru Wu (Purdue) -
Abstract: Let Omega be a compact Riemann surface with boundary, and V a Hilbert space. We prove the existence of flat hermitian metrics on Omega x V with given boundary values. The result generalizes Lempert's theorem that had Omega be the unit disc. It also generalizes results of Donaldson and Coifman-Semmes to the case of infinite rank bundles but only on Riemann surfaces.

The heat kernel on a cone

When: Tue, February 12, 2019 - 12:00pm
Where: Math 2300
Speaker: Mirna Pinsky (UMD) -

Dualities between Complex PDEs and Planar Flows

When: Tue, February 19, 2019 - 4:00pm
Where: MATH 2300
Speaker: Julius Ross (UIC (Chicago)) -
Abstract: I will describe a surprising duality between a case of the Dirichlet problem for the Complex Homogeneous Monge-Ampere Equation and a planar flow coming from fluid mechanics called the Hele-Shaw flow. Using this we are able to prove new things about both this PDE and renowned flow. I will present this in a way that suggests that it is a special case of something much more general, and end with a discussion as to what this may be. All of this is work with David Witt-Nystrom.

The heat kernel on a cone II

When: Tue, March 5, 2019 - 12:30pm
Where: Math 2300
Speaker: Mirna Pinsky (UMD) -

Myers type theorems for solitons

When: Tue, March 12, 2019 - 3:30pm
Where: 1308.0
Speaker: Homare Tadano (Tokyo University of Science) -

Stability and Nonlinear PDE in mirror symmetry

When: Tue, April 2, 2019 - 4:30pm
Where: Kirwan Hall 3206
Speaker: Tristan Collins (MIT) -
Abstract: A longstanding problem in mirror symmetry has been to understand the relationship between the existence of solutions to certain geometric nonlinear PDES (the special Lagrangian equation, and the deformed Hermitian-Yang-Mills equation) and algebraic notions of stability, mainly in the sense of Bridgeland. I will discuss progress in this direction through ideas originating in infinite dimensional GIT. This is joint work with S.-T. Yau.

Remarks on a paper of Richard Hamilton

When: Tue, April 30, 2019 - 4:15pm
Where: Math 2300
Speaker: Yanir Rubinstein (UMD) -
Abstract: In 1988 Richard Hamilton studied the Ricci flow on the 2-sphere and discovered the non-compact cigar soliton that Pereleman has called "an important example" in his celebrated work. In that same paper Hamilton also discovered compact singular solitons called teardrop solitons and these have been inspirational in their own right in the literature of conical Riemann surfaces. Together with K. Zhang we show that rather surprisingly the two constructions are related: the former is the blow-up limit of the latter in a sense.