Informal Geometric Analysis Seminar

University of Maryland

PREVIOUS YEARS: 2012-2013, 2013-2014, 2014-2015, 2015-2016, 2016-2017, 2017-2018.

- September 7, 2 PM, MATH 2300 (Friday, NOTE SPECIAL DATE AND TIME!)

Paolo Piccione (Sao Paolo)

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

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). - October 16, SPECIAL PLACE AND TIME: 4:30 PM in Math 3206

Dan Coman (Syracuse)

*Title: Universality results for zeros of random holomorphic sections*

Abstract: Consider a sequence of singular Hermitian holomorphic line bundles on a compact Kaehler manifold. We prove a universality result which shows that the asymptotic distribution of zeros of random holomorphic sections is independent of the choice of probability measure on the space of holomorphic sections. We give several applications of this result, in particular to the distribution of zeros of random polynomials. The results are joint with T. Bayraktar and G. Marinescu. - October 23, 4:30 PM, MATH 3206 (special time and place)

Norman Levenberg (Indiana)

*Title: Pluripotential theory and convex bodies*

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. - November 13, 4 PM (special time)

Bo Berndtsson (Chalmers)

*Title: TBA*

Abstract: TBA. - December 4,
Kuang-Ru Wu (Purdue)

*Title: A Dirichlet problem for flat hermitian metrics*

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. - December 10, 2 PM, MATH 3206 (Monday, NOTE SPECIAL DATE,TIME and LOCATION!)

Yuchen Liu (Yale)

*Title: Openness of uniform K-stability in families of Q-Fano varieties*

Abstract: K-stability is the algebraic notion which is supposed to characterize whether a Fano variety admits a K\"ahler-Einstein metric. One important feature of the notion of K-stability is that it is supposed to give a nicely behaved moduli space. To construct the K-moduli space of Q-Fano varieties as an algebraic space, one important step is to prove the openness of K-(semi)stable locus in families. In this talk, I will explain the proof of openness of uniform K-stability in families of Q-Fano varieties. This is achieved via showing the lower semi-continuity of delta-invariant, an interesting invariant introduced by Fujita and Odaka as a variant of Tian's alpha-invariant. This is a joint work with Harold Blum. - February 5, 12:30-2 pm, SPECIAL TIME

Yuxiang Ji (UMD)

*Title: Logarithmic convexity of push-forward measures (after Graham)*

Abstract: TBA. - February 12, 12:30-2 pm, SPECIAL TIME

Mirna Pinski (UMD)

*Title: The heat kernel on a cone*

Abstract: Starting with the known heat kernel for the Laplacian in one dimension, using a parametrix construction, we determine the asymptotics of the heat kernel for the operator $H_\kappa = -\partial_r^2 + \frac{\kappa(r)}{r^2}$ on the manifold with a conic singularity. We discuss the construction in the case of dimension one. (following work by E. Mooers). - February 19,

Julius Ross (UIC)

*Title: Dualities between Complex PDEs and Planar Flows*

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. - March 5, 12:30-2 pm, SPECIAL TIME

Mirna Pinski (UMD)

*Title: The heat kernel on a cone II*

Abstract: Starting with the known heat kernel for the Laplacian in one dimension, using a parametrix construction, we determine the asymptotics of the heat kernel for the operator $H_\kappa = -\partial_r^2 + \frac{\kappa(r)}{r^2}$ on the manifold with a conic singularity. We discuss the construction in the case of dimension one. (following work by E. Mooers). - March 12, 3:30 PM, MATH 1308 (SPECIAL TIME AND PLACE)

Homare Tadano (Tokyo University of Science)

*Title: Myers type theorems for solitons*

Abstract: TBA. - March 19, Spring break.
- April 2, SPECIAL TIME AND PLACE: 4:30 PM, MATH 3206

Tristan Collins (MIT)

*Title: Stability and Nonlinear PDE in mirror symmetry*

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. - April 22, 2:00 PM, joint with G & T seminar, MATH 3206

Ziquan Zhuang (Princeton University)

*Title: Sharpness of Tian's criterion of K-stability*

Abstract: K-stability is an algebraic notion that captures the existence of K\"ahler-Einstein metric on Fano varieties. A famous criterion of Tian states that a Fano variety of dimension n whose alpha invariant is greater than n/(n+1) is K-stable. In this talk, I will discuss some recent refinement of this criterion by K. Fujita and present the construction of some (singular) Fano varieties with alpha invariant n/(n+1) that is not K-stable to show that Tian's criterion is sharp. This is joint work with Yuchen Liu.

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