Informal Geometric Analysis Archives for Academic Year 2020

Maslov Class of a Lagrangian Immersion

When: Tue, September 3, 2019 - 4:00pm
Where: Kirwan Hall 2300
Speaker: Mirna Pinsky (UMD) -
Abstract: I will present the paper by Jean-Marie Morvan in which he shows the relationship between the Maslov class and the mean curvature vector field of a Lagrangian submanifold of the Euclidean space.

Solutions to the first-order complex-valued ODE in finding the BPS states

When: Tue, September 17, 2019 - 4:00pm
Where: Kirwan Hall 2300
Speaker: Yuxiang Ji (UMD) -
Abstract: Let P(z) be a complex polynomial with no multiple root and z(t) a path beginning and ending at roots of P. We study the solutions to the first-order ODE sqrt(P(z)) dz/dt=alpha, where t is a real parameter and alpha is a phase. This ODE helps find the BPS states in Physics. Moreover, this equation is useful in studying the one-dimensional initial value problem for the special Lagrangian equation in complex geometry.

Lagrangian mean curvature flow

When: Thu, October 3, 2019 - 4:00pm
Where: Kirwan Hall 2300
Speaker: Mirna Pinski (UMD) -
Abstract: Lagrangian mean curvature flow equation and some generalizations.

Schauder Estimates for Conical Laplace Equations

When: Tue, October 15, 2019 - 3:30pm
Where: MATH2300
Speaker: Yuxiang Ji (UMD) -
Abstract: Let g_\beta be the standard conical Kaehler metric on C^n for some \beta in (0, 1), we consider the conical Laplace equation with the background metric g_\beta: \Delta_\beta u=f in the unit geodesic ball. I will introduce some results on the estimates for the H\"{o}lder continuity of the second derivatives of the solution u.Then as a corollary, a sharp Schauder estimate for the Laplace equation is obtained.

C^{1,1} Regularity of Geodesics of Singular Kahler Metrics

When: Tue, October 22, 2019 - 3:30pm
Where: Kirwan Hall 2300
Speaker: Nick McCleerey (Northwestern) -
Abstract: I'll talk about some recent joint work with Jianchun Chu concerning regularity of geodesics of plurisubharmonic functions, in particular when the boundary functions are assumed to have ``strong" singularities. I'll then show how this can be used to study some geodesics in nef and big classes.

The Gromov-Wasserstein Problem for Vector Bundle Automorphisms

When: Tue, November 19, 2019 - 3:30pm
Where: Kirwan Hall 2300
Speaker: Jakob Hultgren (UMD) -
Abstract: By elementary linear algebra, any complex matrix in the special linear group can be factored into a product of elementary matrixes, i.e. matrixes with ones on the diagonal and no more than one non-zero element outside the diagonal. The corresponding factorisation problem for SLn valued holomorphic functions on Stein manifolds is called the Gromov-Wasserstein problem and was solved by Ivarsson and Kutzschebauch in 2008. In this talk I will adress a 'vector bundle analog' of this problem. In particular, I will provide a theorem ruling out topological obstructions. This is joint work with Erlend F Wold at University of Oslo.

Schauder estimates for linear conical elliptic and parabolic equations

When: Thu, November 21, 2019 - 3:30pm
Where: Kirwan Hall 2300
Speaker: Yuxiang Ji (UMD) -
Abstract: I will present a paper by Bin Guo and Jian Song in which they derive interior Schauder estimates for linear elliptic and parabolic equations with background Kaehler metric of conical singularities along a divisor of simple normal crossings.

Milnor fibres and Lagrangian almost mean curvature flow

When: Tue, December 3, 2019 - 3:30pm
Where: Kirwan Hall 2300
Speaker: Mirna Pinsky (UMD) -
Abstract: I will present some details of the proof from the paper:

Special Lagrangians, stable bundles and mean curvature flow by R. P. Thomas and S.-T. Yau

Introduction to non-Archimedean methods in the study of canonical metrics

When: Tue, January 28, 2020 - 3:30pm
Where: Kirwan 2300
Speaker: Mingchen Xia (Chalmers ) -
Abstract: According to the celebrated Chen-Cheng papers, the existence of cscK metrics on a polarized Kahler manifold is characterized by a stability condition formulated in terms of geodesic rays in the space of Kahler potentials. Within the space of geodesic rays, a small algebraic portion, which can be identified with a space of non-Archimedean potentials, are believed to be enough to define a strong enough stability condition that implies the existence of cscK metric. These lead to the attempt of applying non-Archimedean methods into the study of Kahler geometry. In this talk, I will introduce results about canonical metrics obtained so far by non-Archimedean methods.

Convergence of Bergman measure

When: Tue, February 25, 2020 - 3:30pm
Where: Kirwan Hall 2300
Speaker: Sanal Shivaprasad (University of Michigan) -
Abstract: We consider certain degenerating families of complex manifolds, each carrying a canonical measure (for example, the Bergman measure on a compact Riemann surface of genus at least one). We show that the measure converges, in a suitable sense, to a measure on a non-Archimedean space, in the sense of Berkovich. No knowledge of non-Archimedean geometry will be assumed.