Informal Geometric Analysis Archives for Fall 2020 to Spring 2021
Maslov Class of a Lagrangian Immersion
When: Tue, September 3, 2019 - 4:00pmWhere: 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 - 4:00pmWhere: 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 - 4:00pmWhere: Kirwan Hall 2300
Speaker: Mirna Pinski (UMD) - https://stat.umd.edu/people/all-directory/item/1338-mpinsky1.html
Abstract: Lagrangian mean curvature flow equation and some generalizations.
Schauder Estimates for Conical Laplace Equations
When: Tue, October 15, 2019 - 3:30pmWhere: 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 - 3:30pmWhere: Kirwan Hall 2300
Speaker: Nick McCleerey (Northwestern) - https://sites.math.northwestern.edu/~njm2/
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 - 3:30pmWhere: Kirwan Hall 2300
Speaker: Jakob Hultgren (UMD) - http://math.umd.edu/~hultgren/
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 - 3:30pmWhere: 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 - 3:30pmWhere: 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 - 3:30pmWhere: 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 - 3:30pmWhere: Kirwan Hall 2300
Speaker: Sanal Shivaprasad (University of Michigan) - http://www-personal.umich.edu/~sanal/
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.
Ambrose and Calabi Type Theorems via m--Bakry--Émery Ricci Curvature
When: Tue, March 10, 2020 - 3:30pmWhere: 2300.0
Speaker: Tadano Homare (Tokyo University of Science) - https://www.tus.ac.jp/en/fac/p/index.php?6e37
Abstract: I will introduce some Ambrose and Calabi type compactness criteria for complete Riemannian manifolds via $m$--Bakry--\'{E}mery Ricci curvature with positive and negative $m$. Our theorems generalize Myers and Ambrose type compactness criteria due to M. Fern\'{a}ndez-L\'{o}pez and E. Garc\'{i}a-R\'{i}o, M. Limoncu, H. Tadano, and J.-Y Wu when $m > 0$, as well as improve Myers type compactness criterion due to W. Wylie when $m < 0$. The key ingredients in proving our results are the Bochner formula via Witten--Laplacian and the Riccati comparison theorem due to P. Mastrolia, M. Rimoldi, and G. Veronelli.
Problems related to the Chern scalar curvature
When: Tue, March 17, 2020 - 3:30pmWhere: MATH 2300
Speaker: Mehdi Lejmi (CUNY) - https://fsw01.bcc.cuny.edu/mehdi.lejmi/
Abstract: On an almost-Hermitian manifold, the Chern connection is the unique Hermitian connection with J-anti-invariant torsion. In this talk, we compare the Chern scalar curvature to the Riemannian scalar curvature. Moreover, we study problems related to to the Chern scalar curvature like the analog of the Yamabe problem or the critical metrics of the total Chern scalar curvature in a conformal class. This is a joint work with Caner Koca.