Informal Geometric Analysis Seminar
University of Maryland

• September 3
  Mirna Pinsky (UMD)
  Title: Maslov Class of a Lagrangian Immersion
  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.


• September 17
  Yuxiang Ji (UMD)
  Title: Solutions to the first-order complex-valued ODE in finding the BPS states
  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.


• October 3 (Thursday!)
  Mirna Pinsky (UMD)
  Title: Lagrangian mean curvature flow
  Abstract: Lagrangian mean curvature flow equation and some generalizations.


• October 15
  Yuxiang Ji (UMD)
  Title: Schauder Estimates for Conical Laplace Equations
  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 Holder continuity of the second derivatives of the solution u.Then as a corollary, a sharp Schauder estimate for the Laplace equation is obtained.


• October 22
  Nicholas McCleerey (Northwestern)
  Title: TBA
  Abstract: TBA


• November 19
  Jakob Hultgren (UMD)
  Title: The Gromov-Wasserstein Problem for Vector Bundle Automorphisms
  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.


• November 21, 3:30 PM
  Yuxiang Ji (UMD)
  Title: TBA
  Abstract: TBA


• November 26
  Mirna Pinsky (UMD)
  Title: TBA
  Abstract: TBA


• January 28
  Mingchen Xia (Chalmers)
  Title: Introduction to non-Archimedean methods in the study of canonical metrics
  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.


• February 25
  Sanal Shivaprasad (Michigan)
  Title: Convergence of Bergman measure
  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.


• March 10
  Homare Tadano (Tokyo)
  Title: Ambrose and Calabi Type Theorems via $m$--Bakry--\'{E}mery Ricci Curvature
  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.


• March 17
  Mehdi Lejmi (CUNY) CANCELLED
  Title: Problems related to the Chern scalar curvature
  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.



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