9/4: organizational meeting
9/11: no meeting (no classes scheduled)
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9/18: Shuhao Li (Stony Brook University)
Title: Open-closed string topology and Lagrangian embeddings
Abstract: The Maslov class is a basic invariant of a Lagrangian submanifold. However, many fundamental questions concerning constraints on the Maslov classes of Lagrangian embeddings (known as Maslov class rigidity) remain open. For example, the existence/non-existence of a Maslov-zero Lagrangian embedding of a given closed manifold L into the standard symplectic vector space C^n is a well-known problem. I will speak about a new approach to these types of questions using a construction of a (curved) deformation of the dg algebra of chains on the based loop space, following ideas from open-closed string topology, and derive new results on the Maslov class rigidity question.
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9/25: Prof. Gueo Grantcharov (Florida International U.)
Title: On the SL(n,H) and strong HKT geometry
Abstract: Abstract: In the talk we'll review the basics of the theory of hypercomplex manifolds and the Hermitian geometry of two particular, but complementary cases - when the manifold has trivial canonical bundle (the SL(n,H) case) and when it admits a strong HKT metric. The second part is based on a joint work with B. Brienza, A. Fino, and M. Verbitsky.
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10/2: Prof. Yueqiao Wu (Johns Hopkins)
Title: Valuative aspects of complete Calabi-Yau manifolds
Abstract: Yau’s solution to the Calabi conjecture asserts that any compact Kahler manifold with vanishing first Chern class admits a unique Ricci-flat Kahler metric in the given class. Suppose now X is a smooth affine variety, which is in particular a Stein manifold, with a non-vanishing holomorphic n-form. Complete Calabi—Yau metrics on X are much less understood. In this talk, I will begin with revisiting Tian—Yau’s construction of a complete Calabi—Yau metric of Euclidean volume growth on the complement of a smooth divisor in a Fano manifold, and explain how this relates to the conjectural connection of such metrics with prescribing Calabi—Yau tangent cones at infinity using valuations. I will then explain how the study of finite generation and K-stability of these valuations suggest every Calabi—Yau tangent cone at infinity is close to a Tian—Yau type compactification of X.
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10/9: Ryan McGowan (Rutgers)
Title: Yau's Liouville Theorem in Kahler Geometry
Abstract: Yau’s Liouville theorem says, roughly, that a Ricci-flat Kähler metric on C^n that is uniformly equivalent to the Euclidean metric must itself have constant coefficients. We will compare the classical proof using C^2 and C^3 estimates with a geometric proof that avoids the C^3 calculation. We will then sketch how a Liouville theorem yields an Evans-Krylov C^{2,\alpha} estimate. The talk draws on the minicourse of Hans-Joachim Hein and Yifan Chen, at the Erdős Center–Alfréd Rényi Institute of Mathematics in Budapest.
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10/16: no meeting
10/23: Prof. Mehdi Lejmi (CUNY)
Title: Pluricanonical metric, Laplacian flows and G_2 structures.
Abstract: In this talk, we will discuss how pluricanonical metrics provide a link between conformal goemetry and spectral theory. In particular, we use pluricanonical metrics to find explicit solutions to Laplacian flows. Moreover,
we prove that a circle quotient of locally conformal parallel G_2 structure is pluricanonical. In particular, we look at a possible topological t-duality between pluricanonical and nearly Kahler structures..
This is based on two joint works: the first with Ethan Addison and Tedi Draghici and the second with Tedi Draghici.
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10/30: [POSTPONED to Spring] Prof. Sven Hirsch (Columbia University)
Title: The spacetime Geroch conjecture.
Abstract: The classical Geroch conjecture states that the only metric on the torus with nonnegative scalar curvature is the flat metric. We prove the analogous result for initial data sets subject to the dominant energy condition. This is joint with Simone Cecchini, Bernhard Hanke, Lukas Schönlinner and Rudolf Zeidler.
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11/6: Prof. Patricia Medina (CUNY)
Title: Sharp Reconstruction Bounds for Diffeomorphisms with Controlled Distortion
Abstract: Motivated by a reconstruction problem arising in dynamical system autoencoders, we investigate how geometric constraints on a diffeomorphism limit the transfer of information between complementary coordinate subspaces. The central question is whether a smooth, orientation-preserving transformation can compensate for the loss of observed coordinates while maintaining prescribed bounds on its differential.
For equal odd observed and hidden dimensions, we establish a sharp reconstruction bound under the assumption that the singular values of the differential lie in a fixed interval [m,M]. The result identifies a precise distortion threshold for exact reconstruction, with explicit affine maps attaining the bound. We also construct nonlinear examples using translated radial rotations, illustrating the distinction between global distortion constraints and reconstruction behavior on bounded sets.
The talk will emphasize the geometric formulation, the main ideas behind the sharp bound, and the role of nonlinear constructions, with the autoencoder architecture serving primarily as motivation.
This is joint work with Hy P. G. Lam.
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11/13: Prof. Tang-Kai Li (Columbia University)
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11/20: Prof. Sadok Kallel (Universite de Lille, France, and American University of Sharjah, UAE)
Title: TBA
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11/27: no meeting this week
12/4: Prof. Shengzhen Ning (University of Maryland)
Title: TBA
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12/11: Prof. Jesse Madnick (Seton Hall University)
Title: TBA
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