2/20: no meeting this week
2/27: Prof. Manuel Rivera (Purdue U.)
Title: String topology via the coHochschild complex
Abstract: I will describe a tractable chain-level model for the free loop space of a simplicial complex. The construction is based on coHochschild homology theory for coalgebras and does not assume any restrictions on the fundamental group or the commutative ring of coefficients.
I will then describe a way of lifting Poincaré duality to the chain level by adapting the formalism of Pre-Calabi Yau structures (developed independently by Tradler/Zeinalian and Kontsevich/Takeda/Vlassopoulos) to our setting.
Combining the two ingredientes above we produce algebraic formulae for string topology of (possibly non-simply connected) homology manifolds. From these formulae one can observe that assuming locality conditions on the chain-level lift of Poincare duality are necessary to recover the loop coproduct, an operation that can distinguish homotopy equivalent non-homeomorphic manifolds. This is joint work with Alex Takeda.
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3/6: Prof. David Pham (CUNY)
Title: Twists, Codazzi Tensors, and the 6-sphere
Abstract: Starting with an almost Hermitian manifold (M,g,J,\omega) and an automorphism \psi of TM, one can "twist" the existing structure to obtain a new almost Hermitian structure in a rather natural way. In this talk, we will examine these \psi-twisted almost Hermitian structures with particular emphasis on questions regarding the integrability of the twisted almost complex structure J^\psi and the Riemannian geometry of the twisted metric g^\psi. By studying the latter, we identity a certain class of automorphisms of the tangent bundle with nice transformation properties. We call these automorphisms g-Codazzi maps because of their close relationship with Codazzi tensors. The aforementioned results are ultimately applied to the standard nearly Kahler structure on the 6-sphere where we prove a nonintegrability result for the class of g-Codazzi maps.
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3/13: Prof. Junsheng Zhang (NYU)
Title: Gromov-Hausdroff limits of immortal Kahler-Ricci flows
Abstract: We show that the normalized Kahler-Ricci flow on a compact Kahler manifold with semiample canonical bundle converges in the Gromov-Hausdorff topology to the metric completion of the twisted Kahler-Einstein metric on the canonical model, as conjectured by Song-Tian. This is based on joint work with Man-Chun Lee and Valentino Tosatti.
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3/20: no classes at CUNY (no meeting)
3/27: Prof. Bin Guo (Rutgers)
Title: Geometric estimates on Kahler spaces
Abstract: We will discuss the role of complex Monge-Ampere equations as auxiliary equations in deriving sharp analytic and geometric estimates in Kahler geometry. By studying Green's functions, we will explore how to derive estimates for diameters and establish uniform Sobolev inequalities on Kahler manifolds, which depend only on the entropy of the volume form and are independent of the lower bound of the Ricci curvature.
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4/3: no meeting (Spring break)
4/10: no meeeting
4/17: Prof. Ralph Gomez (Swarthmore College)
Title: Transformations in Sasaki-Einstein Geometry
Abstract: In this talk we will spend some time reviewing a construction introduced by C.Boyer and K. Galicki in constructing Sasaki-Einstein manifolds. We will then give a new perspective on this construction by implementing the Berglund-H\"ubsch transpose from BHK mirror symmetry. Once we have discussed this, we will see how this sheds light on so-called "twins" in Sasaki-Einstein rational homology 7-spheres.
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4/24: Prof. Christoforos Neofytidis (University of Cyprus)
Title: A topological interpretation of numbers
Abstract: I will explain how an arbitrary finite set of numbers containing zero can be understood with topology, namely, as the set of degrees of maps between two closed manifolds in any dimension greater than two.
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5/1: Prof. Max Hallgren (Rutgers)
Title: Expanding soliton models for Kahler-Ricci flow near conical singularities
Abstract: Hamilton proved that every compact Riemannian manifold admits a unique short-time solution to the Ricci flow. Since then, much work has been devoted to extending this theory to singular initial data and to understanding the geometry of the resulting flow near singular points. In this talk, I will discuss recent joint work with Longteng Chen and Lucas Lavoyer on Ricci flows whose initial data is a Kahler space with isolated conical singularities. I will explain how expanding solitons arise as models for the flow near the singular set, and I will also discuss the relationship between our solutions and the Kahler–Ricci flow through singularities constructed by Song and Tian.
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5/8: Prof. Mingyang Li (Simons Center)
Title: Poincare Einstein metrics with complex geometry and cusp formations.
Abstract: Poincare-Einstein metrics are a natural class of complete Einstein metrics with negative Einstein constant, though their construction is often difficult. In this talk, I will discuss four-dimensional Poincare-Einstein metrics with complex geometry. In this setting, the Einstein equation reduces to a Toda type nonlinear elliptic equation, yielding an infinite-dimensional, nonperturbative construction of Poincare-Einstein metrics. I will then explain how the same framework extends to metrics with various cusp ends and how it can be used to study degenerations of Poincare-Einstein metrics. This provides a unified analytic perspective on several noncompact Einstein geometries through elliptic boundary value problems. Joint work with Hongyi Liu.
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5/15: Prof. Robert Bryant (Duke U.)
Title: The affine Bonnet problem
Abstract:
The Bonnet problem in Euclidean surface theory is well-known: Given a metric \(g\) on an oriented surface \(M^2\) and a function \(H\), when (and in how many ways) can \((M,g)\) be isometrically immersed in \(\mathbb{R}^3\) with mean curvature \(H\)? For generic data \((g,H)\), such an isometric immersion is impossible and, in the case that it does exist, the immersion is unique. Bonnet showed that, aside from the famous case of surfaces of constant mean curvature, there is a finite dimensional moduli space of \((g,H)\) for which the space of such Bonnet immersions has positive dimension.
The corresponding problem in affine theory is still not solved. After reviewing the Euclidean results on this problem by O. Bonnet, J. Radon, É. Cartan, and A. Bobenko, I will give a report on affine analogs of these results. In particular, I will consider the classification of the data \((g,H)\) for which the space of affine Bonnet immersions has positive dimension, showing a surprising connection with integrable systems in the case of data with the highest possible dimension of solutions.
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