# Real planar curves - algebraic, geometric, and topological aspects

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The study of real planar curves dates back to antiquity, where the ancient Greeks studied curves defined on the plane cut out by polynomials of two variables. We’ll provide a friendly overview to beautiful formulas of Plücker which govern the “shape” of planar curves. We will discuss the Shapiro—Shapiro conjecture and connections to the real Schubert calculus, and end by presenting some new conjectures and computational evidence joint with Frank Sottile.

# Generalizations of homogeneous Einstein manifolds.

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I will explain something of the theory of homogeneous Einstein metrics and why certain generalizations of this equation occur naturally in the study of homogeneous spaces.

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# Superconnections in geometry

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It is well-known that on a non-projective complex manifold, a

coherent sheaf may not have a resolution by a complex of holomorphic vector

bundles. Nevertheless, J. Block showed that such resolution always exists if

we allow anti-holomorphic flat superconnections which generalize complexes

of holomorphic vector bundles. Block's result makes it possible to study

coherent sheaves with differential geometric and analytic tools. For

example, in a joint work with J.M Bismut, S, Shen, and I, we give an

analytic proof of the Grothendieck-Riemann-Roch theorem for coherent sheaves

on complex manifolds. In this talk I will present the ideas and applications

of anti-holomorphic flat superconnections. I will also talk about analogous

constructions of superconnections in other areas of geometry.

# Investigating New Relations between Intrinsic and Extrinsic Geometric Quantities

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In a work on the geometry of minimal submanifolds written in 1968, Shiing-Shen Chern invited more efforts and reflections to identify relationships between intrinsic and extrinsic curvature invariants of submanifolds in various ambient spaces. After 1993, when Bang-Yen Chen introduced the first of his curvature invariants, namely scal - inf(sec), a lot of work has been done to explore this avenue, which represents an active research area. We will survey some of these results obtained in the last three decades, and conclude our talk with new relationships between intrinsic and extrinsic curvature invariants.

# Generalizing Lie theory to higher dimensions - the De Rham theorem on simplices and cubes

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There is a generalization of Lie theory from Lie algebras to differential graded Lie algebras. Ordinary Lie theory involves first order ordinary differential equations. Higher Lie theory may be understood as a non-linear generalization of the de Rham theorem on simplicial complexes (in Dupont's formulation), as against graphs. In this talk, we present an alternate approach to this theory, using the more elementary de Rham theorem on cubical complexes.

Along the way, we will need an interesting relationship between cubical and simplicial complexes, which has recently become better known due to its use in Lurie's theory of straightening for infinity categories.

# On the formality of sphere bundles

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A manifold is called formal if it has the same rational homotopy type

as the cohomology ring. We first consider the formality of a sphere bundle

over a formal manifold. In this case the formality is entirely determined by

the Bianchi-Massey tensor, which is a 4-tensor on a subspace of the

cohomology ring, introduced by Crowley and Nordstrom. As a special case, we

will see that if a manifold and its unit tangent bundle are both formal,

then the manifold has either Euler characteristic zero or rational

cohomology ring generated by one element. Finally we discuss the case of

a general base manifold, and give an obstruction to formality.

# On the moduli spaces of ALH*-gravitational instantons

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Gravitational instantons are defined as non-compact hyperKahler

4-manifolds with L^2 curvature decay. They are all bubbling limits of K3

surfaces and thus serve as stepping stones for understanding the K3 metrics.

In this talk, we will focus on a special kind of them called

ALH*-gravitational instantons. We will explain the Torelli theorem, describe

their moduli spaces and some partial compactifications of the moduli spaces.

This talk is based on joint works with T. Collins, A. Jacob, R. Takahashi,

X. Zhu and S. Soundararajan.

Earlier time and joint seminar with Differential Geometry Seminar.

# Homological Mirror Symmetry for Theta Divisors

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Symplectic geometry is a relatively new branch of geometry.

However, a string theory-inspired duality known as “mirror symmetry” reveals

more about symplectic geometry from its mirror counterparts in complex

geometry. M. Kontsevich conjectured an algebraic version of mirror symmetry

called “homological mirror symmetry” (HMS) in his 1994 ICM address. HMS

results were then proved for symplectic mirrors to Calabi-Yau and Fano

manifolds. Those mirror to general type manifolds have been studied in more

recent years, including my research. In this talk, we will introduce HMS

through the example of the 2-torus T^2. We will then outline how it relates

to HMS for a hypersurface of a 4-torus T^4, in joint work with Haniya Azam,

Heather Lee, and Chiu-Chu Melissa Liu. From there, we generalize to

hypersurfaces of higher dimensional tori, otherwise known as “theta

divisors.” This is also joint with Azam, Lee, and Liu.

Joint with Differential Geometry Seminar.