Real planar curves - algebraic, geometric, and topological aspects

Speaker: 

Thomas Brazelton

Institution: 

Harvard University

Time: 

Thursday, January 18, 2024 - 12:00pm to 12:50pm

Host: 

Location: 

RH 340P

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.

Superconnections in geometry

Speaker: 

Zhaoting Wei

Institution: 

Texas A&M University-Commerce

Time: 

Monday, March 11, 2024 - 4:00pm

Location: 

RH 340N

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

Speaker: 

Bogdan Suceava

Institution: 

CSU Fullerton

Time: 

Monday, December 11, 2023 - 4:00pm to 5:00pm

Host: 

Location: 

RH 340N

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

Speaker: 

Ezra Getzler

Institution: 

Northwestern

Time: 

Monday, November 27, 2023 - 4:00pm to 5:00pm

Host: 

Location: 

RH 340N

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

Speaker: 

Jiawei Zhou

Institution: 

BIMSA, Beijing

Time: 

Monday, November 13, 2023 - 4:00pm

Location: 

RH 340N

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

Speaker: 

Yu-Shen Lin

Institution: 

Boston University

Time: 

Monday, June 5, 2023 - 3:30pm

Location: 

RH 340N

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

Speaker: 

Catherine Cannizzo

Institution: 

UC Riverside

Time: 

Tuesday, April 11, 2023 - 4:00pm

Location: 

ISEB 1200

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.

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