Past Seminars by Series

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2020
Thu Mar 05

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
An invitation to contact homology
Erkao Bao, Scientist at the company Houzz in Palo Alto

Contact homology is an invariant of the contact structure, which is an odd-dimensional counterpart of a symplectic structure. It was proposed by Eliashberg, Givental and Hofer in 2000. The application of contact homology and its variants include distinguishing contact structures, knot invariants, the Weinstein conjecture and generalization, and calculating Gromov-Witten invariants. In this talk, I will start with the notion of contact structures, then give a heuristic definition of the contact homology as an infinite dimensional Morse homology, and finally explain the major difficulties to make the definition rigorous. This is a joint work with Ko Honda.

2019
Thu Dec 19

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Differential Geometry and Sympletic Topology
TBATBA
Tue Dec 17

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Differential Geometry and Sympletic Topology
TBATBA
Thu Dec 12

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Differential Geometry and Sympletic Topology
TBATBA
Tue Dec 10

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Differential Geometry and Sympletic Topology Seminar
TBATBA
Thu Dec 05

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Einstein's gravity and stability of black holes
Pei-Ken Hung, MIT

Though Einstein's fundamental theory of general relativity has already celebrated its one hundredth birthday, there are still many outstanding unsolved problems. The Kerr stability conjecture is one of the most important open problems, which posits that the Kerr metrics are stable solutions of the vacuum Einstein equation. Over the past decade, there have been huge advances towards this conjecture based on the study of wave equations in black hole spacetimes and structures in the Einstein equation. In this talk, I will discuss the recent progress in the stability problems with special focus on the wave gauge.

Tue Dec 03

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Differential Geometry and Sympletic Topology
TBATBA
Thu Nov 28

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Differential Geometry and Sympletic Topology
TBATBA
Mon Nov 25

Differential Geometry and Symplectic Topology Seminar

10:10am - Vincent Hall 203A
Differential Signatures and Algebraic Curves
Michael Ruddy,, Max Planck Institute

For the action of a group on the plane, the group equivalence problem for curves can be stated as: given two curves, decide if they are related by an element of the group. The signature method, using differential invariants, to answer the local group equivalence problem for smooth curves and its application to image science has been extensively studied. For planar algebraic curves under subgroups of the general linear group, we show that this provides a method to associate a unique algebraic curve to each equivalence class, the algebraic curve's signature curve. However, computing the implicit equation of the signature curve is a challenging problem. In this talk we consider signatures of algebraic curves, show how to compute the degree without computing its defining polynomial explicitly, and present some results on the structure of signature curves for generic algebraic curves of fixed degree. Additionally we show that this leads to a method to solve the group equivalence problem for algebraic curves using numerical algebraic geometry.

Thu Nov 21

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Geometry of degenerating Calabi-Yau manifolds
Ruobing Zhang, Stony Brook

This talk concerns a family of "collapsing" Ricci-flat Kähler manifolds, namely Calabi-Yau manifolds, converging to a lower dimensional limit, which develop singularities arising in various contexts such as metric Riemannian geometry, complex geometry and degenerating nonlinear equations. A primary aspect is to formulate how well behaved or badly behaved such spaces can be in terms of the recently developed regularity theory. Under the above framework, our next focus is on a longstanding fundamental problem which is to understand singularities of collapsing Ricci-flat metrics along an algebraically degenerating family. We will give accurate characterizations of such metrics and explain possible generalizations.

Tue Nov 19

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Differential Geometry and Sympletic Topology
TBATBA
Thu Nov 14

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Differential Geometry and Sympletic Topology
TBATBA
Tue Nov 12

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Differential Geometry and Sympletic Topology
TBATBA
Thu Nov 07

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Differential Geometry and Sympletic Topology
TBATBA
Tue Nov 05

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Differential Geometry and Sympletic Topology
TBATBA
Thu Oct 31

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Differential Geometry and Sympletic Topology
TBATBA
Tue Oct 29

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Differential Geometry and Sympletic Topology
TBATBA
Thu Oct 24

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Differential Geometry and Sympletic Topology
TBATBA
Tue Oct 22

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Differential Geometry and Sympletic Topology
TBATBA
Thu Oct 17

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Differential Geometry and Sympletic Topology
TBATBA
Tue Oct 15

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Differential Geometry and Sympletic Topology
TBATBA
Thu Oct 10

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Differential Geometry and Sympletic Topology
TBATBA
Tue Oct 08

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Differential Geometry and Sympletic Topology
TBATBA
Thu Oct 03

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Differential Geometry and Sympletic Topology
TBATBA
Tue Oct 01

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Differential Geometry and Sympletic Topology
TBATBA
Thu Sep 26

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Differential Geometry and Sympletic Topology
TBATBA
Tue Sep 24

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Differential Geometry and Sympletic Topology
TBATBA
Thu Sep 19

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
D=4, N=1 Compactifications of Maximal Supergravities via Generalised Geometry - Kahler potentials, superpotentials and moduli
David Tennnyson, Imperial College London

We analyse compactifications of 11 dimensional or type II supergravity down to 4 dimensional Minkowski space for generic flux and generic internal Killing spinors. We note the failure of conventional differential geometry to capture the generic features of the theory and show that the correct formalism comes in the form of a closed form Leibniz algebroid - or as we call it in the physics community, generalised geometry. Our structure is similar to the generalised geometry of Hitchin, but now the structure group is the non-compact exceptional group E_{7(7)}x R^{+}. It turns out that having N=1 supersymmetry in the effective theory on Minkowski space is equivalent to an integrable SU(7) structure on the generalised tangent bundle. We provide the tensors that define the SU(7) structure and give the integrability conditions. Finally we provide an expression for the Kahler potential on the space of structures, the superpotential of the lower dimensional theory, and we explore the moduli of these structures giving explicit answers in certain cases.

Thu Apr 25

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Harmonic surfaces and simple loops
Vlad Markovic - Ordway Visitor, Caltech
Tue Apr 23

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Self-homeomorphisms of reducible 3-manifolds and applications in topology, geometry and dynamics.
Christoforos Neofytidis, University of Geneva

We recall the self-homeomorphisms of a closed oriented reducible 3-manifold. Using this description, we discuss various problems in low-dimensional topology and dynamics, such as the existence of Anosov tori in 3-manifolds (joint work with Shicheng Wang), the simplicial volume of mapping tori of 3-manifolds (joint work Michelle Bucher) and the virtual Betti numbers of mapping tori of 3-manifolds.

2018
Thu Dec 20

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Projective Geometry, Complex Hyperbolic Space, and Geometric Transitions
Steve Trettel, UC Santa Barbara

The natural analog of Teichmuller theory for hyperbolic manifolds in dimension 3 or greater is trivialized by Mostow Rigidity, so mathematicians have worked to understand more general deformations. Two well studied examples, convex real projective structures and complex hyperbolic structures, have been investigated extensively and provide independently developed deformation theories. Here we will discuss a surprising connection between the these, and construct a one parameter family of geometries deforming complex hyperbolic space into a new geometry built out of real projective space and its dual. This connects the aforementioned deformation theories and provides geometric motivation for a representation-theoretic observation of Cooper, Long, and Thistlethwaite

Thu Dec 06

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Enumerative geometry: old and new
Felix Janda, University of Michigan

For as long as people have studied geometry, they have counted
geometric objects. For example, Euclid's Elements starts with the
postulate that there is exactly one line passing through two distinct
points in the plane. Since then, the kinds of counting problems we are
able to pose and to answer has grown. Today enumerative geometry is a
rich subject with connections to many fields, including combinatorics,
physics, representation theory, number theory and integrable systems.

In this talk, I will show how to solve several classical counting
questions. Then I will describe a more modern problem with roots in
string theory which has been the subject of intense study for the last
two decades, namely the study of the Gromov-Witten invariants of the
quintic threefold, a Calabi-Yau manifold. I will explain a recent
break-through in understanding the higher genus invariants that stems
from a seemingly unrelated problem related to the study of holomorphic
differentials on Riemann surfaces.

Fri Oct 19

Differential Geometry and Symplectic Topology Seminar

2:30pm - Vincent Hall 6
The Smooth 4-dimensional Poincare Conjecture and Dehn surgery on links
Alex Zupan, University of Nebraska

The smooth version of the 4-dimensional Poincare Conjecture (S4PC) states that every homotopy 4-sphere is diffeomorphic to the standard 4-sphere. One way to attack the S4PC is to examine a restricted class of 4-manifolds. For example, Gabai's proof of Property R implies that every homotopy 4-sphere built with one 2-handle and one 3-handle is standard. In this talk, we consider homotopy 4-spheres X built with two 2-handles and two 3-handles, which are uniquely determined by the attaching link L for the 2-handles in the 3-sphere. We prove that if one of the components of L is the connected sum of a torus knot T(p,2) and its mirror (a generalized square knot), then X is diffeomorphic to the standard 4-sphere. This is joint work with Jeffrey Meier.

Thu Oct 04

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Stability of Ricci solitons
Huaidong Cao - Ordway Visitor, Lehigh University

In this talk we continue our discussion of the previous week on stability of
Ricci solitons, especially in four dimensions.

Thu Sep 27

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Second variation of Perelman's entropy and stability of Ricci solitons
Huaidong Cao, Ordway Visitor, Lehigh University

Einstein metrics are critical points of the well-known classical Hilbert action
(namely the total scalar curvature functional) under volume normalization. Likewise,
Ricci solitons are critical points of Perelman's entropy.
In this talk, we shall discuss the second variation of Perelman's entropy and stability
of compact Ricci solitons. It turns out the stability for positive Einstein manifolds is
related to two eigenvalue estimates: the first eigenvalue of the Laplacian on functions,
and that of the Lichnerowicz Laplacian on symmetric 2-tensors.

Thu Sep 20

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Monotonicity formulas and Type I singularities
Huaidong Cao, Ordway Visitor, Lehigh University

In this talk we shall introduce Huisken's monotonicity formula for the mean curvature flow and Perelman's monotonicity formulas for the Ricci flow. We shall discuss their applications, including the role they play in studying Type-I singularities of the flows.