Past Seminars by Series

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2019
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.

Thu May 03

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Bochner-Kahler metrics (after R. Bryant)
Rui Loja Fernandes - Ordway Visitor, UIUC

In a seminal paper, R. Bryant gave a local description of Bochner-Kahler metrics and a classification of germs of such metrics. In this talk I will sketch a new approach to this classification using the theory of integrability of Lie algebroids. It provides new insight and allows for more precise results, extending the work of Bryant. Our approach can also be used to solve other geometric classification problems (e.g., symplectic connections with special holonomy). This is on-going joint work with Ivan Struchiner (U Sao Paulo, Brazil).

Thu Apr 19

Differential Geometry and Symplectic Topology Seminar

1:35pm - Vincent Hall 570
How to Make a Black Hole
Xianliang An, University of Toronto

Black holes are predicted by Einstein's theory of general relativity, and now we have ample observational evidence for their existence. However theoretically there are many unanswered questions about how black holes come into being. In this talk, with tools from hyperbolic PDE, quasilinear elliptic equations, geometric analysis and dynamical systems, we will prove that, through a nonlinear focusing effect, initially low-amplitude and diffused gravitational waves can give birth to a black hole region in our universe. This result extends the 1965 Penrose’s singularity theorem and it also proves a conjecture of Ashtekar on black-hole thermodynamics. Open problems and new directions will also be discussed.

Mon Mar 26

Differential Geometry and Symplectic Topology Seminar

4:30pm - Vincent Hall 570
Isometric embedding via strongly symmetric positive systems
Jeanne Clelland, University of Colorado

(Joint work with Gui-Qiang Chen, Marshall Slemrod, Dehua Wang, and Deane Yang)

In this talk, I will give an outline of our new proof for the local existence of a smooth isometric embedding of a smooth 3-dimensional Riemannian manifold with nonzero Riemannian curvature tensor into 6-dimensional Euclidean space. Our proof avoids the sophisticated microlocal analysis used in earlier proofs by Bryant-Griffiths-Yang and Nakamura-Maeda; instead, it is based on a new local existence theorem for a class of nonlinear, first-order PDE systems that we call "strongly symmetric positive." These are a subclass of the class of symmetric positive systems, which were introduced by Friedrichs in order to study certain PDE systems that do not fall under one of the standard types (elliptic, hyperbolic, and parabolic).

As in earlier proofs, we construct solutions via the Nash-Moser implicit function theorem, which requires showing that the linearization of the isometric embedding PDE system near an approximate embedding has a smooth solution that satisfies "smooth tame estimates." We accomplish this in two steps:
(1) Show that the approximate embedding can be chosen so that the reduced linearized system becomes strongly symmetric positive after a carefully chosen change of variables.
(2) Show that any such system has local solutions that satisfy smooth tame estimates.

The main advantage of our approach is that step (2) is much more straightforward than similar results for other classes of PDE systems used in prior proofs, while step (1) requires only linear algebra.

The talk will focus on the main ideas of the proof; technical details will be kept to a minimum.

Thu Mar 08

Differential Geometry and Symplectic Topology Seminar

1:25pm - Vincent Hall 570
Stable equivalence of smoothly knotted surfaces
Dave Auckly, Kansas State University

t is well known that there are smoothly inequivalent, objects in 4-dimensions that are topologically equivalent. Fairly general results exist stating that such objects become smoothly equivalent after some number of stabilizations. Until this past summer the only thing known about the number of stabilizations needed was an infinite collection of examples where one stabilization was enough.

This talk will present the proof of a theorem demonstrating that when the easiest topological invariants are trivial two smooth surfaces become smoothly isotopic after just one stabilization.
This is joint work with Kim, Melvin, Ruberman and Schwartz.

We will also present joint work with Ruberman aimed at the difference between the homotopy groups of the diffeomorphism group and the homotopy groups of the homeomorphism group.

Thu Mar 01

Differential Geometry and Symplectic Topology Seminar

1:35pm - Vincent Hall 570
Topology of Enumerative Problems: Inflection Points on Cubic Curves
Weiyan Chen, University of Minnesota
Thu Feb 22

Differential Geometry and Symplectic Topology Seminar

1:30pm - Vincent Hall 570
On Symplectic Fillings and Milnor Fibers of Quotient Surface Singularities
Professor Jongil Park, Seoul National University

Abstract
One of active research areas in symplectic 4-manifolds is to cassify symplectic fillings of certain 3-manifolds equipped with a contact structure. Among them, people have long studied symplectic fillings of the link of a quotient surface singularity. Note that the link of a quotient surface singularity carries a canonical contact structure which is also known as the Milnor fillable contact structure. One the other hand, algebraic geometers also have studied Milnor fibers as a general fiber of
smoothings for a quotient surface singularity.
In this series of talks, I’d like to review basics on symplectic fillings and Milnor fibers of quotient surface singularities. Then I will explain some recent works on symplectic fillings and Milnor fibers of quotient surface singularities. The 1st talk is a review of known results on symplectic fillings and Milnor fibers of quotient surface singularities, and the 2nd talk is about identifying minimal symplectic fillings with Milnor fibers of quotient surface singularities (joint work with Heesang
Park, Dongsoo Shin and Giancarlo Urz´ua). In the 3rd talk, I’ll also explain how we interpret minimal symplectic fillings of quotient surface singularities as positive allowable Lefschetz fibrations (joint work with Hakho Choi). A tentative schedule is the following:
Talk 1: 2/8(Thur) A review of known results on symplectic fillings and Milnor fibers of quotient surface singularities
Talk 2: 2/15(Thur) Identifying minimal symplectic fillings with Milnor fibers of quotient surface singularities
Talk 3: 2/22(Thur) A Lefschetz fibration viewpoint on minimal
symplectic fillings of quotient surface singularities

Thu Feb 15

Differential Geometry and Symplectic Topology Seminar

1:30pm - Vincent Hall 570
On Symplectic Fillings and Milnor Fibers of Quotient Surface Singularities
Professor Jongil Park, Seoul National University

One of active research areas in symplectic 4-manifolds is to cassify symplectic fillings of certain 3-manifolds equipped with a contact structure. Among them, people have long studied symplectic fillings of the link of a quotient surface singularity. Note that the link of a quotient surface singularity carries a canonical contact structure which is also known as the Milnor fillable contact structure. One the other hand, algebraic geometers also have studied Milnor fibers as a general fiber of smoothings for a quotient surface singularity.
In this series of talks, I’d like to review basics on symplectic fillings and Milnor fibers of quotient surface singularities. Then I will explain some recent works on symplectic fillings and Milnor fibers of quotient surface singularities. The 1st talk is a review of known results on symplectic fillings and Milnor fibers of quotient surface singularities, and the 2nd talk is about identifying minimal symplectic fillings with Milnor fibers of quotient surface singularities (joint work with Heesang
Park, Dongsoo Shin and Giancarlo Urz´ua). In the 3rd talk, I’ll also explain how we interpret minimal symplectic fillings of quotient surface singularities as positive allowable Lefschetz fibrations (joint work with Hakho Choi). A tentative schedule is the following:
Talk 1: 2/8(Thur) A review of known results on symplectic fillings and Milnor fibers of quotient surface singularities
Talk 2: 2/15(Thur) Identifying minimal symplectic fillings with Milnor fibers of quotient surface singularities
Talk 3: 2/22(Thur) A Lefschetz fibration viewpoint on minimal
symplectic fillings of quotient surface singularities

Thu Feb 08

Differential Geometry and Symplectic Topology Seminar

1:30pm - Vincent Hall 570
On Symplectic Fillings and Milnor Fibers of Quotient Surface Singularities
Professor Jongil Park, Seoul National University

Abstract
One of active research areas in symplectic 4-manifolds is to cassify symplectic fillings of certain 3-manifolds equipped with a contact structure. Among them, people have long studied symplectic fillings of the link of a quotient surface singularity. Note that the link of a quotient surface singularity carries a canonical contact structure which is also known as the Milnor fillable contact structure. One the other hand, algebraic geometers also have studied Milnor fibers as a general fiber of
smoothings for a quotient surface singularity.
In this series of talks, I’d like to review basics on symplectic fillings and Milnor fibers of quotient surface singularities. Then I will explain some recent works on symplectic fillings and Milnor fibers of quotient surface singularities. The 1st talk is a review of known results on symplectic fillings and Milnor fibers of quotient surface singularities, and the 2nd talk is about identifying minimal symplectic fillings with Milnor fibers of quotient surface singularities (joint work with Heesang
Park, Dongsoo Shin and Giancarlo Urz´ua). In the 3rd talk, I’ll also explain how we interpret minimal symplectic fillings of quotient surface singularities as positive allowable Lefschetz fibrations (joint work with Hakho Choi). A tentative schedule is the following:
Talk 1: 2/8(Thur) A review of known results on symplectic fillings and Milnor fibers of quotient surface singularities
Talk 2: 2/15(Thur) Identifying minimal symplectic fillings with Milnor fibers of quotient surface singularities
Talk 3: 2/22(Thur) A Lefschetz fibration viewpoint on minimal
symplectic fillings of quotient surface singularities