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Friday, January 25, 2019 - 14:00 ,
Location: Skiles 006 ,
Peter Lambert-Cole ,
Georgia Insitute of Technology ,
Organizer: Peter Lambert-Cole

The classical degree-genus formula computes the genus
of a nonsingular algebraic curve in the complex projective plane.
The well-known Thom conjecture posits that this is a lower bound
on the genus of smoothly embedded, oriented and connected surface
in CP^2.
The conjecture was first proved twenty-five years ago by
Kronheimer and Mrowka, using Seiberg-Witten invariants. In this
talk, we will describe a new proof of the conjecture that combines
contact geometry with the novel theory of bridge trisections of
knotted surfaces. Notably, the proof completely avoids any gauge
theory or pseudoholomorphic curve techniques.

Friday, November 30, 2018 - 14:00 ,
Location: Skiles 006 ,
Surena Hozoori ,
Georgia Institute of Technology ,
shozoori3@gatech.edu ,
Organizer: Surena Hozoori

In post-geometrization low dimensional topology, we expect to be able to relate any topological theory of 3-manifolds to the Riemannian geometry of those manifolds. On the other hand, originated from reformalization of classical mechanics, the study of contact structures has become a central topic in low dimensional topology, thanks to the works of Eliashberg, Giroux, Etnyre and Taubes, to name a few. Yet we know very little about how Riemannian geometry fits into the theory.In my oral exam, I will talk about "Ricci-Reeb realization problem" which asks which functions can be prescribed as the Ricci curvature of a "Reeb vector field" associated to a contact manifold. Finally motivated by Ricci-Reeb realization problem and using the previous study of contact dynamics by Hofer-Wysocki-Zehnder, I will prove new topological results using compatible geometry of contact manifolds. The generalization of these results in higher dimensions is the first known results achieving tightness based on curvature conditions.

Friday, October 19, 2018 - 14:00 ,
Location: Skiles 006 ,
Sudipta Kolay ,
Georgia Tech ,
Organizer: Sudipta Kolay

I will discuss some applications of the holonomic approximation theorem to questions about immersions, embeddings, and singularities.

Friday, October 12, 2018 - 14:00 ,
Location: Skiles 006 ,
Sudipta Kolay ,
Georgia Tech ,
Organizer: Sudipta Kolay

One of the general methods of proving h-principle is holonomic
aprroximation. In this series of talks, I will give a proof of holonomic
approximation theorem,
and talk about some of its applications.

Friday, September 21, 2018 - 14:00 ,
Location: Skiles 006 ,
Peter Lambert-Cole ,
Georgia Insitute of Technology ,
Organizer: Peter Lambert-Cole

The Oka-Grauert principle is one of the first examples of an
h-principle. It states that for a Stein domain X and a complex Lie
group G, the topological and holomorphic classifications of principal
G-bundles over X agree. In particular, a complex vector bundle over X
has a holomorphic trivialization if and only if it has a continuous
trivialization. In these talks, we will discuss the complex geometry of
Stein domains, including various characterizations of Stein domains,
the classical Theorems A and B, and the Oka-Grauert principle.

Friday, September 14, 2018 - 13:55 ,
Location: Skiles 006 ,
Peter Lambert-Cole ,
Georgia Insitute of Technology ,
Organizer: Peter Lambert-Cole
The Oka-Grauert principle is one of the first examples of an
h-principle. It states that for a Stein domain X and a complex Lie
group G, the topological and holomorphic classifications of principal
G-bundles over X agree. In particular, a complex vector bundle over X
has a holomorphic trivialization if and only if it has a continuous
trivialization. In these talks, we will discuss the complex geometry of
Stein domains, including various characterizations of Stein domains,
the classical Theorems A and B, and the Oka-Grauert principle.

Wednesday, May 2, 2018 - 14:00 ,
Location: Skiles 006 ,
Hyunki Min ,
Georgia Tech ,
hmin38@gatech.edu ,
Organizer: Hyun Ki Min

Understanding contact structures on
hyperbolic 3-manifolds is one of the major open problems in the area of contact
topology. As a first step, we try to classify tight contact structures on a specific hyperbolic 3-manifold. In this talk, we will review the previous classification
results and classify tight contact structures on the Weeks manifold, which
has the smallest hyperbolic volume. Finally,
we will discuss how to generalize this method to classify tight contact structures
on some other hyperbolic 3-manifolds.

Friday, March 16, 2018 - 14:00 ,
Location: Skiles 006 ,
Jen Hom ,
Georgia Tech ,
Organizer: Jennifer Hom

In this series of talks, we will study the relationship between the Alexander module and the bordered Floer homology of the Seifert surface complement. In particular, we will show that bordered Floer categorifies Donaldson's TQFT description of the Alexander module.