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Friday, April 19, 2019 - 14:00 ,
Location: Skiles 006 ,
Arash Yavari and Fabio Sozio, School of Civil and Environmental Engineering ,
Georgia Tech ,
Organizer: Igor Belegradek

We formulate a geometric nonlinear theory of the mechanics of accretion. In this theory the material manifold of an accreting body is represented by a time-dependent Riemannian manifold with a time-independent metric that at each point depends on the state of deformation at that point at its time of attachment to the body, and on the way the new material isadded to the body. We study the incompatibilities induced by accretion through the analysis of the material metric and its curvature in relation to the foliated structure of the accreted body. Balance laws are discussed and the initial-boundary value problem of accretion is formulated. The particular cases where the growth surface is either fixed or traction-free are studied and some analytical results are provided. We numerically solve several accretion problems and calculate the residual stresses in nonlinear elastic bodies induced from accretion.

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

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.