Seminars and Colloquia by Series

Nonlinear Mechanics of Accretion

Series
Geometry Topology Working Seminar
Time
Friday, April 19, 2019 - 14:00 for 2 hours
Location
Skiles 006
Speaker
Arash Yavari and Fabio Sozio, School of Civil and Environmental Engineering – Georgia Tech
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.

Bridge trisections and minimal genus

Series
Geometry Topology Working Seminar
Time
Friday, January 25, 2019 - 14:00 for 2 hours
Location
Skiles 006
Speaker
Peter Lambert-Cole – Georgia Insitute of Technology
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.

An Oral Exam: Curvature, Contact Topology and Reeb Dynamics

Series
Geometry Topology Working Seminar
Time
Friday, November 30, 2018 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Surena Hozoori – Georgia Institute of Technology –
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.

Holonomic Approximation Theorem I

Series
Geometry Topology Working Seminar
Time
Friday, October 12, 2018 - 14:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 006
Speaker
Sudipta Kolay – Georgia Tech
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.

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