Seminars and Colloquia by Series

Infinite energy cascades and modified scattering for the cubic Schr\"odinger on product spaces

Series
PDE Seminar
Time
Thursday, April 17, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Zaher Hani – New York University
We consider the cubic nonlinear Schr\"odinger equation posed on the product spaces \R\times \T^d. We prove the existence of global solutions exhibiting infinite growth of high Sobolev norms. This is a manifestation of the "direct energy cascade" phenomenon, in which the energy of the system escapes from low frequency concentration zones to arbitrarily high frequency ones (small scales). One main ingredient in the proof is a precise description of the asymptotic dynamics of the cubic NLS equation when 1\leq d \leq 4. More precisely, we prove modified scattering to the resonant dynamics in the following sense: Solutions to the cubic NLS equation converge (as time goes to infinity) to solutions of the corresponding resonant system (aka first Birkhoff normal form). This is joint work with Benoit Pausader (Princeton), Nikolay Tzvetkov (Cergy-Pontoise), and Nicola Visciglia (Pisa).

Turan Number of the Generalized Triangle

Series
Graph Theory Seminar
Time
Thursday, April 17, 2014 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Liana Yepremyan – McGill University (Montreal) and Georgia Tech
Generalized triangle T_r is an r-graph with edges {1,2,…,r}, {1,2,…,r-1, r+1} and {r,r+1, r+2, …,2r-2}. The family \Sigma_r consists of all r-graphs with three edges D_1, D_2, D_3 such that |D_1\cap D_2|=r-1 and D_1\triangle D_2\subset D_3. In 1989 it was conjectured by Frankl and Furedi that ex(n,T_r) = ex(n,\Sigma_r) for large enough n, where ex(n,F) is the Tur\'{a}n function. The conjecture was proven to be true for r=3, 4 by Frankl, Furedi and Pikhurko respectively. We settle the conjecture for r=5,6 and show that extremal graphs are blow-ups of the unique (11, 5, 4) and (12, 6, 5) Steiner systems. The proof is based on a technique for deriving exact results for the Tur\'{a}n function from “local stability" results, which has other applications. This is joint work with Sergey Norin.

Bi-parameter singular integrals: recent results and examples

Series
Analysis Seminar
Time
Wednesday, April 16, 2014 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Henri Martikainen – Georgia Tech
We discuss bi-parameter Calderon-Zygmund singular integrals from the point of view of modern probabilistic and dyadic techniques. In particular, we discuss their structure and boundedness via dyadic model operators. In connection to this we demonstrate, via new examples, the delicacy of the problem of finding a completely satisfactory product T1 theorem. Time permitting related non-homogeneous bi-parameter results may be mentioned.

Monoids in the braid and mapping class groups from contact topology

Series
Geometry Topology Seminar
Time
Wednesday, April 16, 2014 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jeremy Van Horn-Morris – University of Arkansas
A monoidal subset of a group is any set which is closed under the product (and contains the identity). The standard example is Dehn^+, the set of maps whcih can be written as a product of right-handed Dehn twists. Using open book decompositions, many properties of contact 3-manifolds are encoded as monoidal subsets of the mapping class group. By a related construction, contact topology also produces a several monoidal subsets of the braid group. These generalize the notion of positive braids and Rudolphs ideas of quasipositive and strongly quasipositive. We'll discuss the construction of these monoids and some of the many open questions.

Topics in Ergodic Theory VII: Ruelle's Entropy Inequality.

Series
Dynamical Systems Working Seminar
Time
Wednesday, April 16, 2014 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles
Speaker
Rafael de la Llave – Georgia Tech
We prove Ruelle's Entropy Inequality for C^1 maps. This is part of a reading seminar geared towards understanding of Smooth Ergodic Theory. (The study of dynamical systems using at the same time tools from measure theory and from differential geometry)It should be accesible to graduate students and the presentation is informal. The first goal will be a proof of the Oseledets multiplicative ergodic theorem for random matrices. Then, we will try to cover the Pesin entropy formula, invariant manifolds, etc.

Black Holes and Strong Dynamical Gravity

Series
Research Horizons Seminar
Time
Wednesday, April 16, 2014 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dr. Laguna – School of Physics
Numerical relativity has opened the door to unveil phenomena associated with strong dynamical gravity. I will present results from three studies of black holes that have been only possible thanks to state of the art computational tools and powerful computer hardware.

Stein fillings of contact manifolds supported by planar open books.

Series
Dissertation Defense
Time
Wednesday, April 16, 2014 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Amey Kaloti – Georgia Tech –
In this thesis we study topology of symplectic fillings of contact manifolds supported by planar open books. We obtain results regarding geography of the symplectic fillings of these contact manifolds. Specifically, we prove that if a contact manifold $(M,\xi)$ is supported by a planar open book, then Euler characteristic and signature of any Stein filling of $(M,\xi)$ is bounded. We also prove a similar finiteness result for contact manifolds supported by spinal open books with planar pages. Moving beyond the geography of Stein fillings, we classify fillings of some lens spaces.In addition, we classify Stein fillings of an infinite family of contact 3-manifolds up to diffeomorphism. Some contact 3-manifolds in this family can be obtained by Legendrian surgeries on $(S^3,\xi_{std})$ along certain Legendrian 2-bridge knots. We also classify Stein fillings, up to symplectic deformation, of an infinite family of contact 3-manifolds which can be obtained by Legendrian surgeries on $(S^3,\xi_{std})$ along certain Legendrian twist knots. As a corollary, we obtain a classification of Stein fillings of an infinite family of contact hyperbolic 3-manifolds up to symplectic deformation.

HYPERBOLIC SYSTEMS OF BALANCE LAWS WITH DISSIPATION

Series
PDE Seminar
Time
Tuesday, April 15, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Constantine Dafermos – Brown University
ABSTRACT: The lecture will outline a research program which aims at establishing the existence and long time behavior of BV solutions for hyperbolic systems of balance laws, in one space dimension, with partially dissipative source, manifesting relaxation. Systems with such structure are ubiquitous in classical physics.

Pfaffian Orientations, Flat Embeddings, and Steinberg’s Conjecture

Series
Dissertation Defense
Time
Tuesday, April 15, 2014 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Peter Whalen – Georgia Institute of Technology –
The first result of this thesis is a partial result in the direction of Steinberg's Conjecture. Steinberg's Conjecture states that any planar graph without cycles of length four or five is three colorable. Borodin, Glebov, Montassier, and Raspaud showed that planar graphs without cycles of length four, five, or seven are three colorable and Borodin and Glebov showed that planar graphs without five cycles or triangles at distance at most two apart are three colorable. We prove a statement that implies the first of these theorems and is incomparable with the second: that any planar graph with no cycles of length four through six or cycles of length seven with incident triangles distance exactly two apart are three colorable. We are next concerned with the study of Pfaffian orientations. A theorem proved by William McCuaig and, independently, Neil Robertson, Paul Seymour, and Robin Thomas provides a good characterization for whether or not a bipartite graph has a Pfaffian orientation as well as a polynomial time algorithm for that problem. We reprove this characterization and provide a new algorithm for this problem. First, we generalize a preliminary result needed to reprove this theorem. Specifically, we show that any internally 4-connected, non-planar bipartite graph contains a subdivision of K3,3 in which each path has odd length. We then make use of this result to provide a much shorter proof of this characterization using elementary methods. In the final piece of the thesis we investigate flat embeddings. A piecewise-linear embedding of a graph in 3-space is flat if every cycle of the graph bounds a disk disjoint from the rest of the graph. We first provide a structural theorem for flat embeddings that indicates how to build them from small pieces. We then present a class of flat graphs that are highly non-planar in the sense that, for any fixed k, there are an infinite number of members of the class such that deleting k vertices leaves the graph non-planar.

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