Seminars and Colloquia by Series

Introduction to Center Manifold Theory

Series
Dynamical Systems Working Seminar
Time
Thursday, April 14, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 170
Speaker
Jiayin Jin – Georgia Tech
In this talk, I will state the main results of center manifold theory for finite dimensional systems and give some simple examples to illustrate their applications. This is based on the book “Applications of Center Manifold Theory” by J. Carr.

The slicing problems for sections of proportional dimensions

Series
School of Mathematics Colloquium
Time
Thursday, April 14, 2016 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Alexander Koldobskiy – University of Missouri, Columbia –
We consider the following problem. Does there exist an absolute constant C such that for every natural number n, every integer 1 \leq k \leq n, every origin-symmetric convex body L in R^n, and every measure \mu with non-negative even continuous density in R^n, \mu(L) \leq C^k \max_{H \in Gr_{n-k}} \mu(L \cap H}/|L|^{k/n}, where Gr_{n-k} is the Grassmannian of (n-k)-dimensional subspaces of R^n, and |L| stands for volume? This question is an extension to arbitrary measures (in place of volume) and to sections of arbitrary codimension k of the hyperplace conjecture of Bourgain, a major open problem in convex geometry. We show that the above inequality holds for arbitrary origin-symmetric convex bodies, all k and all \mu with C \sim \sqrt{n}, and with an absolute constant C for some special class of bodies, including unconditional bodies, unit balls of subspaces of L_p, and others. We also prove that for every \lambda \in (0,1) there exists a constant C = C(\lambda) so that the above inequality holds for every natural number, every origin-symmetric convex body L in R^n, every measure \mu with continuous density and the codimension of sections k \geq \lambda n. The latter result is new even in the case of volume. The proofs are based on a stability result for generalized intersections bodies and on estimates of the outer volume ratio distance from an arbitrary convex body to the classes of generalized intersection bodies.

The Kelmans-Seymour conjecture V: no contractible edges or triangles (finding TK_5)

Series
Graph Theory Seminar
Time
Wednesday, April 13, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Yan Wang – Math, GT
Let G be a 5-connected nonplanar graph. To show the Kelmans-Seymour conjecture, we keep contracting a connected subgraph on a special vertex z until the following happens: H does not contain K_4^-, and for any subgraph T of H such that z is a vertex in T and T is K_2 or K_3, H/T is not 5-connected. In this talk, we study the structure of these 5-separations and 6-separations, and prove the Kelmans-Seymour conjecture.

New results on zeroes of stationary Gaussian functions

Series
Analysis Seminar
Time
Wednesday, April 13, 2016 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Naomi Feldheim – Stanford University
We consider (complex) Gaussian analytic functions on a horizontal strip, whose distribution is invariant with respect to horizontal shifts (i.e., "stationary"). Let N(T) be the number of zeroes in [0,T] x [a,b]. First, we present an extension of a result by Wiener, concerning the existence and characterization of the limit N(T)/T as T approaches infinity. Secondly, we characterize the growth of the variance of N(T). We will pose to discuss analogues of these results in a few other settings, such as zeroes of real-analytic Gaussian functions and winding of planar Gaussian functions, pointing out interesting similarities and differences. For the last part, we consider the "persistence probability" (i.e., the probability that a function has no zeroes at all in some region). Here we present results in the real setting, as even this case is yet to be understood. Based in part on joint works with Jeremiah Buckley and Ohad Feldheim.

What is....Compressive Sensing?

Series
Research Horizons Seminar
Time
Wednesday, April 13, 2016 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Prof. Michael Lacey – School of Mathematics, Georgia Institute of Technology –

Please Note: Food and Drinks will be provided before the seminar.

Compressive sensing is a (relatively) new paradigm in data analysis that is having a large impact on areas from signal processing, statistics, to scientific computing. I am teaching a special topics on the subject in the Fall term, in support of the GT-IMPACT program. The talk will list some basic principles in the subject, stating some Theorems, and using images, and sounds to illustrate these principles.

Virus-Immune Dynamics in Age-Structured HIV Model

Series
Mathematical Biology Seminar
Time
Wednesday, April 13, 2016 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Cameron Browne – U. of Louisiana
Mathematical modeling of viruses, such as HIV, has been an extensive area of research over the past two decades. For HIV, some important factors that affect within-host dynamics include: the CTL (Cytotoxic T Lymphocyte) immune response, intra-host diversity, and heterogeneities of the infected cell lifecycle. Motivated by these factors, I consider several extensions of a standard virus model. First, I analyze a cell infection-age structured PDE model with multiple virus strains. The main result is that the single-strain equilibrium corresponding to the virus strain with maximal reproduction number is a global attractor, i.e. competitive exclusion occurs. Next, I investigate the effect of CTL immune response acting at different times in the infected-cell lifecycle based on recent studies demonstrating superior viral clearance efficacy of certain CTL clones that recognize infected cells early in their lifecycle. Interestingly, explicit inclusion of early recognition CTLs can induce oscillatory dynamics and promote coexistence of multiple distinct CTL populations. Finally, I discuss several directions of ongoing modeling work attempting to capture complex HIV-immune system interactions suggested by experimental data.

The shape sphere: a new vista on the three body problem (David Alcaraz conference: Video conference)

Series
CDSNS Colloquium
Time
Tuesday, April 12, 2016 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Richard Montgomery – Univ. California Santa Cruz
Video Conference David Alcaraz confernce. Newton's famous three-body problem defines dynamics on the space of congruence classes of triangles in the plane. This space is a three-dimensional non-Euclidean rotationally symmetric metric space ``centered'' on the shape sphere. The shape sphere is a two-dimensional sphere whose points represent oriented similarity classes of planar triangles. We describe how the sphere arises from the three-body problem and encodes its dynamics. We will see how the classical solutions of Euler and Lagrange, and the relatively recent figure 8 solution are encoded as points or curves on the sphere. Time permitting, we will show how the sphere pushes us to formulate natural topological-geometric questions about three-body solutions and helps supply the answer to some of these questions. We may take a brief foray into the planar N-body problem and its associated ``shape sphere'' : complex projective N-2 space.

Low-Budget PDE Solver with Painting Applications

Series
Applied and Computational Mathematics Seminar
Time
Monday, April 11, 2016 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Byungmoon Kim – Adobe Research
This talk will tell the story on using simulation for painting. I will tell a few of projects that had simulation and painting involved. One is iPad-based ultra-low-cost real time simulation of old photography process to compute effects that modern day users may find interesting. The other is more full-blown fluid simulation for painting using highest-end GPU. Even with massive processing power of GPU, real time high fidelity painting simulation is hard since computation budget is limited. Basically we should deal with large errors. It may sound odd if someone says that very low-accuracy simulation is interesting - but this is very true. In particular, we tried to pull most pressure effect out from about 10 Jacobi iterations that we could afford. I would like to share my experience on improving fixed number of fixed point iterations.

Iterated Quotients of Ring Spectra and Hopf-Galois Extensions

Series
Geometry Topology Seminar
Time
Monday, April 11, 2016 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jonathan Beardsley – Johns Hopkins University
Given an action by a loop space on a structured ring spectrum we describe how to produce its associated quotient ring spectrum. We then describe how this structure may be leveraged to produce intermediate Hopf-Galois extensions of ring spectra, analogous to the way one produces intermediate Galois extensions from normal subgroups of a Galois group. We will give many examples of this structure in classical cobordism spectra and in particular describe an entirely new construction of the complex cobordism spectrum which bears a striking resemblance to Lazard's original construction of the Lazard ring by iterated extensions.

Dynamical systems tools for Solar sails

Series
CDSNS Colloquium
Time
Monday, April 11, 2016 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Angel Jorba – Univ. of Barcelona
Dynamical systems have proven to be a useful tool for the design of space missions. For instance, the use of invariant manifolds is now common to design transfer strategies. Solar Sailing is a proposed form of spacecraft propulsion, where large membrane mirrors take advantage of the solar radiation pressure to push the spacecraft. Although the acceleration produced by the radiation pressure is smaller than the one achieved by a traditional spacecraft it is continuous and unlimited. This makes some long term missions more accessible, and opens a wide new range of possible applications that cannot be achieved by a traditional spacecraft. In this presentation we will focus on the dynamics of a Solar sail in a couple of situations. We will introduce this problem focusing on a Solar sail in the Earth-Sun system. In this case, the model used will be the Restricted Three Body Problem (RTBP) plus Solar radiation pressure. The effect of the solar radiation pressure on the RTBP produces a 2D family of "artificial'' equilibria, that can be parametrised by the orientation of the sail. We will describe the dynamics around some of these "artificial'' equilibrium points. We note that, due to the solar radiation pressure, the system is Hamiltonian only for two cases: when the sail is perpendicular to the Sun - Sail line; and when the sail is aligned with the Sun - sail line (i.e., no sail effect). The main tool used to understand the dynamics is the computation of centre manifolds. The second example is the dynamics of a Solar sail close to an asteroid. Note that, in this case, the effect of the sail becomes very relevant due to the low mass of the asteroid. We will use, as a model, a Hill problem plus the effect of the Solar radiation pressure, and we will describe some aspects of the natural dynamics of the sail.

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