Seminars and Colloquia by Series

Optimization of Network Dynamics: Attributes and Artifacts

Series
School of Mathematics Colloquium
Time
Thursday, February 4, 2016 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Prof. Dr. Adilson E. Motter – Northwestern University –
The recent interest in network modeling has been largely driven by the prospect that network optimization will help us understand the workings of evolution in natural systems and the principles of efficient design in engineered systems. In this presentation, I will reflect on unanticipated properties observed in three classes of network optimization problems. First, I will discuss implications of optimization for the metabolic activity of living cells and its role in giving rise to the recently discovered phenomenon of synthetic rescues. I will then comment on the problem of controlling network dynamics and show that theoretical results on optimizing the number of driver nodes often only offer a conservative lower bound to the number actually needed in practice. Finally, I will discuss the sensitive dependence of network dynamics on network structure that emerges in the optimization of network topology for dynamical processes governed by eigenvalue spectra, such as synchronization and consensus processes. It follows that optimization is a double-edged sword for which desired and adverse effects can be exacerbated in network systems due to the high dimensionality of their phase spaces.

Homogeneous solutions to the incompressible Euler equation

Series
PDE Seminar
Time
Wednesday, February 3, 2016 - 16:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Roman Shvydkoy – University of Illinois, Chicago
In this talk we describe recent results on classification and rigidity properties of stationary homogeneous solutions to the 3D and 2D Euler equations. The problem is motivated be recent exclusions of self-similar blowup for Euler and its relation to Onsager conjecture and intermittency. In 2D the problem also arises in several other areas such as isometric immersions of the 2-sphere, or optimal transport. A full classification of two dimensional solutions will be given. In 3D we reveal several new classes of solutions and prove their rigidity properties. In particular, irrotational solutions are characterized by vanishing of the Bernoulli function; and tangential flows are necessarily 2D axisymmetric pure rotations. In several cases solutions are excluded altogether. The arguments reveal geodesic features of the Euler equation on the sphere. We further discuss the case when homogeneity corresponds to the Onsager-critical state. We will show that anomalous energy flux at the singularity vanishes, which is suggestive of absence of extreme $0$-dimensional intermittencies in dissipative flows.

A Discrete Quadratic Carleson Theorem

Series
Analysis Seminar
Time
Wednesday, February 3, 2016 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Michael Lacey – Gatech
We will describe sufficient conditions on a set $\Lambda \subset [0,2\pi) $ so that the maximal operator below is bound on $\ell^2(Z)$. $$\sup _{\lambda \in \Lambda} \Big| \sum_{n\neq 0} e^{i \lambda n^2} f(x-n)/n\Big|$$ The integral version of this result is an influential result to E.M. Stein. Of course one should be able to take $\Lambda = [0,2\pi) $, but such a proof would have to go far beyond the already complicated one we will describe. Joint work with Ben Krause.

Dual complexes of unirational varieties

Series
Algebra Seminar
Time
Monday, February 1, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Dustin Cartwright – UT Knoxville
The dual complex of a semistable degeneration records the combinatorics of the intersections in the special fiber. More generally, one can associate a polyhedral dual complex to any toroidal degeneration. It is natural to ask for connections between the geometry of an algebraic variety and the combinatorial properties of its dual complex. In this talk, I will explain one such result: The dual complex of an n-dimensional uniruled variety has the homotopy type of an (n-1)-dimensional simplicial complex. The key technical tool is a specialization map to dual complexes and a balancing condition for these specialization.

Lipschitz metric for a nonlinear wave equation

Series
CDSNS Colloquium
Time
Monday, February 1, 2016 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Geng Chen – Georgia Tech
The nonlinear wave equation: u_{tt} - c(u)[c(u)u_x]_x = 0 is a natural generalization of the linear wave equation. In this talk, we will discuss a recent breakthrough addressing the Lipschitz continuous dependence of solutions on initial data for this quasi-linear wave equation. Our earlier results showed that this equation determines a unique flow of conservative solution within the natural energy space H^1(R). However, this flow is not Lipschitz continuous with respect to the H^1 distance, due to the formation of singularity. To prove the desired Lipschitz continuous property, we constructed a new Finsler type metric, where the norm of tangent vectors is defined in terms of an optimal transportation problem. For paths of piecewise smooth solutions, we carefully estimated how the distance grows in time. To complete the construction, we proved that the family of piecewise smooth solutions is dense, following by an application of Thom's transversality theorem. This is a collaboration work with Alberto Bressan.

A central limit theorem for temporally non-homogenous Markov chains with applications to dynamic programming

Series
Stochastics Seminar
Time
Thursday, January 28, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Alessandro Arlotto – Duke University
We prove a central limit theorem for a class of additive processes that arise naturally in the theory of finite horizon Markov decision problems. The main theorem generalizes a classic result of Dobrushin (1956) for temporally non-homogeneous Markov chains, and the principal innovation is that here the summands are permitted to depend on both the current state and a bounded number of future states of the chain. We show through several examples that this added flexibility gives one a direct path to asymptotic normality of the optimal total reward of finite horizon Markov decision problems. The same examples also explain why such results are not easily obtained by alternative Markovian techniques such as enlargement of the state space. (Joint work with J. M. Steele.)

Cross-immunoreactivity causes antigenic cooperation

Series
School of Mathematics Colloquium
Time
Thursday, January 28, 2016 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Leonid Bunimovich – Georgia Institute of Technology –
Hepatitis C virus (HCV) has the propensity to cause chronic infection. HCV affects an estimated 170 million people worldwide. Immune escape by continuous genetic diversification is commonly described using a metaphor of "arm race" between virus and host. We developed a mathematical model that explained all clinical observations which could not be explained by the "arm race theory". The model applied to network of cross-immunoreactivity suggests antigenic cooperation as a mechanism of mitigating the immune pressure on HCV variants. Cross-immunoreactivity was observed for dengue, influenza, etc. Therefore antigenic cooperation is a new target for therapeutic- and vaccine- development strategies. Joint work with P.Skums and Yu. Khudyakov (CDC). Our model is in a sense simpler than old one. In the speaker's opinion it is a good example to discuss what Math./Theor. Biology is and what it should be. Such (short) discussion is expected. NO KNOWLEDGE of Biology is expected to understand this talk.

The Kelmans-Seymour conjecture I: Special Separations

Series
Graph Theory Seminar
Time
Wednesday, January 27, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Yan Wang – Math, GT
Seymour and, independently, Kelmans conjectured in the 1970s that every 5-connected nonplanar graph contains a subdivision of K_5. This conjecture was proved by Ma and Yu for graphs containing K_4^-, and an important step in their proof is to deal with a 5-separation in the graph with a planar side. In order to establish the Kelmans-Seymour conjecture for all graphs, we need to consider 5-separations and 6-separations with less restrictive structures. We will talk about special 5-separations and 6-separations, including those with an apex side. Results will be used in subsequently to prove the Kelmans-Seymour conjecture.

One Bit Sensing, RIP bounds and Empirical Processes

Series
Analysis Seminar
Time
Wednesday, January 27, 2016 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Michael Lacey – Gatech
A signal is a high dimensional vector x, and a measurement is the inner product . A one-bit measurement is the sign of . These are basic objects, as will be explained in the talk, with the help of some videos of photons. The import of this talk is that one bit measurements can be as effective as the measurements themselves, in that the same number of measurements in linear and one bit cases ensure the RIP property. This is explained by a connection with variants of classical spherical cap discrepancy. Joint work with Dimtriy Bilyk.

Birkhoff conjecture and spectral rigidity of planar convex domains.

Series
Job Candidate Talk
Time
Wednesday, January 27, 2016 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jacopo De Simoi – Paris Diderot University –
Dynamical billiards constitute a very natural class of Hamiltonian systems: in 1927 George Birkhoff conjectured that, among all billiards inside smooth planar convex domains, only billiards in ellipses are integrable. In this talk we will prove a version of this conjecture for convex domains that are sufficiently close to an ellipse of small eccentricity. We will also describe some remarkable relation with inverse spectral theory and spectral rigidity of planar convex domains. Our techniques can in fact be fruitfully adapted to prove spectral rigidity among generic (finitely) smooth axially symmetric domains which are sufficiently close to a circle. This gives a partial answer to a question by P. Sarnak.

Pages