Seminars and Colloquia by Series

Induced subgraphs in graphs without linear and polynomial-size anticomplete sets

Series
Graph Theory Seminar
Time
Thursday, February 8, 2018 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Sophie Spirkl – Princeton University
The celebrated Erdos-Hajnal conjecture states that for every graph H, there is a constant c > 0 such that every graph G that does not contain H as an induced subgraph has a clique or stable set of size at least n^c, where n = |V(G)|. One approach for proving this conjecture is to prove that in every H-free graph G, there are two linear-size sets A and B such that either there are no edges between A and B, or every vertex in A is adjacent to every vertex in B. As is turns out, this is not true unless both H and its complement are trees. In the case when G contains neither H nor its complement as an induced subgraph, the conclusion above was conjectured to be true for all trees (Liebenau & Pilipczuk), and I will discuss a proof of this for a class of tree called "caterpillars". I will also talk about results and open questions for some variants, including allowing one or both of A and B to have size n^c instead of linear size, and requiring the bipartite graph between A and B to have high or low density instead of being complete or empty. In particular, our results improve the bound on the size of the largest clique or stable that must be present in a graph with no induced five-cycle. Joint work with Maria Chudnovsky, Jacob Fox, Anita Liebenau, Marcin Pilipczuk, Alex Scott, and Paul Seymour.

A constrained optimization problem for the Fourier transform

Series
Analysis Seminar
Time
Wednesday, February 7, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dominique Maldague – UC Berkeley –
Among functions $f$ majorized by indicator functions $1_E$, which functions have maximal ratio $\|\widehat{f}\|_q/|E|^{1/p}$? I will briefly describe how to establish the existence of such functions via a precompactness argument for maximizing sequences. Then for exponents $q\in(3,\infty)$ sufficiently close to even integers, we identify the maximizers and prove a quantitative stability theorem.

Dehn surgery on the figure 8 knot

Series
Geometry Topology Student Seminar
Time
Wednesday, February 7, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hyun Ki Min – GaTech
The figure 8 knot is the simplest hyperbolic knot. In the late 1970s, Thurston studied how to construct hyperbolic manifolds from ideal tetrahedra. In this talk, I present the Thurston’s theory and apply this to the figure 8 knot. It turns out that every Dehn surgery on the figure 8 knot results in a hyperbolic manifold except for 10 exceptional surgery coefficients. If time permits, I will also introduce the classification of tight contact structures on these manifolds. This is a joint work with James Conway.

Topology of the set of singularities of viscosity solutions of the Hamilton-Jacobi equation

Series
PDE Seminar
Time
Tuesday, February 6, 2018 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Albert Fathi – Georgia Tech –
This is a joint work with Piermarco Cannarsa and Wei Cheng. We study the properties of the set S of non-differentiable points of viscosity solutions of the Hamilton-Jacobi equation, for a Tonelli Hamiltonian. The main surprise is the fact that this set is locally arc connected—it is even locally contractible. This last property is far from generic in the class of semi-concave functions. We also “identify” the connected components of this set S. This work relies on the idea of Cannarsa and Cheng to use the positive Lax-Oleinik operator to construct a global propagation of singularities (without necessarily obtaining uniqueness of the propagation).

The Fast Slepian Transform

Series
Applied and Computational Mathematics Seminar
Time
Monday, February 5, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Mark A. Davenport – Georgia Institute of Technology
The discrete prolate spheroidal sequences (DPSS's) provide an efficient representation for discrete signals that are perfectly timelimited and nearly bandlimited. Due to the high computational complexity of projecting onto the DPSS basis - also known as the Slepian basis - this representation is often overlooked in favor of the fast Fourier transform (FFT). In this talk I will describe novel fast algorithms for computing approximate projections onto the leading Slepian basis elements with a complexity comparable to the FFT. I will also highlight applications of this Fast Slepian Transform in the context of compressive sensing and processing of sampled multiband signals.

Discrete stochastic Hamilton Jacobi equation

Series
CDSNS Colloquium
Time
Monday, February 5, 2018 - 11:15 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Renato Iturriaga – CIMAT
We present a discrete setting for the viscous Hamilton Jacobi equation, and prove convergence to the continuous case.

KAM for quasi-linear and fully nonlinear PDEs

Series
CDSNS Colloquium
Time
Monday, February 5, 2018 - 10:10 for 1 hour (actually 50 minutes)
Location
skiles 005
Speaker
Riccardo Montalto – University of Zurich
In this talk I will present some results concerning the existence and the stability of quasi-periodic solutions for quasi-linear and fully nonlinear PDEs. In particular, I will focus on the Water waves equation. The proof is based on a Nash-moser iterative scheme and on the reduction to constant coefficients of the linearized PDE at any approximate solution. Due to the non-local nature of the water waves equation, such a reduction procedure is achieved by using techniques from Harmonic Analysis and microlocal analysis, like Fourier integral operators and Pseudo differential operators.

Crash Course in Ergodic Theory

Series
Research Horizons Seminar
Time
Friday, February 2, 2018 - 15:53 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Leonid Bunimovich – GA Tech
Some basic problems, notions and results of the Ergodic theory will be introduced. Several examples will be discussed. It is also a preparatory talk for the next day colloquium where finite time properties of dynamical and stochastic systems will be discussed rather than traditional questions all dealing with asymptotic in time properties.

Invariant Manifolds Near L1 and L2 Points in the Restricted Three-Body Problem

Series
Dynamical Systems Working Seminar
Time
Friday, February 2, 2018 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 271
Speaker
Gladston Duarte – University of Barcelona & GT –
In a given system of coordinates, the Restricted Three-Body Problem has some interesting dynamical objects, for instance, equilibrium points, periodic orbits, etc. In this work, some connections between the stable and unstable manifolds of periodic orbits of this system are studied. Such connections let one explain the movement of Quasi-Hilda comets, which describe an orbit that sometimes can be approximated by an ellipse of semi-major axis greater than Jupiter's one, sometimes smaller. Using a computer algebra system, one can compute an approximation to those orbits and its manifolds and investigate the above mentioned connections. In addition, the Planar Circular model is used as a base for the fitting of the orbit of comet 39P/Oterma, whose data were collected from the JPL Horizons system. The possibility of using other models is also discussed.

Local Differential Privacy for Physical Sensor Data and Sparse Recovery

Series
ACO Student Seminar
Time
Friday, February 2, 2018 - 13:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Audra McMillan – Math, University of Michigan –
Physical sensors (thermal, light, motion, etc.) are becoming ubiquitous and offer important benefits to society. However, allowing sensors into our private spaces has resulted in considerable privacy concerns. Differential privacy has been developed to help alleviate these privacy concerns. In this talk, we’ll develop and define a framework for releasing physical data that preserves both utility and provides privacy. Our notion of closeness of physical data will be defined via the Earth Mover Distance and we’ll discuss the implications of this choice. Physical data, such as temperature distributions, are often only accessible to us via a linear transformation of the data. We’ll analyse the implications of our privacy definition for linear inverse problems, focusing on those that are traditionally considered to be "ill-conditioned”. We’ll then instantiate our framework with the heat kernel on graphs and discuss how the privacy parameter relates to the connectivity of the graph. Our work indicates that it is possible to produce locally private sensor measurements that both keep the exact locations of the heat sources private and permit recovery of the ``general geographic vicinity'' of the sources. Joint work with Anna C. Gilbert.

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