Georgia Institute of Technology College of Sciences

Let P be a set of points and L be a set of lines in the plane, what can we say about the number of incidences between P and L,    I(P, L):= |\{ (p, l)\in P\times L, p\in L\}| ?

The problem changes drastically when we consider a thickening version, i.e. when P is a set of unit balls and L is a set of tubes of radius 1. Furstenberg set conjecture can be viewed as an incidence problem for tubes. It states that a set containing an s-dim subset of a line in every direction should have dimension at least  (3s+1)/2 when s>0. 

We will survey a sequence of results by Orponen, Shmerkin and a recent joint work with Ren that leads to the solution of this conjecture

A major question in dynamical systems is to understand the mechanisms driving global instability in the 3 Body Problem (3BP), which models the motion of three bodies under Newtonian gravitational interaction. The 3BP is called restricted if one of the bodies has zero mass and the other two, the primaries, have strictly positive masses $m_0, m_1$. In the region of the phase space where the massless body is far from the primaries, the problem can be studied as a (fast) periodic perturbation of the 2 Body Problem (2BP), which is integrable.

We prove that the restricted 3BP exhibits topological instability: for any values of the masses $m_0, m_1$ (except $m_0 = m_1$), we build orbits along which the angular momentum of the massless body (conserved along the flow of the 2BP) experiences an arbitrarily large variation. In order to prove this result we show that a degenerate Arnold diffusion mechanism takes place in the restricted 3BP. Our work extends previous results by Delshams, Kaloshin, De la Rosa and Seara for the a priori unstable case $m_1< 0$, where the model displays features of the so-called a priori stable setting. This is joint work with Marcel Guardia and Tere Seara.

Abstract: Motivated by applications in Reinforcement Learning (RL), this talk focuses on the Stochastic Appproximation (SA) method to find fixed points of a contractive operator. First proposed by Robins and Monro, SA is a popular approach for solving fixed point equations when the information is corrupted by noise. We consider the SA algorithm for operators that are contractive under arbitrary norms (especially the l-infinity norm). We present finite sample bounds on the mean square error, which are established using a Lyapunov framework based on infimal convolution and generalized Moreau envelope. We then present our more recent result on concentration of the tail error, even when the iterates are not bounded by a constant. These tail bounds are obtained using exponential supermartingales in conjunction with the Moreau envelop and a novel bootstrapping approach. Our results immediately imply the state-of-the-art sample complexity results for a large class of RL algorithms.

Bio: Siva Theja Maguluri is Fouts Family Early Career Professor and Associate Professor in the H. Milton Stewart School of Industrial and Systems Engineering at Georgia Tech. He obtained his Ph.D. and MS in ECE as well as MS in Applied Math from UIUC, and B.Tech in Electrical Engineering from IIT Madras. His research interests span the areas of Control, Optimization, Algorithms and Applied Probability and include Reinforcement Learning theory and Stochastic Networks. His research and teaching are recognized through several awards including the  Best Publication in Applied Probability award, NSF CAREER award, second place award at INFORMS JFIG best paper competition, Student best paper award at IFIP Performance, CTL/BP Junior Faculty Teaching Excellence Award, and Student Recognition of Excellence in Teaching: Class of 1934 CIOS Award.

Fox et al introduced the model of c-closed graphs, a distribution-free model motivated by one of the most universal signatures of social networks, triadic closure. Even though enumerating maximal cliques in an arbitrary network can run in exponential time, it is known that for c-closed graph, enumerating maximal cliques and maximal complete bipartite graphs is always fast, i.e., with complexity being polynomial in the number of vertices in the network. In this work, we investigate further by enumerating maximal blow-ups of an arbitrary graph H in c-closed graphs. We prove that given any finite graph H, the number of maximal blow-ups of H in any c-closed graph on n vertices is always at most polynomial in n. When considering maximal induced blow-ups of a finite graph H, we provide a characterization of graphs H when the bound is always polynomial in n. A similar general theorem is also proved when H is infinite.