Georgia Institute of Technology College of Sciences

Recent research trends have explored curious analogies between the theory of graphs and Riemann surfaces. To each graph we can associate a real metric torus, known as its Jacobian. It was previously known that isomorphisms of graph Jacobians yield isomorphisms of the associated graphic matroids, partially mirroring a classical algebraic geometry result known as the Torelli theorem. However, the result on graphs is not as strong as a direct analogue of the Riemann surface result would be, nor does it use as much data. I will discuss how the graph Torelli theorem can be refined to incorporate additional data as with Riemann surfaces, in which case it produces isomorphisms of graphs. If time permits, I will describe further recent work in this direction.

We will start by introducing the Teichmüller space of a surface, which parametrizes the possible conformal structures it supports. By defining this space analytically, we can equip it with the Lp metrics, of which the Teichmüller and Weil-Petersson metrics are special cases. We will discuss the incompleteness of the Lp metrics on Teichmüller space and what we know about their completions.

We consider the Bayesian approach to the linear Gaussian inference problem of inferring the initial condition of a linear dynamical system from noisy output measurements taken after the initial time. In practical applications, the large dimension of the dynamical system state poses a computational obstacle to computing the exact posterior distribution. Model reduction offers a variety of computational tools that seek to reduce this computational burden. In particular, balanced truncation is a control-theoretic approach to model reduction which obtains an efficient reduced-dimension dynamical system by projecting the system operators onto state directions which trade off the reachability and observability of state directions.  We define an analogous balanced truncation procedure for the Bayesian inference setting based on the trade off between prior uncertainty and data information. The resulting reduced model inherits desirable theoretical properties for both the control and inference settings: numerical demonstrations on two benchmark problems show that our method can yield near-optimal posterior covariance approximations with order-of-magnitude state dimension reduction.

We consider a mechanical system consisting of a rotator and a pendulum, subject to a small, conformally symplectic perturbation. The resulting system has energy dissipation. We provide explicit conditions on the dissipation parameter, so that the resulting system exhibits Arnold diffusion. More precisely, we show that there are diffusing orbits along which the energy of the rotator grows by an amount independent of the smallness parameter. The fact that Arnold diffusion may play a role  in  systems with small dissipation was conjectured by Chirikov. Our system can be viewed as a simplified  model for an energy harvesting device, in which context the energy growth translates into generation of electricity.
Joint work with S.W. Akingbade and T-M. Seara.

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.

It is known that if $\{x_n\}$ is a frame for a separable Hilbert space, then there exist some sequences $\{y_n\}$ such that $x= \sum x_n$, and this sum converges in the norm of H. This equation is called the reconstruction formula of x. In this talk, we will talk about the existence of frames that admit absolutely convergent and unconditionally convergent reconstruction formula. Some characterizations of such frames will also be presented. Finally, we will present an extension of this problem about the unconditional convergence of Gabor expansion in Modulation spaces.

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.

We discuss some recent results in graph coloring that show somewhat stronger conclusions in a similar parameter range to traditional coloring theorems. We consider the standard setup of list coloring but ask for a decomposition of the lists into pairwise-disjoint list colorings. The area is new and we discuss many open problems.

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.