Georgia Institute of Technology College of Sciences

Accelerated gradient methods play a central role in optimization, achieving the optimal convergence rates in many settings. While many extensions of Nesterov's original acceleration method have been proposed, it is not yet clear what is the natural scope of the acceleration concept. In this work, we study accelerated methods from a continuous-time perspective. We show there is a Bregman Lagrangian functional that generates a large class of accelerated methods in continuous time, including (but not limited to) accelerated gradient descent, its non-Euclidean extension, and accelerated higher-order gradient methods. We show that in continuous time, these accelerated methods correspond to traveling the same curve in spacetime at different speeds. This is in contrast to the family of rescaled gradient flows, which correspond to changing the distance in space. We show how to implement both the rescaled and accelerated gradient methods as algorithms in discrete time with matching convergence rates. These algorithms achieve faster convergence rates for convex optimization under higher-order smoothness assumptions. We will also discuss lower bounds and some open questions. Joint work with Ashia Wilson and Michael Jordan.

I'll describe a way to construct an A-infinity category associated to a contact manifold, analogous to a Fukaya category for a symplectic manifold. The objects of this category are Legendrian submanifolds equipped with augmentations. Currently we're focusing on standard contact R^3 but we're hopeful that we can extend this to other contact manifolds. I'll discuss some properties of this contact Fukaya category, including generation by unknots and a potential application to proving that ``augmentations = sheaves''. This is joint work in progress with Tobias Ekholm and Vivek Shende.

The knot group has played a central role in classical knot theory and has many nice properties, some of which fail in interesting ways for knotted surfaces. In this talk we'll introduce an invariant of knotted surfaces called ribbon genus, which measures the failure of a knot group to 'look like' a classical knot group. We will show that ribbon genus is equivalent to a property of the group called Wirtinger deficiency. Then we will investigate some examples and conclude by proving a connection with the second homology of the knot group.

After briefly describing my research interests, I’ll speak on two results that involve points moving around on surfaces. The first result shows how to “hear the shape of a billiard table.” A point bouncing around a polygon encodes a sequence of edges. We show how to recover geometric information about the table from the collection of all such bounce sequences. This is joint work with Calderon, Coles, Davis, and Oliveira. The second result answers the question, “Given n distinct points in a closed ball, when can a new point be added in a continuous fashion?” We answer this question for all values of n and for all dimensions. Our results generalize the Brouwer fixed point theorem, which gives a negative answer when n=1. This is joint work with Chen and Gadish.

The n-dimensional L^p Brunn-Minkowski inequality for p<1 , in particular the log-Brunn-Minkowski inequality, is proposed by Boroczky-Lutwak-Yang-Zhang in 2013, based on previous work of Firey and Lutwak . When it came out, it promptly became the major problem in convex geometry. Although some partial results on some specific convex sets are shown to be true, the general case stays wide open. In this talk I will present a breakthrough on this conjecture due to E. Milman and A Kolesnikov, where we can obeserve a beautiful interaction of PDE, operator theory, Riemannian geometry and all sorts of best constant estimates. They showed the validity of the local version of this inequality for orgin-symmtric convex sets with a C^{2} smooth boundary and strictly postive mean curvature, and for p between 1-c/(n^{3/2}) and 1. Their infinitesimal formulation of this inequality reveals the deep connection with the poincare-type inequalities. It turns out, after a sophisticated transformation, the desired inequality follows from an estimate of the universal constant in Poincare inequality on convex sets.

We prove a criterion for nondoubling parabolic measure to satisfy a weak reverse H¨older inequality on a domain with time-backwards ADR boundary, following a result of Bennewitz-Lewis for nondoubling harmonic measure.

Many problems in algebraic geometry involve counting solutions to geometric problems. The number of intersection points of two projective planar curves and the number of lines on a cubic surface are two classical problems in this enumerative geometry. Using A1-homotopy theory, one can gain new insights to old enumerative problems. We will outline some results in A1-enumerative geometry, including the speaker’s current work on Bézout’s Theorem.

We generalize the Lp spectral cluster bounds of Sogge for the Laplace-Beltrami operator on compact Riemannian manifolds to systems of orthonormal functions. We show that these bounds are optimal on any manifold in a very strong sense. These spectral cluster bounds follow from Schatten-type bounds on oscillatory integral operators and their optimality follows by semi-classical analysis.

Erdős and Nešetřil conjectured in 1985 that every graph with maximum degree Δ can be strong edge colored using at most (5/4)Δ^2 colors. In this talk we discuss recent progress made in the case of Δ=4, and go through the method used to improve the upper bound to 21 colors, one away from the conjectured 20.

We discuss the construction of spectral sequences and some of their applications to algebraic geometry, including the classic Leray spectral sequence. We will draw a lot of diagrams and try to avoid doing anything involving lots of indices for a portion of the talk.

Abstract: Queueing systems are studied in various asymptotic regimes because they are hard to study in general. One popular regime of study is the heavy-traffic regime, when the system is loaded very close to its capacity. Heavy-traffic behavior of queueing systems is traditionally studied using fluid and diffusion limits. In this talk, I will present a recently developed method called the 'Drift Method', which is much simpler, and is based on studying the drift of certain test functions. In addition to exactly characterizing the heavy-traffic behavior, the drift method can be used to obtain lower and upper bounds for all loads. In this talk, I will present the drift method, and its successful application in the context of data center networks to resolve a decade-old conjecture. I will also talk about ongoing work and some open problems. Bio: Siva Theja Maguluri is an Assistant Professor in the School of Industrial and Systems Engineering at Georgia Tech. Before that, he was a Research Staff Member in the Mathematical Sciences Department at IBM T. J. Watson Research Center. He obtained his Ph.D. from the University of Illinois at Urbana-Champaign in Electrical and Computer Engineering where he worked on resource allocation algorithms for cloud computing and wireless networks. Earlier, he received an MS in ECE and an MS in Applied Math from UIUC and a B.Tech in Electrical Engineering from IIT Madras. His research interests include Stochastic Processes, Optimization, Cloud Computing, Data Centers, Resource Allocation and Scheduling Algorithms, Networks, and Game Theory. The current talk is based on a paper that received the best publication in applied probability award, presented by INFORMS Applied probability society.

In this talk, we introduce rather exotic algebraic structures called hyperrings and hyperfields. We first review the basic definitions and examples of hyperrings, and illustrate how hyperfields can be employed in algebraic geometry to show that certain topological spaces (underlying topological spaces of schemes, Berkovich analytification of schemes, and real schemes) are homeomorphic to sets of rational points of schemes over hyperfields.

In this talk I will present a proof of a generalization of a theorem by Siegel, about the existence of an analytic conjugation between an analytic map, $f(z)=\Lambda z +\hat{f}(z)$, and a linear map, $\Lambda z$, in $\mathbb{C}^n$. This proof illustrates a standar technique used to deal with small divisors problems. I will be following the work of E. Zehnder. This is a continuation of last week talk.