Georgia Institute of Technology College of Sciences

The SL2 skein algebra of a surface is built from diagrams of curves on the surface. To multiply two diagrams, we draw one diagram on top of the other and then resolve the crossings with the Kauffman bracket. If we replace SL2 with another quantum group, we replace curves by embedded graphs on the surface. Recently, Thang Le showed that the SL2 skein algebra has a nice decomposition into simpler algebras whenever the surface has an ideal triangulation. This triangular decomposition is a powerful tool and should help us to study other skein algebras if we are able to show that the necessary ingredients exist. In this talk, I will explain what these ingredients are and how to find them for the SL3 skein algebra of trivalent webs on a surface.

Our mantra throughout the talk will be simple, "Train tracks approximate simple closed curves." Our goal will be to explore some examples of train tracks, draw some meaningful pictures, and develop an analogy between train tracks and another well known method of approximation. No great knowledge of anything is required for this talk as long as one is willing to squint their eyes at their computer's screen a bit at times.

Whitney proved in 1931 that 4-connected planar triangulations are Hamiltonian. Hakimi, Schmeichel, and Thomassen conjectured in 1979 that if $G$ is a 4-connected planar triangulation with $n$ vertices, then $G$ contains at least $2(n-2)(n-4)$ Hamiltonian cycles, with equality if and only if $G$ is a double wheel. On the other hand, a recent result of Alahmadi, Aldred, and Thomassen states that there are exponentially many Hamiltonian cycles in 5-connected planar triangulations. In this paper, we consider 4-connected planar $n$-vertex triangulations $G$ that do not have too many separating 4-cycles or have minimum degree 5. We show that if $G$ has $O(n/\log_2 n)$ separating 4-cycles then $G$ has $\Omega(n^2)$ Hamiltonian cycles, and if $\delta(G) \ge 5$ then $G$ has $2^{\Omega(n^{1/4})}$ Hamiltonian cycles. Both results improve previous work. Moreover, the proofs involve a "double wheel" structure, providing further evidence to the above conjecture.

Poincare Conjecture, undoubtedly, is the most influential and challenging problem in the world of Geometry and Topology. Over a century, it has left it’s mark on developing the rich theory around it. In this talk I will give a brief history of the development of Topology and then I will focus on the Exotic behavior of manifolds. In the last part of the talk, I will concentrate more on the theory of 4-manifolds.

Traditionally, the sandpile group is defined on a graph and the Matrix-Tree Theorem says that this group's size is equal to the number of spanning trees. An extension of the Matrix-Tree Theorem gives a relationship between the sandpile group and bases of an arithmetic matroid. I provide a family of combinatorially meaningful maps between these two sets.  This generalizes a bijection given by Backman, Baker, and Yuen and extends work by Duval, Klivans, and Martin.

Traditionally, the sandpile group is defined on a graph and the Matrix-Tree Theorem says that this group's size is equal to the number of spanning trees. An extension of the Matrix-Tree Theorem gives a relationship between the sandpile group and bases of an arithmetic matroid. I provide a family of combinatorially meaningful maps between these two sets.  This generalizes a bijection given by Backman, Baker, and Yuen and extends work by Duval, Klivans, and Martin.

Please note the unusual time/day.

A finite connected acyclic graph is called a tree. Both properties - connectivity and being acyclic - make very good sense in higher dimensions as well. This has led Gil Kalai (1983) to define the notion of a $d$-dimensional hypertree for $d > 1$. The study of hypertrees is an exciting area of research, and I will try to give you a taste of the many open questions and what we know (and do not know) about them. No specific prior background is assumed.

The talk is based on several papers. The list of coauthors on these papers includes Roy Meshulam, Mishael Rosenthal, Yuval Peled, Lior Aronshtam, Tomsz Luczak, Amir Dahari, Ilan Newman and Yuri Rabinovich.

This talk will introduce a precise high-dimensional asymptotic theory for Boosting (AdaBoost) on separable data, taking both statistical and computational perspectives. We will consider the common modern setting where the number of features p and the sample size n are both large and comparable, and in particular, look at scenarios where the data is asymptotically separable. Under a class of statistical models, we will provide an (asymptotically) exact analysis of the generalization error of AdaBoost, when the algorithm interpolates the training data and maximizes an empirical L1 margin. On the computational front, we will provide a sharp analysis of the stopping time when boosting approximately maximizes the empirical L1 margin. Our theory provides several insights into properties of Boosting; for instance, the larger the dimensionality ratio p/n, the faster the optimization reaches interpolation. At the heart of our theory lies an in-depth study of the maximum L1-margin, which can be accurately described by a new system of non-linear equations; we analyze this margin and the properties of this system, using Gaussian comparison techniques and a novel uniform deviation argument. Time permitting, I will present analogous results for a new class of boosting algorithms that correspond to Lq geometry, for q>1. This is based on joint work with Tengyuan Liang.

The goal of this talk is to present a summary of Sam Payne's 2009 paper "Analytification is the limit of all tropicalizations" (Math. Res. Lett. 16, no. 3 543–556). We will introduce Berkovich analytic spaces, tropicalization of projective varieties, and tropicalization of closed subvarieties of toric varieties, as well as the connections between these concepts. We will try to present many examples.

Note: Part I will focus on tropicalization of affine varieties and Berkovich analytic spaces, Part II will focus on tropicalization of toric varieties and discuss Sam Payne's theorem.

Learning latent structures in noisy data has been a central task in statistical and computational sciences. For applications such as ranking, matching and clustering, the structure of interest is non-convex and, furthermore, of combinatorial nature. This talk will be a gentle introduction to selected models and methods for statistical inference of such combinatorial structures. I will particularly discuss some of my recent research interests.

In many optimization problems, a feasible solution induces a multi-dimensional cost vector. For example, in load-balancing a schedule induces a load vector across the machines. In k-clustering, opening k facilities induces an assignment cost vector across the clients. In this paper we consider the following minimum norm optimization problem : given an arbitrary monotone, symmetric norm, find a solution which minimizes the norm of the induced cost-vector. This generalizes many fundamental NP-hard problems. We give a general framework to tackle the minimum norm problem, and illustrate its efficacy in load balancing and, time permitting, in the clustering setting.

In many optimization problems, a feasible solution induces a multi-dimensional cost vector. For example, in load-balancing a schedule induces a load vector across the machines. In k-clustering, opening k facilities induces an assignment cost vector across the clients. In this paper we consider the following minimum norm optimization problem : given an arbitrary monotone, symmetric norm, find a solution which minimizes the norm of the induced cost-vector. This generalizes many fundamental NP-hard problems. We give a general framework to tackle the minimum norm problem, and illustrate its efficacy in load balancing and, time permitting, in the clustering setting.

(The speaker is an ACO alum; after the lecture, the speaker will engage with the ACO students for 30-45 minutes.)