Georgia Institute of Technology College of Sciences

Khovanov homology is a powerful and computable homology theory for links which extends to tangles and tangle cobordisms. It is closely, but perhaps mysteriously, related to many flavors of Floer homology. Szabó has constructed a combinatorial spectral sequence from Khovanov homology which (conjecturally) converges to a Heegaard Floer-theoretic object. We will discuss work in progress to extend Szabó’s construction to an invariant of tangles and surfaces in the four-sphere.

Many real-world problems reduce to optimization problems that are solved by iterative methods. In this talk, I focus on recently developed efficient algorithms for solving large-scale optimization problems that arises in medical imaging and image processing. In the first part of my talk, I will introduce the Bregman Operator Splitting with Variable Stepsize (BOSVS) algorithm for solving nonsmooth inverse problems. The proposed algorithm is designed to handle applications where the matrix in the fidelity term is large, dense, and ill-conditioned. Numerical results are provided using test problems from parallel magnetic resonance imaging. In the second part, I will focus on the Euler's Elastica-based model which is non-smooth and non-convex, and involves high-order derivatives. I introduce two efficient alternating minimization methods based on operator splitting and alternating direction method of multipliers, where subproblems can be solved efficiently by Fourier transforms and shrinkage operators. I present the analytical properties of each algorithm, as well as several numerical experiments on image inpainting problems, including comparison with some existing state-of-art methods to show the efficiency and the effectiveness of the proposed methods.

The discrete truncated moment problem considers the question whether given a discrete subsets $K \subset \mathbb{R}$ and a sequence of real numbers one can find a measure supported on $K$ whose (power) moments are exactly these numbers. The truncated moment is a challenging problem. We derive a minimal set of necessary and sufficient conditions. This simple problem is surprisingly hard and not treatable with known techniques. Applications to the truncated moment problem for point processes, the so-called relizability or representability problem are given. The relevance of this problem for statistical mechanics in particular the theory of classic liquids, is explained. This is a joint work with M. Infusino, J. Lebowitz and E. Speer.

In this talk, I will refine the concept of the symmetry group of a geometric object through its symmetry groupoid, which incorporates both global and local symmetries in a common framework. The symmetry groupoid is related to the weighted differential invariant signature of a submanifold, that is introduced to capture its fine grain equivalence and symmetry properties. The groupoid/signature approach will be connected to recent developments in signature-based recognition and symmetry detection of objects in digital images, including jigsaw puzzle assembly.

In this introductory talk we present some basic results in ergodic theory, due to Poincare, von Neumann, and Birkhoff. We will also discuss many examples of dynamical systems where the theory can be applied. As motivation for a broad audience, we will go over the connection of the theory with some classical problems in statistical mechanics (part 3 of 3).

Concentration inequalities are fundamental tools in probabilistic combinatorics and theoretical computer science for proving that functions of random variables are typically near their means. Of particular importance is the case where f(X) is a function of independent random variables X=(X_1,...,X_n). Here the well-known bounded differences inequality (also called McDiarmid's or Hoeffding--Azuma inequality) establishes sharp concentration if the function f does not depend too much on any of the variables. One attractive feature is that it relies on a very simple Lipschitz condition (L): it suffices to show that |f(X)-f(X')| \leq c_k whenever X,X' differ only in X_k. While this is easy to check, the main disadvantage is that it considers worst-case changes c_k, which often makes the resulting bounds too weak to be useful. In this talk we discuss a variant of the bounded differences inequality which can be used to establish concentration of functions f(X) where (i) the typical changes are small although (ii) the worst case changes might be very large. One key aspect of this inequality is that it relies on a simple condition that (a) is easy to check and (b) coincides with heuristic considerations as to why concentration should hold. Indeed, given a `good' event G that holds with very high probability, we essentially relax the Lipschitz condition (L) to situations where G occurs. The point is that the resulting typical changes c_k are often much smaller than the worst case ones. If time permits, we shall illustrate its application by considering the reverse H-free process, where H is 2-balanced. We prove that the final number of edges in this process is concentrated, and also determine its likely value up to constant factors. This answers a question of Bollobás and Erdös.

Abstract: Abstract: Let p(x,xi) be a complex-valued polynomial of degree 2 on R^{2n}, and let P be the corresponding non-self-adjoint Weyl quantization. We will discuss some results on the relationship between the classical Hamilton flow exp(H_p) and the L^2 operator theory for the Schrödinger evolution exp(-iP), under a positivity condition of Melin and Sjöstrand.

It was shown by V.E. Zakharov that the equations for water waves can be posed as a Hamiltonian PDE, and that the equilibrium solution is an elliptic stationary point. This talk will discuss two aspects of the water wave equations in this context. Firstly, we generalize the formulation of Zakharov to include overturning wave profiles, answering a question posed by T. Nishida. Secondly, we will discuss the question of Birkhoff normal forms for the water waves equations in the setting of spatially periodic solutions, including the function space mapping properties of these transformations. This latter is joint work with C. Sulem.

A matroid is a combinatorial abstraction of an independence structure, such as linear independence among vectors and cycle-free-ness among edges of a graph. An algebraic variety is a solution set of a system of polynomial equations, and it has a polyhedral shadow called a tropical variety. An irreducible algebraic variety gives rise to a matroid via algebraic independence in its coordinate ring. In this talk I will show that the tropical variety is compatible with the algebraic matroid structure. I will also discuss some open problems on algebraic matroids and how they behave under operations on tropical varieties.

1. One day before the election, the statistics site 538 predicted a 70% chance of a Clinton victory. How do we judge the quality of probabilistic prediction models? Ultimately every quant finance model has a probabilistic prediction model at its core, for instance the geometric Brownian Motion is the core of Black-Scholes. I will explain the Basel Traffic Ligths Framework and then I'll ask the audience to think how the framework can be extended. 2. Multi-factor local volatility. I will explain Dupire's local volatility model and ask how this model can be extended to a multi-factor framework. 3. Model overfitting. There are objective criteria for statistical model overfitting, such as AIC. Such criteria don't exist for risk-neutral derivatives pricing models.

We will review the definition of the Chekanov-Eliashberg differentialgraded algebra for Legendrian knots in R^3 and look at examples tounderstand a few of the invariants that come from Legendrian contacthomology.

The classic 1935 paper of Erdos and Szekeres entitled ``A combinatorial problem in geometry" was a starting point of a very rich discipline within combinatorics: Ramsey theory. In that paper, Erdos and Szekeres studied the following geometric problem. For every integer n \geq 3, determine the smallest integer ES(n) such that any set of ES(n) points in the plane in general position contains n members in convex position, that is, n points that form the vertex set of a convex polygon. Their main result showed that ES(n) \leq {2n - 4\choose n-2} + 1 = 4^{n -o(n)}. In 1960, they showed that ES(n) \geq 2^{n-2} + 1 and conjectured this to be optimal. Despite the efforts of many researchers, no improvement in the order of magnitude has been made on the upper bound over the last 81 years. In this talk, we will sketch a proof showing that ES(n) =2^{n +o(n)}.

Andew Suk will discuss some of the techincal details in his colloquium talk about the Erdos-Szekeres convex polygon problem. This is mainly an informal discussion.

This is an information session about research opportunities related to GT MAP activities. If you are a math graduate student, please join for free pizza as well.

At the intersection of computability and algebraic geometry, the following question arises: does an integral polynomial system of equations have any integral solutions? Famously, the combined work of Robinson, Davis, Putnam, and Matiyasevich answers this in the negative. Nonetheless, algorithms have played in increasing role in the development of algebraic geometry and its many applications. I address some research related to this general theme and some outstanding questions.