Georgia Institute of Technology College of Sciences

The topological entropy of a subshift is the exponential growth rate of the number of words of different lengths in its language. For subshifts of entropy zero, finer growth invariants constrain their dynamical properties. In this talk we will survey how the complexity of a subshift affects properties of the ergodic measures it carries. In particular, we will see some recent results (joint with B. Kra) relating the word complexity of a subshift to its set of ergodic measures as well as some applications.

The Bergman fan is a tropical linear space with trivial valuations describing a matroid combinatorially as it corresponds to a matroid. In this talk, based on a plenty of examples, we study the definition of the Bergman fan and their subdivisions. The talk will be closed with the recent result of the Maclagan-Yu's paper (https://arxiv.org/abs/1908.05988) that the fine subdivision of the Bergman fan of any matroid is r-1 connected where r is the rank of the matroid.

The talk is concerned with low multilinear rank approximations to antisymmetric tensors, that is, multivariate arrays for which the entries change sign when permuting pairs of indices. Such tensors play a major role in quantum chemistry. We show which ranks can be attained by an antisymmetric tensor and discuss the adaption of existing approximation algorithms to preserve antisymmetry, most notably a Jacobi-type algorithm. Particular attention is paid to the special case when choosing the rank equal to the order of the tensor. It is shown that this case can be addressed with an unstructured rank-1 approximation. This allows for the straightforward application of the higher-order power method, for which we discuss effective initialization strategies. This is a joint work with Daniel Kressner (EPFL).

I will outline the construction of some knot concordance invariants based on the Heegaard Floer homology of double branched coverings. The construction builds on some ideas developed by Hendricks, Manolescu, Hom and Lidman. This is joint work with Andras Stipsicz, and Sungkyung Kang.

Surfaces are some of the most basic examples of spaces. Although topologists have studied surfaces for a long time, they continue to fascinate. I'll give an overview of the study of surfaces over the past 150 years by highlighting work of seven mathematicians. We'll discuss the classification of surfaces, and we'll also discuss mapping class groups, which are collections of symmetries of surfaces. I'll also give the flavor of four of my own research projects about surfaces, one for each of four broad mathematical areas: group theory, geometry, topology, and dynamics.

We show that the dynamics of nonlinear dynamical systems with many degrees of freedom (possibly infinitely many) can be similar to that of ordered system in a surprising fashion. To this aim, in the literature one typically uses techniques from perturbation theory, such as KAM theorem or Nekhoroshev theorem. Unfortunately they are known to be ill-suited for obtaining results in the case of many degrees of freedom. We present here a probabilistic approach, in which we focus on some observables of physical interest (obtained by averaging on the probability distribution on initial data) and for several models we get results of stability on long times similar to Nekhoroshev estimates. We present the example of a nonlinear chain of particles with alternating masses, an hyper-simplified model of diatomic solid. In this case, which is similar to the celebrated Fermi-Pasta-Ulam model and is widely studied in the literature, we show the progress with respect to previous results, and in particular how the present approach permits to obtain theorems valid in the thermodynamic limit, as this is of great relevance for physical implications.

Building on previous work of Rozansky and Willis, we generalise Rasmussen’s s-invariant to connected sums of $S^1 \times S^2$. Such an invariant can be computed by approximating the Khovanov-Lee complex of a link in $\#^r S^1 \times S^2$ with that of appropriate links in $S^3$. We use the approximation result to compute the s-invariant of a family of links in $S^3$ which seems otherwise inaccessible, and use this computation to deduce an adjunction inequality for null-homologous surfaces in a (punctured) connected sum of $\bar{CP^2}$. This inequality has several consequences: first, the s-invariant of a knot in the three-sphere does not increase under the operation of adding a null-homologous full twist. Second, the s-invariant cannot be used to distinguish $S^4$ from homotopy 4-spheres obtained by Gluck twist on $S^4$. We also prove a connected sum formula for the s-invariant, improving a previous result of Beliakova and Wehrli. We define two s-invariants for links in $\#^r S^1 \times S^2$. One of them gives a lower bound to the slice genus in $\natural^r S^1 \times B^3$ and the other one to the slice genus in $\natural^r D^2 \times S^2$ . Lastly, we give a combinatorial proof of the slice Bennequin inequality in $\#^r S^1 \times S^2$.

In this talk I will give an introduction to certain aspects of geometric Littlewood-Paley theory, which is an area of harmonic analysis concerned with deriving regularity properties of sets and measures from the analytic behavior of associated operators. The work we shall describe has been carried out in collaboration with Fedor Nazarov, Maria Carmen Reguera, Xavier Tolsa, and Michele Villa.

Function classes are collections of Boolean functions on a finite set. Recently, a method of studying function classes via commutative algebra, by associating a squarefree monomial ideal to a function class, was introduced by Yang. I will describe this connection, as well as some free resolutions and Betti numbers for these ideals for an interesting collection of function classes, corresponding to intersection-closed posets. This is joint work with Chris Eur, Greg Yang, and Mengyuan Zhang.

We consider a physical periodic Ehrenfests' Wind-Tree model where a moving particle is a hard ball rather than (mathematical) point particle. Some dynamics and statistical properties of this model are studied. Moreover, it is shown that it has a new superdiffusive regime where the diffusion coefficient $D(t)\sim(\ln t)^2$ of dynamics seems to be never observed before in any model.

In this talk I'll first give an background overview of Bourgain's approach to prove the invariance of the Gibbs measure for the periodic cubic nonlinear Schrodinger equation in 2D and of the para-controlled calculus of Gubinelli-Imkeller and Perkowski in the context of parabolic stochastic equations. I will then present our resolution of the long-standing problem of proving almost sure global well-posedness (i.e. existence /with uniqueness/) for the periodic nonlinear Schrödinger equation (NLS) in 2D on the support of the Gibbs measure, for any (defocusing and renormalized) odd power nonlinearity. Consequently we get the invariance of the Gibbs measure. This is achieved by a new method we call /random averaging operators /which precisely captures the intrinsic randomness structure of the problematic high-low frequency interactions at the heart of this problem. This is work with Yu Deng (USC) and Haitian Yue (USC).

In both biological brains and artificial neural networks, the representational geometry - the shape and distribution of activity - at different layers in an artificial network or across different populations of neurons in the brain, can reveal important signatures of the underlying computations taking place. In this talk, I will describe how we are developing strategies for comparing and aligning neural representations, using a combination of tools from computational geometry and optimal transport.

Determining when two objects have “the same shape” is difficult; this difficulty depends on the dimension we are working in. While many of the same techniques work to study things in dimensions 5 and higher, we can better understand dimensions 1, 2, and 3 using other methods. We can think of 4-dimensional space as the “bridge” between low-dimensional behavior and high-dimensional behavior.

One way to understand the possibilities in each dimension is to examine objects called cobordisms: if an (n+1)-dimensional space has an ``edge,” which is called a boundary, then that boundary is itself an n-dimensional space. We say that two n-dimensional spaces are cobordant if together they form the boundary of an (n+1)-dimensional space. Using the idea of spaces related by cobordism, we can form an algebraic structure called a group. In this way, we can attempt to understand higher dimensions using clues from lower dimensions.

In this talk, I will discuss different types of cobordism groups and how to study them using tools from a broad range of mathematical areas.

I will discuss the prime decomposition of three-manifolds. First I will define the connect sum operation, irreducible and prime 3-manifolds. Then using the connect sum operation as "multiplication," I will show any closed oriented three-manifold decomposes uniquely into prime factors using spheres. If time permits, I will show another way of decomposing using discs.

The condition number of a matrix A is the quantity κ(A) = smax(A)/smin(A), where smax(A), smin(A) are the largest and smallest singular values of A, respectively. Let A be a random n × n matrix with i.i.d, mean zero, unit variance, subgaussian entries. We will discuss a result by Litvak, Tikhomirov and Tomczak-Jaegermann which states that, in this setting, the condition number satisfies the small ball probability estimate

P{κ(A) ≤ n/t} ≤ 2 exp(−ct^2), t ≥ 1, where c > 0 is a constant depending only on the subgaussian moment.

A sequence A of positive integers is r-Ramsey complete if for every r-coloring of A, every sufficiently large integer can be written as a sum of the elements of a monochromatic subsequence. Burr and Erdos proposed several open problems in 1985 on how sparse can an r-Ramsey complete sequence be and which polynomial sequences are r-Ramsey complete. Erdos later offered cash prizes for two of these problems. We prove a result which solves the problems of Burr and Erdos on Ramsey complete sequences. The proof uses tools from probability, combinatorics, and number theory. 

Joint work with David Conlon.

A sequence A of positive integers is r-Ramsey complete if for every r-coloring of A, every sufficiently large integer can be written as a sum of the elements of a monochromatic subsequence. Burr and Erdos proposed several open problems in 1985 on how sparse can an r-Ramsey complete sequence be and which polynomial sequences are r-Ramsey complete. Erdos later offered cash prizes for two of these problems. We prove a result which solves the problems of Burr and Erdos on Ramsey complete sequences. The proof uses tools from probability, combinatorics, and number theory. 

Joint work with David Conlon.

Fictitious play is one of the simplest and most natural dynamics for two-player zero-sum games. Originally proposed by Brown in 1949, the fictitious play dynamic has each player simultaneously best-respond to the distribution of historical plays of their opponent. In 1951, Robinson showed that fictitious play converges to the Nash Equilibrium, albeit at an exponentially-slow rate, and in 1959, Karlin conjectured that the true convergence rate of fictitious play after k iterations is O(k^{-1/2}), a rate which is achieved by similar algorithms and is consistent with empirical observations. Somewhat surprisingly, Daskalakis and Pan disproved a version of this conjecture in 2014, showing that an exponentially-slow rate can occur, although their result relied on adversarial tie-breaking. In this talk, we show that Karlin’s conjecture holds if ties are broken lexicographically and the game matrix is diagonal. We also show a matching lower bound under this tie-breaking assumption. This is joint work with Jake Abernethy and Andre Wibisono.