Georgia Institute of Technology College of Sciences

Organizational meeting.

In this talk, we consider the Cauchy problem of a modified two-component Camassa-Holm shallow water system. We first establish local well-possedness of the Cauchy problem of the system. Then we present several blow-up results of strong solutions to the system. Moreover, we show the existence of global weak solutions to the system. Finally, we address global conservative solutions to the system. This talk is based on several joint works with C. Guan, K. H. Karlsen, K. Yan and W. Tan.

Recap of generic chaining from last time and more discussion about it. Then, the lower Dudley bound (Theorem 2.1.1) and the Bernoulli upper bound (4.1.2) and statement of the Bernoulli conjecture (lower bound) will be covered from The Generic Chaining book.

We survey research spanning more than 20 years on what starts out to be a very simple problem: Representing a poset as the inclusion order of circular disks in the plane. More generally, we can speak of spherical orders, i.e., posets which are inclusion orders of balls in R^d for some d. Surprising enough, there are finite posets which are not sphere orders. Quite recently, some elegant results have been obtained for circle orders, lending more interest to the many open problems that remain.

The main goal is to characterize the dilatation of an outer automorphisms of free groups. It is known that for any automorphism, its dilatation is a weak Perron number. The converse was recently shown by Thurston; for every weak Perron number, there is an automorphism represented by a train track map. We will discuss Thurston's theorem and its proof.

The classification theorem for a C_0 operator describes its quasisimilarity class by means of its Jordan model. The purpose of this talk will be to investigate when the relation between the operator and its model can be improved to similarity. More precisely, when the minimal function of the operator T can be written as a product of inner functions satisfying the so-called (generalized) Carleson condition, we give some natural operator theoretic assumptions on T that guarantee similarity.

We will present a classical proof of the center stable manifold.

The study of random Cayley graphs of finite groups is related to the investigation of Expanders and to problems in Combinatorial Number Theory and in Information Theory. I will discuss this topic, describing the motivation and focusing on the question of estimating the chromatic number of a random Cayley graph of a given group with a prescribed number of generators. The investigation of this problem combines combinatorial, algebraic and probabilistic tools. Several intriguing questions that remain open will be mentioned as well.

The Burgers equation is a basic hydrodynamic model describing the evolution of the velocity field of sticky dust particles. When supplied with random forcing it turns into an infinite-dimensional random dynamical system that has been studied since late 1990's. The variational approach to Burgers equation allows to study the system by analyzing optimal paths in the random landscape generated by the random force potential. Therefore, this is essentially a random media problem. For a long time only compact cases of Burgers dynamics on the circle or a torus were understood well. In this talk I discuss the Burgers dynamics on the entire real line with no compactness or periodicity assumption. The main result is the description of the ergodic components and One Force One Solution principle on each component. Joint work with Eric Cator and Kostya Khanin.

I'll discuss some work on rigorous computation of invariant manifolds and computer assisted proof of the existence of transverse connecting orbits for differential equations. I'm also interested in how these computations can be used to obtain global topological data, such as the chain groups and boundary maps of Morse Theory.

Cheeger's fundamental inequality states that any edge-weighted graph has a vertex subset $S$ such that its expansion (a.k.a. conductance of $S$ or the sparsity of the cut $(S,\bar{S})$)is bounded as follows: \phi(S) = w(S,S') /w(S) \leq \sqrt{2 \lambda_2},where $w$ is the total edge weight of a subset or a cut and $\lambda_2$ is the second smallest eigenvalue of thenormalized Laplacian of the graph.We study natural generalizations of the sparsest cut in a graph: (i) a partition ofthe vertex set into $k$ parts that minimizes the sparsity of the partition (defined as the ratio of theweight of edges between parts to the total weight of edges incident to the smallest $k-1$ parts); (ii) a collection of $k$ disjoint subsets $S_1, \ldots, S_k$ that minimize $\max_{i \in [k]} \phi(S_i)$; (iii) a subset of size $O(1/k)$ of the graph with minimum expansion. Our main results are extensions of Cheeger's classical inequality to these problems via higher eigenvalues of the graph Laplacian.In particular, for the sparsest $k$-partition, we prove that the sparsity is at most $8\sqrt{\lambda_k} \log k$where $\lambda_k$ is the $k^{th}$ smallest eigenvalue of the normalized Laplacian matrix.For the $k$ sparse cuts problem we prove that there exist$ck$ disjoint subsets $S_1, \ldots, S_{(1 - \eps)k}$, such that \max_i \phi(S_i) \leq C \sqrt{\lambda_{k} \log k}where $C>0$ are suitable absolute constants; this leads to a similar bound for the small-set expansion problem, namely for any $k$, there is a subset $S$ whoseweight is at most a $\bigO(1/k)$ fraction of the total weight and $\phi(S) \le C \sqrt{\lambda_k \log k}$. The latter two results are the best possible in terms of the eigenvalues up to constant factors. Our results are derived via simple and efficient algorithms, and can themselves be viewed as generalizations of Cheeger's method.Based on joint work with Prasad Raghavendra, Prasad Tetali and Santosh Vempala.

We study an Achlioptas-process version of the random k-SAT process: a bounded number of k-CNF clauses are drawn uniformly at random at each step, and exactly one added to the growing formula according to a particular rule. We prove the existence of a rule that shifts the satisfiability threshold. This extends a well-studied area of probabilistic combinatorics and random graphs to random CSP's. In particular, while a rule to delay the 2-SAT threshold was known previously, this is the first proof of a rule to shift the threshold of a CSP that is NP-hard. We then propose a gap decision problem based upon this semi-random model with the aim of investigating the hardness of the random k-SAT decision problem.