Georgia Institute of Technology College of Sciences

Two of the most influential theorems in discrete mathematics state, respectively, that diagonal Ramsey numbers grow exponentially and that error-correcting codes for noisy channels exist up to the information limit. The former, proved by Erdős in 1947 using random graphs, led to the development of the probabilistic method in combinatorics. The latter, proved by Shannon in 1948 using random codes, is one of the founding results of coding theory. Since then, the probabilistic method has been a cornerstone in the development of both Ramsey theory and coding theory. In this talk, we highlight a few important applications of the probabilistic method in these two parallel but interconnected worlds. We then present new results on Ramsey numbers of graphs and hypergraphs and codes correcting deletion errors, all based on probabilistic ideas.

This talk is concerned with α-Hölder-continuous weak solutions of the incompressible Euler equations. A great deal of recent effort has led to the conclusion that the space of Euler flows is flexible when α is below 1/3, the famous Onsager regularity. We show how convex integration techniques can be extended above the Onsager regularity to all α<1/2 in the case of the forced Euler equations. This leads to a new non-uniqueness theorem for any initial data. This work is joint with Aynur Bulut and Manh Khang Huynh.

Weighted inequalities for singular integral operators are central in the study of non-homogeneous harmonic analysis. Two weight inequalities for singular integral operators, in-particular attracted attention as they can be essential in the perturbation theory of unitary matrices, spectral theory of Jacobi matrices and PDE's. In this talk, I will discuss several results concerning the two weight inequalities for various Calder\'on-Zygmund operators in both Euclidean setting and in the more generic setting of spaces of homogeneous type in the sense of Coifman and Weiss.

The two-weight conjecture for singular integral operators T was first raised by Nazarov, Treil and Volberg on finding the real variable characterization of the two weights u and v so that T is bounded on the weighted $L^2$ spaces. This conjecture was only solved completely for the Hilbert transform on R until recently. In this talk, I will describe our result that resolves a part of this conjecture for any Calder\'on-Zygmund operator on the spaces of homogeneous type by providing a complete set of sufficient conditions on the pair of weights. We will also discuss the existence of similar analogues for multilinear Calder\'on-Zygmund operators.

Algebraic Combinatorics originated in Algebra and Representation Theory, yet its objects and methods turned out applicable to other fields from Probability to Computer Science. Its flagship hook-length formula for the number of Standard Young Tableaux, or the beautiful Littlewood-Richardson rule have inspired large areas of study and development. We will see what lies beyond the reach of such nice product formulas and combinatorial interpretations and enter the realm of Computational Complexity and Asymptotics. We will also show how an 80 year old open problem on Kronecker coefficients of the Symmetric group lead to the disprove of the wishful approach towards the resolution of the algebraic P vs NP Millennium problem.

In the first talk of this series we introduced the definition of Chebyshev polynomials on compact subsets of the complex plane and discussed some properties. This week, after a short review of  the first talk, we will start to discuss asymptotic properties of Chebyshev polynomials and how they are related with logarithmic potential theory. Our main focus will be the necessary concepts from potential theory needed in the study of asymptotic properties of Chebyshev polynomials.  

Large-scale computing systems are massively important, using over 1% of the world's electricity. It is vital that these systems be both fast and resource-efficient, and scheduling theory is a key tool towards that goal. However, prior scheduling theory is not equipped to handle large multiserver systems, with little extant theoretical analysis, and no optimality results.

I focus on two important multiserver scheduling systems: The one-server-per-job (M/G/k) model, and the multiple-servers-per-job (MSJ) model. In the M/G/k, we prove the first optimality result, demonstrating that the Shortest Remaining Processing Time (SRPT) policy yields asymptotically optimal mean response time in the heavy traffic limit. In the MSJ model, we prove the first mean response analysis for any scheduling policy, for a novel policy called ServerFilling. Moreover, we introduce the ServerFilling-SRPT policy, for which we present the first asymptotic optimality result for the MSJ model. Each result progresses by proving novel bounds on relevant work, and using novel methods to convert those bounds to bounds on mean response time. These results push the state of the art of scheduling theory ever closer to applicability to today's large-scale computing systems.