Georgia Institute of Technology College of Sciences

Understanding the behavior of large physical systems is a problem of fundamental importance in mathematical physics. Analysis of systems of many interacting particles is key for understanding various phenomena from physical sciences (e.g. gases in non-equilibrium, galactic dynamics) to social sciences (e.g. modeling social networks). Similarly, the description of systems of weakly nonlinear interacting waves, referred to as wave turbulence theory, finds a wide range of applications from solid state physics and water waves to plasma theory and oceanography. However, with the size of the system of interest being extremely large, deterministic prediction of its behavior is practically impossible, and one resorts to an averaging description. In this talk, we will discuss about kinetic theory, which is a mesoscopic framework to study the qualitative properties of large systems. As we will see, the main idea behind kinetic theory is that, in order to identify averaging quantities of large systems, one studies their asymptotic behavior as the size of the system tends to infinity, with the hope that the limiting effective equation will reveal properties observed in a system of large, but finite size. We will focus on the Boltzmann equation, which is the effective equation for systems of interacting particles, and its higher order extensions, as well as the kinetic wave equation which describes systems of many nonlinearly interacting waves.

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.

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.