Part II

Research Talks by Math Institutes

October 13, 2016 – Promenade Room 101A
09:00 – 09:30: ICERM - Katie Oliveras: "Understanding Water Waves – Using nonlinear techniques to improve"
09:35 – 10:05: IPAM - Lorenzo Boninsegna : “Complex Energy Landscapes: Theory and Applications”
10:10 – 10:40: IAS/PCMI - Dimitri Shlyakhtenko: "Random Matrices"
10:45 – 11:00: Break
11:00 – 12:00: Q&A with Institute Representatives

ICERM – Katie Oliveras
“Understanding Water Waves – Using nonlinear techniques to improve”
Euler’s equations for fluid motion are a set of nonlinear differential equations that describe water waves. Due to their complicated nature, most engineers use mathematical models that have been derived as approximations to these equations. While working with the simpler, approximate equations yields insight into the behavior of water waves, important and critical information about the nonlinear dynamics is often lost. But how much information is lost?
In this talk, I will discuss a new formulation of the fully nonlinear equations due to Ablowitz, Fokas, and Musslimani, and how it has helped has us gain a much deeper understanding of the role that nonlinearities play when modeling water waves. In addition, I will show how we can use ideas ranging from the quadratic formula to complex analysis to solve various inverse problems related to water waves, and compare results with both numerical data and physical experiments.

IPAM – Lorenzo Boninsegna
“Complex Energy Landscapes: Theory and Applications”
An energy landscape can be naively thought of as a detailed topographic chart of the system being studied, where all the information is scrupulously noted, such as the position and the altitude of peaks, valleys and passes, i.e. maxima, minima and saddle points. The system is then a point living in this fictional mountain range, which encodes all the thermodynamic and kinetic information. As such, energy landscapes are the natural language to describe many body systems such as biomolecules or glasses.

Indeed, protein self-organizing folding is nowadays understood in terms of (minimally frustrated) free energy landscapes, and so is glassy dynamics in non - crystalline solids.

Such energy landscapes are complex because of their simultaneous dependence on a large number of degrees of freedom (e.g. position of atoms in the molecule or the glass), whose interplay produces a plethora of fascinating behaviors, e.g. rare events such as barrier crossing and a hierarchy of timescales in the dynamics just to name a few.

All this richness makes simulating and effectively interpreting such complex energy landscapes a formidable task, and standard techniques are not adequate anymore. Designing novel protocols making the analysis accessible is going to call for a joint effort of the community at the interface of many different fields like physics, mathematics, biology, computer science and chemistry.

In this talk I will present an overview of energy landscape theories and a summary of state of art methods to investigate them.

IAS/PCMI - Dimitri Shylakhtenko
“Random Matrices”
This will be a short survey about some aspects of the broad field of random matrices. This field is the topic of the PCMI 2017 session, and the goal of this talk is to describe a few aspects of the subject in hopes of generating interest and attracting applications to the Park City Math Institute summer session.

Back to NSF Mathematics Institutes' Modern Math Workshop (at SACNAS)