Ruth Charney, Brandeis University

Searching for Hyperbolicity

As students, we first encounter groups as algebraic objects, but groups can also be viewed as symmetries of geometric objects. This viewpoint gives rise to powerful tools for studying infinite groups. The work of Max Dehn in the early 1900’s on groups acting on the hyperbolic plane was an early indication of this phenomenon. Dehn’s ideas were vastly generalized in the 1980’s by Cannon and Gromov to a large class of groups, known as Gromov hyperbolic groups. In recent years there has been an effort to push these ideas even further. If a group fails to be Gromov hyperbolic, might it still display some hyperbolic behavior? Might some of the techniques used in hyperbolic geometry still apply? I will talk about some recent work on finding and encoding hyperbolic behavior in infinite groups.

Svitlana Mayboroda, University of Minnesota

The hidden landscape of localization of eigenfunctions

Numerous manifestations of wave localization permeate acoustics, quantum physics, mechanical and energy engineering. It was used in construction of noise abatement walls, LEDs, optical devices, to mention just a few applications. Yet, no systematic methods could predict the exact spatial location and frequencies of the localized waves.

In this talk I will present recent results revealing a new criterion of localization, tuned to the aforementioned questions, and will illustrate our findings in the context of the boundary problems for the Laplacian and bilaplacian, $div A\nabla$, and (continuous) Anderson and Anderson-Bernoulli models on a bounded domain. Via a new notion of “landscape” we connect localization to a certain multi-phase free boundary problem and identify location, shapes, and energies of localized eigenmodes. The landscape further provides sharp estimates on the rate of decay of eigenfunctions and delivers accurate bounds for the corresponding eigenvalues, in the range where both classical Agmon estimates and Weyl law may fail.

Linda Petzold, University of California Santa Barbara (UCSB)

Inference of the Functional Network Controlling Circadian Rhythm

Petzold’s lecture focuses on the use of computing and mathematics to better understand circadian rhythm, the process by which living organisms manage to follow a 24-hour cycle. Difficulties in following this cycle are experienced, for example, in jet lag and shift work. Circadian rhythm disorders are known risk factors for heart disease, obesity, and diabetes, as well as numerous psychiatric and neurodegenerative diseases. In mammals, the Suprachiasmatic Nucleus (SCN), a brain region of about 20,000 neurons, serves as the master circadian clock, coordinating timing throughout the body and entraining the body to daily light cycles. The extent to which cells in the SCN can synchronize and entrain to external signals depends both on the properties of the individual oscillators (neurons) and on the communication network between individual cell oscillators Petzold’s lecture explores both the development of mathematical models and inference of the structure of the network which connects the neurons.

Footnote. Linda Petzold is professor in Mechanical Engineering and Computer Science at the University of California, Santa Barbara. Her email address is petzold@engineering.ucsb.edu.

Mariel Vazquez, University of California, Davis

Understanding DNA Topology

DNA is subject to high levels of condensation in the cell. In order to ensure genome stability, the cell must control changes on DNA geometry and topology induced by DNA packing and by cellular processes such as DNA replication and recombination. We use techniques from knot theory and low-dimensional topology, aided by computational tools, to study the topological state of the genome and the mechanism of enzymes that control and modify the topology of DNA.