What is the "simplest" knot in a given three-manifold Y?
We know that the answer is the unknot when Y=S^3, as the unknot
happens to be the only knot in the three-sphere with the smallest
genus (=0). In this talk, we will discuss the more general notion of
the rational genus of knots. In particular, we will show that the
simple knots are really the "simplest" knots in the lens spaces in
the sense of being a genus minimizer in its homology class. This is
a joint work with Yi Ni.

Thermodynamic formalism as a mathematical theory has its roots in one of the most successful theories of physics -- thermodynamics and statistical mechanics. In its core, the theory of thermodynamic formalism seeks to describe properties of observable "macroscopic" phenomena based on the average behavior of the "microscopic" constituents. In the language of dynamical systems: given a dynamical system $(X, f)$, with $X$ the phase space and $f$ the map defining the dynamics, one seeks to describe properties of functions defined on $X$ (the macroscopic observables) based on the (often averaged, in some well-defined sense) behavior of $f$. In particular, thermodynamic formalism leads to strong results in dimension theory of dynamical systems (e.g. describing fractal dimensions and measures of sets arising as invariant sets of some chaotic dynamical systems). In this first of a series of two talks, we shall present the main ingredients of thermodynamic formalism: topological entropy, metric entropy, topological pressure, and the variational principle for the pressure.

As, beginning with the famous Hofstadter's butterfly, all
numerical studies of spectral and dynamical quantities related to
quasiperiodic operators are actually performed for their rational
frequency approximants, the questions of continuity upon such
approximation are of fundamental importance. The fact that continuity
issues may be delicate is illustrated by the recently discovered
discontinuity of the Lyapunov exponent for non-analytic potentials.

I will review the subject and then focus on work in progress, joint with Avila and Sadel, where we develop a new approach to continuity, powerful enough to handle matrices of any size and leading to a number of strong consequences.

This is the first of (likely) two talks, where an almost entire proof will be presented. For understanding most of the talk knowledge of spectral theory should not be necessary and just knowing some basic harmonic analysis should suffice.

 
Cell polarization, in which intracellular substances localize to a particular spot in response to external stimuli or internal cues, is central to cell physiology, and it underlies processes such as cell motility, cell division and cell differentiation. In this talk, I will present our recent works, using budding yeast as a model system, on how cells initiate symmetry breaking preceding the new bud emergence or mating projection. Along with experimental data, our mathematical modeling and simulations reveal potential mechanisms which underlie a biased cell polarity and pheromone concentration dependent cell morphology.