Research Area: Analysis

Mathematical analysis is the discipline of mathematics which uses the notion of limit of functions as its core concept. Limits are the formalism used to define and study the continuum of real numbers and many other topological spaces, from differentiable manifolds to fractals, from calculus of functions in real and in complex variables to infinite dimensional spaces. Our faculty carries out research in many areas of analysis, described below.

Functional Analysis

Functional analysis is the study of spaces of functions, and more generally of topological vector spaces and their associated structures, by means of topological, analytical and geometric methods. It is a far reaching field which plays a fundamental role in various areas such as partial differential equations, function theory, complex analysis, harmonic analysis and topological group theory, mathematical physics, differential geometry, probability and measure theory, machine learning, among others.

Our current projects in this area concern:

• The construction and study of Banach spaces from tree-like structures which emerged from descriptive set theory,
• The construction and study of topologies on classes of noncommutative analogues of the algebras of Lipschitz functions, and the extension of the Gromov-Hausdorff distance to noncommutative geometry.

Dynamics

In dynamical systems, one considers the pair (X,T) where T is a map from a space X to itself. We can view the map T as moving the points around X and apply it repeatedly, taking the point of view that the space X evolves over time.

There are several different subcategories of dynamical systems based on what kind of structure the set X has, and how much of it is preserved by T. We arrive at other important subcategories of dynamics by consideration of various (semi-)groups acting on X.

In addition to being an important subject in its own right, there are many examples of problems for which solutions became apparent only when the problem is rephrased in dynamical systems terms.

We focus on ergodic theory and symbolic dynamical systems (which model topological or smooth dynamical systems by a discrete space consisting of infinite sequences of abstract symbols and a shift operator).

Statistical Mechanics

Dr. Yin's research has focused on large exponential random graphs and, more recently, an abstraction of the parking scenario. A continuing theme of her research is developing a quantitative theory of phase transitions in statistical mechanics models. In the statistical physics literature, asymptotic phase transition is often associated with loss of analyticity in the limiting normalization constant (free energy density), which gives rise to discontinuities in the observed statistics. In the vicinity of a phase transition, even a tiny change in some local characteristic can result in a dramatic change of the entire system.