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 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, 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 Banach algebras, extending the Gromov-Hausdorff distance and metric geometry to noncommutative geometry.
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).