# Pure mathematics

All branches of applied mathematics require a sound theoretical foundation. Our research in variational analysis, functional analysis, statistical analysis and network theory is particularly relevant to optimisation. We also have strengths in topological groups and combinatorics.

## Current examples of where this knowledge is being applied are in:

- Functional analysis
- Combinatorics
- Topological groups
- Variational analysis

## Variational analysis

Variational analysis research is focused on advancing the theory of stationarity and regularity. We are going to clarify the relationships between various stationarity and regularity concepts and criteria for typical for variational analysis objects, exploring the idea precisely formulated by Borwein and Zhu (Techniques of Variational Analysis. Springer-Verlag, New York, 2005): “An important feature of the new variational techniques is that they can handle nonsmooth functions, sets and multifunctions equally well".

Our capabilities in variational analysis are:

- Contribute to the advancement of the general theory of regularity and stationarity in variational analysis. We are going to examine the existing primal and dual concepts for various typical settings of variational problems, find missing links, establish new relations, and present a classification scheme. It might be necessary to introduce and investigate new derivative-like objects. We are going to consider also specific mappings and sets important for applications as well as specific variational problems and make use of the specifications of the developed regularity and stationarity properties and criteria.
- Based on the general classification scheme of necessary and sufficient criteria for the error bound property, we are going to develop a similar scheme for the metric subregularity and calmness properties of multifunctions. The criteria will be applied to investigating stability of multifunctions. We intend to sharpen the existing criteria for special classes of multifunctions. The main attention will be focused on multifunctions given as inverse images of closed (not necessarily convex) sets with respect to smooth mappings.
- We will extend the necessary and sufficient criteria for the stability of error bounds of convex constraint systems to nonconvex constraint systems and consider stability of metric subregularity and calmness of multifunctions aiming at developing a general stability theory for these concepts.
- Based on the regularity criteria for collections of sets and general error bound criteria, we are going to clarify and sharpen convergence estimates for the alternating and averaged projections and the proximal point methods and other iterative methods for solving optimization and equilibrium programs.

## Topology

Topology (from the Greek τόπος, "place", and λόγος, "study") is the mathematical study of shapes and spaces. It is a major area of mathematics concerned with the most basic properties of space, such as connectedness, continuity and boundary. It is the study of properties that are preserved under continuous deformations including stretching and bending, but not tearing or gluing. The exact mathematical definition is given below. Topology developed as a field of study out of geometry and set theory, through analysis of such concepts as space, dimension, and transformation.

Ideas that are now classified as topological were expressed as early as 1736. Toward the end of the 19th century, a distinct discipline developed, referred to in Latin as the geometria situs ("geometry of place") or analysis situs (Greek-Latin for "picking apart of place"). This later acquired the name topology. By the middle of the 20th century, topology had become an important area of study within mathematics.

Topology has many subfields:

- Point-set topology establishes the foundational aspects of topology and investigates concepts inherent to topological spaces (examples include compactness and connectedness).
- Algebraic topology tries to measure degrees of connectivity using algebraic constructs such as homology and homotopy groups.
- Geometric topology primarily studies manifolds and their embeddings (placements) in other manifolds. A particularly active area is low dimensional topology, which studies manifolds of four or fewer dimensions. This includes knot theory, the study of mathematical knots.