Taught Course Centre

Noncommutative differential geometry

Dr Edwin Beggs

Syllabus

• Differential calculi on noncommutative algebras
• Introduction to Hopf algebras and their calculi
• Calculi on graphs and finite groups
• Covariant derivatives on modules and bimodules, curvature
• Monoidal categories
• Noncommutative vector fields, states and divergences
• CP maps and the KSGNS construction, Hilbert C* bimodules
• The flow of states generated by vector fields
• Parallel transport

An introduction to infinite dimensional analysis

Prof Eugene Lytvynov

Syllabus

• Locally convex topological vector space (l.c.s.) and its dual. Examples: rigged Hilbert spaces, Gelfand triple, space of continuous functions with compact support and signed Radon measures. Configuration space.
• Tensor product of vector spaces, topological tensor product of l.c.s.'s, polynomials on the dual of a l.c.s. Polynomial sequence on the dual of a l.c.s.
• Cylinder algebra and cylinder sigma-algebra on the dual of a l.c.s. Borel sigma-algebra. Probability measure on a Hilbert space and on the dual of a nuclear space (generalised stochastic processes). Probability measure on the space of Radon measures (random measure), probability measure on the configuration space (point process).
• Gaussian measure on an infinite dimensional space, white noise measure.
• Gaussian analysis: multiple stochastic integral, Hermite polynomial sequence over infinite dimensional space, Segal–Wiener–Itô isomorphism between the Gaussian L²-space and the symmetric Fock space. Space of test functionals (Hida space), space of generalised functionals (white noise). Segal–Bargmann transform in infinite dimensions.
• Elements of analysis on configuration spaces: Poisson point process, its Wiener–Itô decomposition, Charlier polynomial sequence over configuration space. Correlation measure of a point process. Determinantal point processes.
• Random measures, gamma random measure, Laguerre polynomials over the space of Radon measures.

Algebraic structures

Prof Tomasz Brzeziński

Abstract

An algebraic structure is a set with operations. Typical and most widespread across mathematics are systems such as a semigroup, monoid, group, ring, field, associative algebra, vector space, affine space or module. In this lecture course we will revisit these well-known systems and look at them from a universal algebra perspective, as well as study some simple algebraic systems which have recently gained prominent position in algebra and topology such as braces, racks or quandles (sets with two operations interacting with each other in prescribed ways). In particular we will explore a little known fact (first described 100 years ago by Pruefer and Baer) that one can give a definition of a group without requesting existence of the neutral element and inverses by using a ternary rather than a binary operation (i.e. an operation with three rather than the usual two inputs). A set with such a suitable ternary operation is known as a heap. By picking an element in a heap, the ternary operation is reduced to the binary group operation, for which the chosen element is the neutral element (the resulting group is known as a retract). We will study properties and examples of heaps and relate them to the properties of corresponding groups (retracts). Next we will look at heaps with an additional binary operation that distributes over the ternary heap operation, known as trusses, relate them to both rings and braces, and study their properties.

Random graph models for complex networks

Dr Angelica Pachon

Abstract

The theory of random graphs was founded by Erdős and Rényi in the late 1950s with the papers "On random graphs I" and "On the evolution of random graphs". Since then, many properties of the Erdős and Rényi random graph have been analysed to answer questions of mathematical concern. During the last few decades, increasing interest in the field of random graphs has been devoted to find models that describe the complexity of real-world networks. This course introduces the theory of random graphs, emphasizing on results for the Erdős–Rényi random graph, as well as for random graph models for complex networks.

Syllabus

• The Erdős–Rényi random graph
• Branching processes
• Cluster growth
• The emergence of the largest component
• Connectivity threshold
• Degree sequence of the Erdős–Rényi random graph
• Other models for complex networks

Noncommutative differential geometry

Dr Edwin Beggs

Syllabus

• Differential calculi on noncommutative algebras
• Introduction to Hopf algebras and their calculi
• Calculi on graphs and finite groups
• Covariant derivatives on modules and bimodules, curvature
• Monoidal categories
• Noncommutative vector fields, states and divergences
• CP maps and the KSGNS construction, Hilbert C* bimodules
• The flow of states generated by vector fields
• Parallel transport

Operator semigroups in Banach spaces and their applications

Dr Dmitri Finkelshtein

Description

Operator semigroups are one-parametric families of linear operators which generalise the exponential function. They naturally arise in the study of evolution equations for functions defined on Banach (or more general) spaces. Operator semigroups play a fundamental role in functional analysis and serve as a powerful tool in PDEs, probability, dynamical systems, mathematical physics, and other areas of mathematics. Understanding the properties and applications of operator semigroups is crucial for researchers in various fields, including applied mathematics, physics, engineering, and biology. The course begins by overview of the needed concepts about Banach spaces and linear (unbounded) operators acting on them, followed by an in-depth study of the generators of operator semigroups and their properties. The course proceeds to cover C₀ semigroups and their spectral properties, compact and analytic semigroup, contraction and positive semigroups along with their applications in probability theory. The course will also briefly cover more advanced topics such as perturbations, approximations and asymptotic properties of operator semigroup and their applications in physics and biology.

Syllabus

• Evolution equations in finite- and infinite-dimensional spaces
• Introduction to unbounded operators in Banach spaces
• Semigroups, generators, and resolvents
• Special classes of semigroups: compact, analytic, contraction, positive etc.
• Perturbation and approximation of semigroups
• Asymptotic properties of semigroups
• Applications in physics and biology

By the end of this course, students will have gained a solid understanding of operator semigroups and their applications, enabling them to tackle complex problems in mathematics, physics, and other related disciplines.

Applied algebraic geometry

Dr Nelly Villamizar

Abstract

This course provides a hands-on introduction to applied algebraic geometry, emphasizing the interplay between commutative and homological algebra, geometry, and combinatorics. The syllabus includes a variety of tools and methods that are valuable for computational applications involving the transformation of complex objects into a sequence of simpler components. The geometric setting will usually be projective space, graded rings and modules. Key topics covered include the dimension and Hilbert polynomials of a variety, free resolutions and regular sequences, Gröbner bases, syzygies, the Stanley–Reisner ring of a simplicial complex, Functors, Ext, Tor, Cohen–Macaulay modules, and multivariate polynomial splines. Through practical examples and exercises using the computer algebra system Macaulay2, students will gain hands-on experience with computational algebra tools and the application of algebraic techniques to solve computational problems.

References

Cox, D. A., Little, J. B., O'Shea, D. (1998). Using Algebraic Geometry (Vol. 185). Springer.

Schenck, H. (2003). Computational Algebraic Geometry (London Mathematical Society Student Texts). Cambridge University Press.

Schenck, H. (2022). Algebraic Foundations for Applied Topology and Data Analysis. Springer International Publishing.

Elements of philosophy of mathematics (a working mathematician perspective)

Prof Tomasz Brzeziński

Umbral calculus: combinatorial, algebraic and analytical aspects

Prof Eugene Lytvynov

Algebraic structures

Prof Tomasz Brzeziński

Abstract

An algebraic structure is a collection of sets with operations. Typical and most widespread across mathematics are systems such as a semigroup, monoid, group, ring, field, associative algebra, vector space or module. In this lecture course we will revisit these well-known systems and look at them from a universal algebra perspective, as well as study some simple algebraic systems which have recently gained prominent position in algebra and topology such as braces, racks or quandles (sets with two operations interacting with each other in prescribed ways). In particular we will explore a little known fact (first described nearly 100 years ago by Pruefer and Baer) that one can give a definition of a group without requesting existence of the neutral element and inverses by using a ternary rather than a binary operation (i.e. an operation with three rather than the usual two inputs). A set with such a suitable ternary operation is known as a heap. By picking an element in a heap, the ternary operation is reduced to the binary group operation, for which the chosen element is the neutral element (the resulting group is known as a retract). We will study properties and examples of heaps and relate them to the properties of corresponding groups (retracts). Next we will look at heaps with an additional binary operation that distributes over the ternary heap operation, known as trusses, relate them to both rings and braces, and study their properties.

For further details, see the course web-page.

Criticality theory for Schrödinger operators

Prof Vitaly Moroz

Abstract

This course is an introduction to Agmon's Criticality Theory of Schrödinger operators. It will focus on two core but not widely known ideas, namely Allegretto–Piepenbrink positivity principle and Phragmen–Lindelöf comparison principle. We will see how these fundamental principles enable to prove a range of Hardy type inequalities, and at the same time provide a powerful tool in the analysis of the structure of positive solutions for large classes of nonlinear elliptic equations.

Linear theory:

1. Allegretto–Piepenbrink positivity principle for linear Schrödinger operators and some corollaries;
2. Connection with Hardy inequalities on bounded and unbounded domains;
3. Phragmen–Lindelöf comparison principle: large and small positive solutions;
4. Weak, strong and critical potentials.

Nonlinear applications:

5. Nonlinear Liouville's theorems, Serrin's critical exponent, fast and slow decay solutions;
6. Classification of singularities of semilinear elliptic equations, local Keller–Osserman bound, removable singularities;
7. Boundary blow-up solutions of semilinear elliptic equations in bounded domains, global Keller–Osserman bound.

The course prerequisites are limited to basic concepts of elliptic PDEs: weak solutions, classical maximum principle, basic understanding of Sobolev spaces.

For further details, see the course web-page.

Distribution-dependent stochastic differential equations

Prof Feng-Yu Wang

Abstract

In 1966 H.P. McKean proposed a distribution-dependent stochastic differential equation (DDSDE for short) to characterize the nonlinear partial differential equation for Kac's caricature of a Maxwellian gas. Since that time, this type of SDEs has attracted increasing attention. In this course we will discuss some important results of the theory of DDSDEs, which include:

• Correspondence of weak solutions and nonlinear Fokker–Planck equations;
• Existence and uniqueness;
• Coupling method and Malliavin calculus;
• Gradient estimate and Bismut formula for Lions derivative;
• Comparison theorems;
• Exponential ergodicity in entropy and Wasserstein distance.

For further details, see the course web-page.