Course Descriptions

Term 1 (Autumn term)

From Optimal Control to Reinforcement Learning: Dr Dante Kalise, Imperial College London

This module follows a single conceptual thread that connects classical optimal control theory with modern reinforcement learning from a mathematical perspective, with a deliberate emphasis on approximation and optimisation methods that make these ideas usable in practice. We begin with the two foundational viewpoints of optimal control, the Pontryagin Maximum Principle (PMP), which characterises optimal trajectories through a boundary value problem, and dynamic programming, which characterises optimal feedback through the Hamilton–Jacobi– Bellman (HJB) PDE. From there, we trace the path that leads to feedback synthesis at scale: receding-horizon control (Model Predictive Control, MPC) as a practical bridge from open-loop optimisation to feedback, high-dimensional approximation of value functions and policies, and finally reinforcement learning as the data-driven, sampling-based version of dynamic programming.
A recurring theme is that all of these frameworks are, at their core, optimisation problems solved with approximation architectures: polynomial bases, tensor decompositions, and neural networks. The course pays close attention to how the choice of architecture interacts with the curse of dimensionality, the structure of the underlying control problem, and the conditioning of the resulting training problem. Students will gain hands-on experience implementing these methods and apply them to problems in feedback stabilisation, planning, games, and learning

• Foundations of optimal control: Optimal control and the Pontyragin Maximum Principle; Dynamic programming and the HJB equation
• Analysis and low-dimensional computation: Viscosity solutions and representation formulas; Numerical methods for HJB I.
• From open loop to feedback: Model Predictive Control
• High-dimensional approximation and learning: The curse of dimensionality and approximation architectures; Data-driven feedback synthesis; Optimisation for learning
• Reinforcement learning: From HJB to RL; Applications and synthesis

Regularity theories for PDEs: De Giorgi-Nash-Moser and Schauder: 

Dr Amelie Loher, University of Oxford

This course will present important ideas in regularity theory that have been established for elliptic equations in response to Hilbert’s 19th problem: are minimisers of energy functionals necessarily smooth? The answer revolves around two steps: first the De Giorgi-Nash-Moser theory, second the Schauder estimates. Together, they yield a positive answer to Hilbert’s 19th problem.

We discuss both of these theories in the context of parabolic equations. When time permits, we may explain how these methods can be adapted to hypoelliptic equations and what consequences this bears in kinetic theory. 

Prerequsities: 

Some knowledge on function spaces (Lebesgue spaces, Sobolev spaces, Hölder spaces) and on functional inequalities (Sobolev and Poincaré) is useful. 

Course Content:

(Revision without proofs)

Functional inequalities: 

- Isoperimetric inequality, Sobolev 

(Main content)

A. Parabolic equations:

1. Heat equation - Fundamental solution
2. Scale-invariant estimates - hole filling estimate, De Giorgi-Nash Hölder estimate, (Green function decay estimates)
3.  Perturbative estimates - Schauder, (Calderon-Zygmund)

(time permitting)

B. Kinetic equations:

1. Kolmogorov equation - Fundamental solution
2. Scale-invariant estimates - De Giorgi-Nash Hölder estimate
3. Perturbative estimates - Schauder

Analysis of the Sub-Laplacian: Dr Steven Flynn, UCL

A hidden symmetry in the analysis of phase space is the Heisenberg group structure. The Heisenberg group plays a unifying role in several branches of mathematics including harmonic analyses, several complex variables, quantum mechanics, and control theory.  It is a model space whose spectral theory is governed by a natural non-elliptic Laplacian, which is as close to close to Euclidean as possible while having some essential differences.

We will study the spectral theory of the sub-Laplacian by proving results in contrast with Euclidean space. We will start with the Fourier Transform and see how the theory of Hermite functions arises naturally. We will move on to the explicit spectral decomposition and then to PDEs involving the sub-Laplacian. With time, we will explore some additinal research topics.

This course is aimed at advanced postgraduates, and will be of particular interest to those who has enjoyed Professor Jens Marklof's course.

1. Heisenberg group and the Stone Von Neumann Theorem
2. Weyl and Fourier Transforms
3. Hermite and Special Hermite Function expansions
4. Spectral theory of the Sub-Laplacian and Twisted Laplacian
5. Fundamental Solution and Hypoelliptic Estimates
6. Restriction Theorems for spectral projections
7. The totally nondispersive Wave and Schrodinger Equations
8. Quantum Limits on the Heisenberg group

Introduction to Buildings: Dr Amy Herron, University of Bristol

Buildings are essentially elaborate geometric realizations of Coxeter groups. While they are important in many areas of math, I will focus on the geometry of spherical and Euclidean buildings.

1. Coxeter groups and complexes
2. Chamber complex definition of buildings and terminology with spherical and Euclidean examples
3. Metric space definition of buildings–proof of equivalence
4. Euclidean buildings are CAT(0)
5. Retractions of buildings onto its apartments
6. Apartments are convex
7. Exotic buildings
8. The spherical building at infinity of a Euclidean building
9. Examples of groups acting on buildings

Prerequisities: Knowledge of group theory, simplicial complexes, and metric spaces

Assessments: Registered students will be expected to give a class presentation (between 30 and 50 minutes) relevant to the topics covered and approved by me. After most classes, I will assign one or two problems related to the lecture.

References: 

Anne Thomas. Geometric and topological aspects of Coxeter groups and buildings. 2018.
Mark Ronan. Lectures on buildings: updated and revised. University of Chicago Press, 2009.
Peter Abramenko and Kenneth S Brown. Buildings: theory and applications, volume 248. Springer Science & Business Media, 2008.

Introduction to logarithmic algebraic geometry: Dr Simon Felten, University of Oxford

Logarithmic techniques are an important tool in algebraic geometry. They are used to study degenerations of algebraic varieties, the behavior of moduli spaces at the boundary, and to relate open varieties with their compactifications. They play a fundamental role in enumerative geometry and some approaches to mirror symmetry. Finally, they were used by Tsuji to prove the Cₛₜ-conjecture by Fontaine and Jannsen.

In this course, we cover the foundations of logarithmic algebraic geometry. Specifically, we discuss toric geometry and monoids, log structures, log rings and their spectra, log smoothness, and the log de Rham complex. We then use these techniques to study degenerating families of algebraic varieties and explore methods to construct such families starting from a given degenerate fiber V. If time permits, we discuss the toric Gross–Siebert mirror construction, which employs these techniques to construct mirror pairs of smooth Calabi–Yau varieties by smoothing a pair of singular Calabi–Yau schemes. We assume that the student is acquainted with the basic concepts of algebraic geometry but do not assume any specialized knowledge.

Geometric measure theory: Dr David Bate, University of Warwick

This course will cover several results on the geometry of non-smooth subsets of Euclidean space. After covering the introductory properties of Hausdorff measure and rectifiable sets, we will present proofs of the following theorems:
• The Besicovitch-Federer projection theorem in the plane concerning the orthogonal projections of purely unrectifiable sets.
• The Marstrand--Mattila characterisations of rectifiable sets via tangent planes.
• The Marstrand projection theorem that orthogonal projections almost always preserve Hausdorff dimension.

Stochastic Interface Models: Dr Florian Schweiger, University of Bath

Stochastic interface models arise in the description of the microscopic properties of interfaces, such as the boundary layer between a drop of oil and the surrounding water. Mathematically, they are given by probability measures on graphs of functions. There are many such models, and a variety of techniques have been developed to study them.

The course will provide an introduction to the area. We will discuss several classes of models, in particular the Gaussian free field, higher-order Gaussian models, and Ginzburg–Landau models, as well as their properties.

Term 2 

Exactly solvable models in statistical mechanics: Dr Thomas Bothner, University of Bristol

This course is suitable for students interested in mathematical statistical mechanics, in particular students interested in the theory of lattice spin systems. We will address several fundamental concepts of the theory through the analysis of selected models. Topics will include the Curie-Weiss model (thermodynamic limit and critical exponents), the Ising model (van Hove theorem, infinite-volume Gibbs states, correlation inequalities, phase diagram, Peierls’ argument, Lee-Yang theorem, Kramers-Wannier duality), the Gaussian free field (infinite-volume Gibbs states, random walk representation) and models with continuous symmetry (Mermin-Wagner theorem). The module will prepare students for reading literature of a wide variety of subjects in which statistical mechanical ideas are used.

The content of this course is based on chapters 2, 3, 8, 9 in S. Friedli, Y. Velenik: Statistical Mechanics of Lattice Systems. A concrete mathematical introduction.

Lecture notes will be available on lecturer's homepage.

Prerequisites:  A solid working knowledge in Real Analysis, Complex Analysis and Probability Theory.

Assessment: There is no exam for this course. If you are taking the module for credit (Bristol students only) you must complete the posted exercises and email lecturer your solutions in Latex typset by the due date.

Symmetric Spaces: 

Dr Ido Grayevsky (University of Bristol) and Dr Raz Slutsky (University of Oxford)

1. Hyperbolic geometry: Hyperbolic plane H^2, isometries 
2. Lattices in Lie groups: SL_2(Z) is a lattice in SL_2(R) 
3. Mostow Rigidity: Proof for H^3 
4. The symmetric space of SL_3(R) - construction 
5. Algebra (Torus, roots, Weyl group) 
6. Different decompositions: Cartan, Iwasawa, Bruhat
7. Some geometric properties of high rank symmetric spaces (Weyl sectors, visual boundary, parabolic subgroups, horoballs, horospheres) 
8. Choose from: Margulis Lemma, Kazhdan Margulis, Lubotzky-Mozes-Raghunathan, Schwartz QI rigidity for rank 1 

PDE-constrained optimisation: Professor Bertram During, University of Warwick

This module provides a rigorous introduction to the theory and numerical analysis of PDE-constrained optimisation in infinite-dimensional function spaces. Such problems arise in a wide range of scientific and engineering applications, including optimal control, inverse problems, parameter identification, and the optimal design of physical systems. The aim is to minimise a cost functional depending on a state function and a control function, subject to the constraint that the state satisfies a partial differential equation.
We will study the existence theory of linear-quadratic elliptic problems (weak solutions, existence of optimal controls, adjoint operators, necessary optimality conditions, Lagrange functional, adjoint as Lagrangian multiplier) as well as basic numerical methods for their solution (gradient method, projected gradient method, active set
strategy).
By the end of the module, you should be able to:
• explain basic concepts in PDE-constrained optimisation,
• show mastery of the existence theory for elliptic optimal control problems,
• derive necessary optimality conditions,
• understand basic numerical methods for their solution.

The Kakeya problem, incidence geometry and the polynomial method: 

Dr Andras Mathe, University of Warwick

The Kakeya problem asks how small a subset of Euclidean space can be if it contains a unit line segment in every direction. Following Besicovitch’s surprising constructions of such sets of measure zero, this question has become a central problem at the interface of geometric measure theory, harmonic analysis, additive combinatorics, and incidence geometry. It has also motivated many powerful techniques for understanding how lines, tubes and algebraic surfaces can interact.
The module will introduce the Kakeya problem and its discretised formulation in terms of thin tubes with many different directions. We will study a selection of earlier arguments in the area, chosen to illustrate the main methods rather than to give a complete historical account.
Topics may include incidence bounds for lines, the polynomial method, polynomial partitioning, and related results such as joints-type theorems. The three-dimensional Kakeya conjecture was recently proved by Hong Wang and Joshua Zahl. We will discuss a proof of this theorem, together with any subsequent simplifications or related developments available at the time. The emphasis throughout will be on geometric and combinatorial methods, while also explaining the problem’s connections with analysis.

Dynamical approaches in number theory: Dr Joel Moreira, University of Warwick

Starting with Furstenberg's ergodic theoretic proof of Szemeredi's theorem in 1977, a rich story of cross-pollination of ideas between ergodic theory, multiplicative number theory and additive combinatorics developed. This course will cover some of the highlights in this story and the connections between them. Topics covered, depending on time and students' interest, may include:

• The development of uniformity seminorms in ergodic theory paralleling Gowers norms in
  additive combinatorics.
• The Green-Tao theorem that the primes contain arbitrarily long arithmetic progressions.
• The subsequent program to study linear equations in primes, and how it required an understanding of the behaviour of certain special dynamical systems (nilsystems).
• Sarnak's influential conjecture on Mobius orthogonality and the relation with an older conjecture of Chowla.
• The special case of Chowla's conjecture proved by Tao,  the relation with Erdos' discrepancy problem, and the relation with Host-Kra uniformity of multiplicative functions.
• Frantzikinakis and Host's extension of Green and Tao's work, showing that aperiodic multiplicative functions are Gowers uniform, and how this connects to the problem of partition regularity of Pythagorean triples.
• The "dynamical prime number theorem" of Bergelson and Richter, which led to a new elementary proof of the classical prime number theorem.
• Infinite sumsets in sets of positive density and in the primes.

Algebraic Stacks:  Dr Federico Bongiorno, Imperial College London

In this course, we will approach the theory of algebraic stacks from a geometric point of view. We will study algebraic stacks through their presentations: schemes endowed with a smooth groupoid, and we will introduce the categorical language only as needed for descent, universal properties, and changing presentations.
Our end goal is to understand the local structure theorem for algebraic stacks of Alper–Hall–Rydh: near a point with linearly reductive stabiliser, an algebraic stack is étale-locally a quotient stack.
The course will place more emphasis on worked examples and geometric intuition rather than complete proofs, and it will periodically explore the relation with algebraic foliations.

Topics to be covered include: 

• Group actions, equivalence relations, and groupoids
• Algebraic spaces as quotients of étale equivalence relations.
• Algebraic stacks via smooth groupoid presentations and descent.
• Points of stacks, tangent spaces, residual gerbes, and minimal presentations.
• Quasi-coherent sheaves, equivariant sheaves, and cohomology on stacks.
• Coarse moduli spaces, the Keel–Mori theorem and rigidification.
• Good moduli spaces and their relation with GIT.
• Local structure theorem in the smooth setting.
• Local structure theorem in the general setting (time permitting)

Prerequisites: This course is accessible to students with a good understanding of basic algebraic geometry, comparable to the level of Chapter 2 and 3 of Hartshorne.

The Coarse Geometry of Graphs: Dr Joseph MacManus, University of Bristol

Coarse graph theory is an emerging and exciting topic, studying the interaction between the coarse metric geometry and classical results in structural graph theory, ultimately hoping to understand the ‘large-scale’ structure of graphs. This subfield was popularised by a recent paper of Georgakopoulos–Papasoglu [2].
Success in coarse graph theory often takes the form of proving coarse versions of classical theorems in graph theory. The benefit of this is that the scope of the theorems increases beyond just graphs, and apply more generally to geodesic metric spaces.
In this course we will introduce the basic rudiments of coarse graph theory from the ground up, proving some important results—both positive and negative—along the way. We will also survey the state-of-the-art, as well as many open problems in the field. Minimal background will be assumed, except for a basic familiarity with groups, graphs, and metric spaces.
Week 1: A primer on coarse geometry; quasi-isometries, coarse embeddings, and the basics of geometric group theory as a source of motivation.

Week 2: Quasi-trees; Proof of Manning’s bottleneck criterion [5]. Discussion of Kerr’s theorem on tree approximation [4].

Week 3: The Coarse Menger Conjecture; a proof of the positive result by Georgakopoulos–Papasoglu for two paths [2].

Week 4: Asymptotic minors and the Fat Minor Conjecture; the positive result for K 3 via the bottleneck criterion [2].

Week 5: The Nguyen–Scott–Seymour graph; counterexamples to the Coarse Menger Conjecture [7] and the Fat Minor Conjecture [1].

Week 6: Asymptotic dimension; introduction to asymptotic dimension with ex- amples, and a survey of results and conjectures on the asymptotic dimension of certain families of graphs.

Week 7: Ends and accessibility; Thomassen–Woess’ definition of an accessible graph [8]. Hamann’s graph-theoretic proof of Dunwoody’s accessibility theorem [3].

Week 8: Quasi-planar graphs; characterisations of (vertex-transitive) quasi- planar graphs, both known and conjectural. A short proof of the Euclidean case of Mess’ quasi-plane theorem [6], illustrating the Varopoulos inequality.

References: 

[1] J . Davies, R. Hickingbotham, F. Illingworth, and R. McCarty, Fat minors cannot be thinned (by quasi-isometries), To appear in Analysis and Geometry in Metric Spaces (2026).
[2] A. Georgakopoulos and P. Papasoglu, Graph minors and metric spaces, Combinatorica 45 (2025), no. 3, 33.
[3] M. Hamann, Accessibility in transitive graphs, Combinatorica 38 (2018), no. 4, 847–859.
[4] A. Kerr, Tree approximation in quasi-trees, Groups, Geometry, and Dynamics 17 (2023), no. 4, 1193–
1233.
[5] J . F. Manning, Geometry of pseudocharacters, Geometry & Topology 9 (2005), no. 2, 1147–1185.
[6] G. Mess, T he Seifert conjecture and groups which are coarse quasiisometric to planes, preprint (1988). [7] T. Nguyen, A. Scott, and P. Seymour, A counterexample to the coarse menger conjecture, J ournal of
Combinatorial Theory, Series B 173 (2025), 68–82.
[8] C. Thomassen and W. Woess, Vertex-transitive graphs and accessibility, J ournal of Combinatorial Theory, Series B 58 (1993), no. 2, 248–268.

Term 3