Mathematics Colloquium 2022-23 Abstracts

6 October 2023: Anne-Sophie Kaloghiros (Brunel) The Calabi problem for Fano 3-folds and applications

Algebraic varieties are geometric shapes given by polynomial equations. They appear naturally in pure and applied mathematics: from conic sections in geometry, to cubic curves in cryptography, or non-uniform rational basis splines in computer-aided graphic design.

To measure distances between points on an algebraic variety, we equip it with a metric - a sophisticated dot product. This then leads to the notion of curvature, and allows us to split algebraic varieties into three basic (universal) types: negatively curved, flat and positively curved varieties. Positively curved varieties are higher dimensional generalisations of a sphere; they are called Fano varieties. Fano varieties appear frequently in applications, because they are often parametrised by rational functions.

For an algebraic variety, the choice of a metric is never unique. One can try to find a special metric with good properties: a “canonical metric". Geometers looked for a suitable condition defining a canonical metric for the first half of the 20th century. In 1957, Calabi proposed that this canonical metric should satisfy both a certain algebraic property (being Kähler) and the Einstein (partial differential) equation. Finding which compact complex manifolds admit such a metric is the object of the Calabi problem, an area of research at the crossroads of algebraic and differential geometry that has been very active for the last decades.

A necessary condition for the existence of such a metric is that the manifold belongs to one of the three basic universal types. Yau and Aubin/Yau confirmed Calabi's prediction and showed that manifolds with negative or flat curvature always admit a Kähler-Einstein metric in the 1970s. By contrast, the Calabi problem is much more subtle for manifolds
with positive curvature: Fano manifolds may or may not admit a Kähler-Einstein metric.

Research on the Calabi problem for Fano manifolds culminated in the formulation and proof of the Yau-Tian-Donaldson conjecture. This conjecture, now a theorem, states that a Fano manifold admits a Kähler-Einstein metric precisely when it satisfies a sophisticated algebro-geometric condition called K-polystability. Surprisingly, the notion of K-polystability also sheds some light on their moduli theory, that is how they behave in families ( another poorly understood aspect of their geometry).

In this talk, I will present an overview of the Calabi problem, and present its solution in small dimension ( in which we have a classification of deformation families of smooth Fano varieties). I will discuss applications to other areas such as moduli theory.

13 October 2023: Rob Silversmith (Warwick) Counting problems in algebraic geometry

Choose five conic plane curves randomly. There are exactly 3264 ways to draw a sixth conic that is tangent to all five. (You may need complex numbers to see all of them.) Counting problems like this one have been studied for hundreds of years, and are part of a rich interplay between geometry and combinatorics. I will discuss a very down-to-earth class of counting problems with connections to many fields, including: string theory, rigid frameworks, polyhedral geometry, matroid theory, and cluster algebras. I will also mention some other recent developments and directions in the field.

20 October 2023: Colva Roney-Dougal (St Andrews) Counting permutation groups

What does a random permutation group look like? This talk will start with a brief survey of how we might go about counting subgroups of the symmetric group S_n, and talk about what is known about "most" subgroups.

To tackle the general problem, it would clearly be helpful to know how many subgroups there are. An elementary argument gives that there are at least 2^{n^2/16} subgroups, and it was conjectured by Pyber in 1993 that up to lower order error terms this is also an upper bound. This talk will present an answer to Pyber's conjecture.

This is joint work with Warwick’s own Gareth Tracey.

27 October 2023: Juergen Branke (Warwick Business School) Bayesian Optimisation and Common Random Numbers

Bayesian optimisation algorithms are global optimisation algorithms for expensive-to-evaluate black-box problems, as they often occur when a solution candidate needs to be evaluated using simulation or physical experiments. They build a surrogate model, usually a Gaussian Process, based on the data collected to far, and then use this surrogate model to decide which new solution candidate to evaluate in the next iteration to maximise the value of information gained.
This makes the algorithm very sample efficient, and in recent years, Bayesian optimisation has become very popular in particular for machine learning hyperparameter tuning and engineering design.

This talk will start with a general introduction to Bayesian optimisation, discussing some of the key open challenges. The second part will then focus on how to effectively exploit common random numbers. Many objective functions (e.g., stochastic simulators) require a random number seed as input. By explicitly reusing a seed, the algorithm can compare two or more solutions under the same randomly generated scenario, such as a common customer stream in a job shop problem, or the same random partition of training data into training and validation set for a machine learning algorithm. Our proposed Knowledge Gradient for Common Random Numbers exploits this and iteratively determines a combination of solution candidate and random seed to evaluate next.

3 November 2023: Henna Koivusalo (Bristol) The Tales of Aperiodic Order

Aperiodic order is at most loosely term to describe discrete point sets (or tilings), which have no translational period but feature some signs of long-range organisation. The tale of the study of aperiodic order is fundamentally intertwined with physics, but as a field of mathematics also lies in the deep shadow of logic. My take on this story will cover the past 60-odd years in approximate chronological order, beginning with first examples of aperiodic tilesets, the Nobel prize-winning discovery of quasicrystal materials, and the quest to find wild quasicrystals, and ending with the unbelievable story, from just earlier this year, of finding the first aperiodic monotile.

Time permitting, I will explain in further detail some results on my favourite method for producing aperiodic order, the cut and project sets, which are obtained by taking an irrational slice through a lattice and projecting it to a lower dimensional subspace. The definition of cut and project sets allows for many interpretations and generalisations, and they can naturally be studied in the context of dynamical systems, discrete geometry, harmonic analysis, or Diophantine approximation, for example, depending on one's own tastes and interests.

10 November 2023: Rob Hollingworth (RIS), Tom Montenegro-Johnson, Randa Herzallah, Impact - what it is, how it's done, and why it's good for you

Impact is about how academics reach out to the wider world. This can arise through working with industry, local or national agencies, or through public understanding and involvement. The success of the Maths Institute in the next REF assessment will be critically dependent on both specific Impact Case Studies and the general role of impact within the department.

This three-part talk will explain what Impact is and what it means to the Maths Institute, it will inform about how you can get involved with Impact activities and what this means for you, and it will give one (or two depending on time) examples of Impact Case studies. The colloquium will specifically address topics of relevance for those at the start of their own impact journey (or who may not even know how impactful their activities could be!).

17 November 2023: Jon Chapman (Oxford) Asymptotics beyond all orders: the devil's invention?

"Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever."
— N. H. Abel.

The lecture will introduce the concept of an asymptotic series, showing how useful divergent series can be, despite Abel's reservations. We will then discuss Stokes' phenomenon, whereby the coefficients in the series appear to change discontinuously. We will show how understanding Stokes' phenomenon is the key which allows us to determine the qualitative and quantitative behaviour of the solution in many practical problems. Examples will be drawn from the areas of surface waves on fluids, crystal growth, dislocation dynamics, and Hele-Shaw flow.

24 November 2023: Aretha Teckentrup (Edinburgh) Deep Gaussian process priors in infinite-dimensional inverse problems

Deep Gaussian processes have proved remarkably successful as a tool for various statistical inference tasks. This success relates in part to the flexibility of these processes and their ability to capture complex, non-stationary behaviours. In this talk, we introduce deep Gaussian processes as prior distributions in infinite-dimensional inverse problems, and demonstrate their superiority in example applications including computational imaging and regression. We will discuss recent algorithmic developments for efficient sampling, as well as recent theoretical results which give crucial insight into the behaviour of the methodology.

1 December 2023: Julian Sahasrabudhe (Cambridge) Diagonal Ramsey numbers and high dimensional geometry

Let R(k) be the kth diagonal Ramsey number: that is, the smallest n for which every 2-colouring of the edges of K_n contains a monochromatic K_k. In recent work with Marcelo Campos, Simon Griffiths and Rob Morris, the speaker showed that R(k) < (4-c)^k, for some absolute constant c>0, which was the first exponential improvement over the bound of Erdős and Szekeres, proved in 1935. In this talk I will discuss the proof and a connection with a conjecture on random variables that take values in high dimensional space. If true, this conjecture has further implications for our understanding of the Ramsey numbers.

8 December 2023: John Gibbon (Imperial) Regularity and multifractality in passive and active turbulent Navier-Stokes-like flows

I will begin with a survey of the regularity properties of the incompressible Navier-Stokes equations (NSEs) – one of the Millenium Clay Prize problems ­ – including the weak solution properties of Leray (1934). I will contrast these with the results that we would like to prove to gain full regularity but have not yet done so. Then I will move on to a brief description of the Multifractal Model (MFM), developed by Parisi and Frisch (1985) to describe homogeneous turbulence. I will show that there exists an intriguing correspondence between the NSEs and the MFM. Finally, I will consider the incompressible Toner-Tu equations (ITT) that describe flocking phenomena in active turbulence. They enjoy many similar properties to those possessed by the NSEs, so many results can be lifted over.

12 January 2024: Chiara Saffirio (Basel) From microscopic to macroscopic scales: many interacting quantum particles and their semiclassical approximation

Systems of interacting particles describing notable physical phenomena, such as Bose-Einstein condensation, superconductivity or superfluidity, exhibit a daunting complexity. This complexity renders the exact many-body theory computationally non-approachable, even for physicists conducting computer experiments and simulations. Therefore, an approximate description using effective macroscopic models is highly useful, and the rigorous study of the regime of validity of such approximations is of primary importance in mathematical physics.

In this talk we will focus on the dynamics of systems made of quantum particles in the mean-field regime, where weakly interacting particles exhibit a collective behavior approximated by an averaged potential in convolution form. We study time scales where the semiclassical description becomes relevant. Through a novel technique based on weak-strong stability principles for partial differential equations, we show that the many-body dynamics is well approximated by a Vlasov equation describing the evolution of the effective probability density of particles on the phase space.

19 January 2024: John MacKay (Bristol) Group actions on L^1 spaces

An important (and classical) way to study groups is through their possible affine isometric actions on Hilbert spaces: this leads to the study of Kazhdan's Property (T) and related notions which have useful applications both in group theory and outside, e.g. in dynamics. But it is natural to consider actions on other Banach spaces too, for example, there have been breakthroughs recently by Oppenheim and de Laat-de la Salle for actions on L^p spaces (1<p<infinity). We'll take a tour through this area, and discuss recent joint work with Drutu where we find new actions on L^1 spaces for groups with hyperbolic features.

26 January 2024: Ben Krause (Bristol) 90 years of pointwise ergodic theory

In this talk I will discuss the major highlights of pointwise ergodic theory, beginning with Birkhoff's classical pointwise ergodic theorem, then turning to Bourgain's work on polynomial ergodic theorems, and concluding with modern directions.

2 February 2024: Vasiliki Evdoridou (Open University) Walking on the circle with Denjoy and Wolff

In Complex Analysis, the behaviour of the iterates of the holomorphic self-maps of the unit disc is described by the well-known Denjoy-Wolff theorem. An important class of such maps are inner functions which, in some sense, map the unit circle to irself. In the case of inner functions, is is of great interest to study the behaviour of boundary points, and a remarkable dichotomy on this behaviour was obtained by Aaronson and Doering&Mañé. In this talk, we see how these results generalise when instead of iterates we consider forward compositions of such functions. This is joint work with A.M. Benini, N. Fagella, P. Rippon and G. Stallard.

9 February 2024: Eric Lauga (Cambridge) Biological flows inside cells

Biology is dominated by transport problems involving fluid flows, from the transport of nutrients and locomotion to flows around plants and the circulatory system of animals. In this talk, I will discuss two instances of flows arising inside living cells. First I will present our work modelling natural cytoplasmic streaming in Drosophila embryo, an elongated multi-nucleated cell that is a classical model system for eukaryotic development and morphogenesis (link). I will next discuss our work on artificial cytoplasmic streaming, rationalising recent experiments that generate artificially induced intracellular flows using focused light localised inside individual cells (link).

16 February 2024: Viveka Erlandsson (Bristol) Counting curves à la Mirzakhani

While it is a classical result that the number of closed geodesics of bounded length on a hyperbolic surface grow exponentially in their length, the situation is very different when one considers certain subsets of closed geodesics. An important breakthrough in this direction was Maryam Mirzakhani’s work counting simple geodesics, proving they grow (asymptotically) polynomially in the length. Since then there has been a lot of interest in this type of questions and I will describe some recent development and generalizations of Mirzakhani’s work.

23 February 2024: Kenneth Falconer (St Andrews) Fractals and intermediate dimensions

We will give a short overview of aspects of fractals such as iterated function systems and self-similar and self-affine sets and introduce Hausdorff and box-counting dimensions. In particular we will look at sets where these dimensions differ. We will then show how Hausdorff and box-counting dimensions can be regarded as particular cases of a spectrum of `intermediate’ dimensions’ and discuss properties and examples of intermediate dimensions.

1 March 2024: Christian Ikenmeyer (Warwick Maths/Computer Science) P versus NP and connections to algebraic geometry, representation theory, and algebraic combinatorics

The P versus NP conjecture is the most famous conjecture at the intersection of mathematics and theoretical computer science. The fundamental open questions in algebraic and geometric complexity theory are variants of the P vs NP conjecture with natural connections to algebraic geometry, representation theory, and algebraic combinatorics. In this talk I will outline some of these conjectures, approaches, and connections.

8 March 2024: Matthew England (Coventry) Recent developments in real quantifier elimination technology

Quantifier Elimination (QE) may be considered as a form of simplification in mathematical logic: given a quantified logical statement QE will produce a statement which is equivalent as does not involve any logical quantifiers (there exists / for all). Real QE refers to the case where the logical atoms are constraints on polynomials over the real numbers: in this case the work of Tarski shows that QE is always possible with algorithms relying on results from algebraic geometry and related areas of mathematics.

The best known method for Real QE was Cylindrical Algebraic Decomposition (CAD) proposed by Collins in the 1970s. However, CAD is known to have doubly exponential complexity, in effect producing a wall beyond which its application is infeasible. In this talk we will introduce QE and CAD, and describe three recent algorithmic advances the author has been involved in which "push back" that doubly exponential wall:

(1) the design of new algorithms in integration with and inspired by SAT-solvers;

(2) the improved optimisation of algorithms available from a proof system implementation; and

(3) the optimisation of algorithms through the use of machine learning to tune algorithm decisions while maintaining exact results.

15 March 2024: Coralia Cartis (Oxford) Dimensionality reduction techniques for nonconvex optimization

Modern applications such as machine learning involve the solution of huge scale nonconvex optimization problems, sometimes with special structure. Motivated by these challenges, we investigate more generally, dimensionality reduction techniques in the variable/parameter domain for local and global optimization that rely crucially on random projections.
We describe and use sketching results that allow efficient projections to low dimensions while preserving useful properties, as well as other tools from random matrix theory and conic integral geometry. We focus on functions with low effective dimensionality, a common occurrence in applications involving overparameterized models and that can serve as an insightful proxy for the training landscape in neural networks. We obtain algorithms that scale well with problem dimension, allow inaccurate information and biased noise, have adaptive parameters and benefit from high-probability complexity guarantees and almost sure convergence.

26 April 2024: J Nathan Kutz (Washington) Data-driven model discovery and physics-informed learning

A major challenge in the study of dynamical systems is that of model discovery: turning data into reduced order models that are not just predictive, but provide insight into the nature of the underlying dynamical system that generated the data. We introduce a number of data-driven strategies for discovering nonlinear multiscale dynamical systems and their embeddings from data. We consider two canonical cases: (i) systems for which we have full measurements of the governing variables, and (ii) systems for which we have incomplete measurements. For systems with full state measurements, we show that the recent sparse identification of nonlinear dynamical systems (SINDy) method can discover governing equations with relatively little data and introduce a sampling method that allows SINDy to scale efficiently to problems with multiple time scales, noise and parametric dependencies. For systems with incomplete observations, we show that the Hankel alternative view of Koopman (HAVOK) method, based on time-delay embedding coordinates and the dynamic mode decomposition, can be used to obtain a linear models and Koopman invariant measurement systems that nearly perfectly captures the dynamics of nonlinear quasiperiodic systems. Neural networks are used in targeted ways to aid in the model reduction process. Together, these approaches provide a suite of mathematical strategies for reducing the data required to discover and model nonlinear multiscale systems.

3 May 2024: Patrick Farrell (Oxford) Computing multiple solutions of nonlinear problems

Many nonlinear problems exhibit multiple solutions—think of an umbrella inverted by the wind, a light switch, or a child's seesaw. Computing the distinct solutions of a nonlinear problem as its parameters are varied is a central task in applied mathematics and engineering. The solutions are captured in a bifurcation diagram, plotting (some functional of) the solutions as a function of the parameters. In this talk I will present a new algorithm, deflated continuation, for this task.

Deflated continuation has three advantages. First, it is capable of computing disconnected bifurcation diagrams; previous algorithms only aimed to compute that part of the bifurcation diagram continuously connected to the initial data. Second, its implementation is very simple: it only requires a minor modification to an existing Newton-based solver. Third, it can scale to very large discretisations if a good preconditioner is available; no auxiliary problems must be solved.

We will present applications to hyperelastic structures, liquid crystals, and Bose-Einstein condensates, among others.

10 May 2024: June Barrow-Green (Open University) Ronald Ross and Hilda Hudson: A collaboration on the mathematical theory of epidemics

In 1916 the physician Ronald Ross published the first of three papers on the mathematical study of epidemiology or, as he called it, ‘pathometry’. The second and third of these papers appeared the following year co-authored with the mathematician Hilda Hudson.   At the time Hudson, who had ranked equivalent to the 7th wrangler in the 1903 Cambridge Mathematical Tripos, was known for her work on Cremona Transformations. So how and why did Hudson, a geometer, end up collaborating with Ross on the theory of epidemics? And what role did she play? In my talk I shall discuss the nature and extent of their collaboration, as well as the genesis, content, and significance of their work.

17 May 2024: Tim Burness (Bristol) Simple groups, fixed point ratios and applications

Let G be a group acting on a finite set X. The fixed point ratio (FPR) of an element g in G is simply the proportion of points in X fixed by g. Calculating, or bounding, FPRs has been a central problem in permutation group theory for many decades, finding numerous applications. In this talk, I will survey some of the main FPR results in the special case where G is a simple group, which is an area where there has been several major advances in recent years. I will also highlight the diverse range of applications, which includes powerful new results on bases for permutation groups, the connectivity of generating graphs and the commuting probability of finite groups.

24 May 2024: Henry Wilton (Cambridge) Rational curvature invariants of 2-dimensional complexes

I will discuss some new invariants of 2-complexes. They are inspired by recent developments in the theory of one-relator groups, but also have the potential to unify the theories of many well-studied families including small-cancellation presentation complexes, CAT(0) 2-complexes and 3-manifold spines, in addition to the motivating examples of one-relator presentation complexes. The fundamental result is that these invariants are the extrema of explicit linear-programming problems, and in particular are rational, computable and realised. The definitions suggest a conjectural “map” of 2-complexes, which I will attempt to describe.

31 May 2024: Mark Cannon (Oxford) Difference of Convex Functions in Robust and Adaptive Model Predictive Control

This talk will provide an overview of problems and methods in control systems research, with a particular focus on data-driven optimisation-based control in applications where safety is critical. Starting with a brief introduction to the main concepts and challenges, the discussion will motivate recent work on Model Predictive Control (MPC) in this context. We will consider how to use differences of convex (DC) functions to derive convex conditions that allow the future control system performance to be optimized as a sequence of convex sub-problems. Key contributions to theoretical results, including feasibility, convergence, optimality and closed loop stability will be discussed.

The talk will explain how DC function representations can be computed directly from data and how model estimation can be performed online simultaneously with control to define safe learning-based control algorithms. We describe an application of these ideas to a problem of controlling the transistions of a tiltwing vertical take-off and landing (VTOL) aircraft subject to wind disturbances and model uncertainty. Results of a case study will be discussed involving a VTOL aircraft model with unknown wind gusts and aerodynamics defined by experimental data.