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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.