Maths Colloquia
The School Colloquia are given by eminent speakers and present an overview of important topics of general interest in the mathematical sciences.
These invited lectures are intended to be accessible to all graduate students and academics in the department. MMath and MSc students may also benefit from these presentations.
For an uptodate programme, please see our calendar of events
Previous talks
2023/24
Date  Speaker  Abstract 

1 May 2024  Helen Wilson (University College London)  Understanding the rheology of suspensions. Materials made from a mixture of liquid and solid are, instinctively, very obviously complex. From dilatancy (the reason wet sand becomes dry when you step on it) to extreme shearthinning (e.g., in quicksand) or shearthickening (e.g., in cornflour oobleck), there is a wide range of behaviours to explain and predict. I'll discuss the seemingly simple case of solid spheres suspended in a Newtonian fluid matrix, which still has plenty of surprises up its sleeve. 
17 April 2024  Simon ChandlerWilde (University of Reading)  Publication strategies, 4* papers and the Research Excellence Framework. In this talk, I will say something, as a former Research Excellence Framework (REF) Output Assessor for the REF 2021 Mathematical Sciences SubPanel, about how output assessment works for REF. I will also make some comments about what makes a 4* paper, and will talk about how that interacts with wider considerations of personal publication strategies, informed by my own work as a researcher across mathematics and its applications, and by my various previous university researchfacing roles. 
13 February 2024  Frank Sottile (Texas A&M University)  Galois groups in enumerative geometry and applications. In 1870 Jordan explained how Galois theory can be applied to problems from enumerative geometry, with the Galois group encoding intrinsic structure of the problem. Earlier, Hermite showed the equivalence of Galois groups with geometric monodromy groups, and in 1979 Harris initiated the modern study of Galois groups of enumerative problems. He posited that a Galois group should be “as large as possible”, in that it will be the largest group preserving internal symmetry in the geometric problem. I will describe this background and discuss some work in a longterm project to compute, study and use Galois groups of geometric problems, including those that arise in applications of algebraic geometry. A main focus is to understand Galois groups in the Schubert calculus, a wellunderstood class of geometric problems that has long served as a laboratory for testing new ideas in enumerative geometry. 
6 December 2023  Andrea Liu (University of Pennsylvania)  How physical systems can learn by themselves. In order for artificial neural networks to learn a task, one must solve an inverse design problem. What network will produce the desired output? We have harnessed AI approaches to design physical systems to perform functions inspired by biology. But artificial neural networks require a computer in order to learn in topdown fashion by minimizing a cost function. By contrast, the brain learns bottom up on its own, with each neuron adjusting itself and its synapses without knowing what all the other neurons are doing, and without the aid of an external computer. We have introduced an approach to bottomup learning that has been realized in physical systems that learn on their own. 
22 November 2023  Alan Sokal (University College London)  Total positivity, the classical moment problem, and enumerative combinatorics. A matrix M of real numbers is called totally positive if every minor of M is nonnegative. Gantmacher and Krein showed in 1937 that a Hankel matrix H = (a_{i+j})_{i,j≥0} of real numbers is totally positive if and only if the underlying sequence (a_{n})_{n≥0} is a Stieltjes moment sequence, i.e., the moments of a positive measure on [0,∞). Moreover, this holds if and only if the ordinary generating function ∑_{n≥0} a_{n }t^{n} can be expanded as a Stieltjestype continued fraction with nonnegative coefficients. So, totally positive Hankel matrices are closely connected with the Stieltjes moment problem and with continued fractions. Here, I will introduce a generalization: a matrix M of polynomials (in some set of indeterminates) will be called coefficientwise totally positive if every minor of M is a polynomial with nonnegative coefficients. And a sequence (a_{n})_{n≥0} of polynomials will be called coefficientwise Hankeltotally positive if the Hankel matrix H = (a_{i+j})_{i,j≥0} associated to (a_{n})_{n≥0} is coefficientwise totally positive. It turns out that many sequences of polynomials arising naturally in enumerative combinatorics are (empirically) coefficientwise Hankeltotally positive. In some cases, this can be proven using continued fractions, by either combinatorial or algebraic methods; I will sketch how this is done. In other cases, this can be done using a more general algebraic method called production matrices. However, in a large number of other cases it remains an open problem. 
18 October 2023  Brian Denton (University of Michigan)  Optimization in the presence of model ambiguity in Markov decision processes. Optimizing sequential decisionmaking under uncertainty is essential in many contexts, including inventory control, finance, healthcare, and many others. One of the most common model formulations is the Markov decision process (MDP). However, ambiguity in the MDP model parameters can introduce challenges because recommendations from MDPs depend on the underlying model, and there are often multiple plausible models. To address this problem, we present a framework in which a decisionmaker considers multiple models of the MDP’s ambiguous parameters and seeks to find a policy that maximizes an aggregate measure of performance with respect to each of these models of the MDP, such as weighted rewards, regret, or worstcase performance. I will discuss connections to other models in the stochastic optimization literature, complexity results, and solution methods for solving these problems. I’ll illustrate the approach with two examples, one in the context of preventative treatment for cardiovascular disease and the other in the context of machine maintenance. Finally, I’ll conclude with a summary of the most important takeaway messages from the study 
2022/23
Date  Speaker  Abstract 

19 April 2023  Gesine Reinert (University of Oxford)  Assessing goodness of fit for network models

22 March 2023  Volker Mehrmann (Technische Universität Berlin)  Energybased mathematical modelling, simulation and control of realworld systems

8 February 2023  Nira Chamberlain OBE (Loughborough University and SNCLavalin)  The mathematics that can stop an AI apocalypse

2020/21
Date  Speaker  Abstract 

20 October 2021  Alain Goriely (Oxford)  Modelling dementia Neurodegenerative diseases such as Alzheimer’s or Parkinson’s are Yet a striking feature of these conditions is the characteristic pattern of invasion throughout the brain, leading to wellcodified disease stages visible to neuropathology and associated with various cognitive deficits and pathologies. In this talk, I will show how we can use mathematical modelling to gain insight into this process and, doing so, gain some understanding on how the brain works. In particular, by looking at protein dynamics on the neuronal network, we can unravel some of the universal features associated with dementia that are driven by both the network topology and protein kinetics. By further coupling this approach with functional models of the brain, we will show that we can explain important aspects of function loss during disease development. 
24 November 2021  Richard Thomas FRS (Imperial  Counting things: For centuries mathematicians have generalised statements like “there is a unique line through any 2 points”, but with increasing technical I will outline two different ways to count curves, assuming only a bit 
8 December 2021  XueMei Li (Imperial),  TBA 
4 May 2022  Karen Vogtmann FRS (Warwick))  TBA 
21 April 2021  Sylvia Serfaty (New York)  Systems of points with Coulomb interactions Large ensembles of points with Coulomb interactions arise in various settings of condensed matter physics, classical and quantum mechanics, statistical mechanics, random matrices and even approximation theory, and they give rise to a variety of questions pertaining to analysis, Partial Differential Equations and probability. We will first review these motivations, then present the ”meanfield” derivation of effective models and equations describing the system at the macroscopic scale. We then explain how to analyze the next order behavior, giving information on the configurations at the microscopic level and connecting with crystallization questions, and finish with the description of the effect of temperature. 
24 March 2021  Christoph Schweigert (Hamburg)  Quantum groups and antipodes In the definition of a group in a basic algebra course, one does not A quantum group is a generalization of a group that describes more general symmetries. A quantum group comes with a map S: H → H that generalizes the map g ↦g^1. No general statement can be made for the second power S^2 of this map; but for the fourth power, an explicit formula is known due to Radford (1976). We explain how this seemingly technical statement is deeply 
24 February 2021  Stefan Hollands (Leipzig)  (In)determinism Inside Black Holes In classical General Relativity, the values of fields (e.g. The boundary of the region within where determinism is unchallenged is called the ``Cauchy horizon'. Penrose has proposed ("strong cosmic censorship conjecture") that this view may be too naive and that the Cauchy horizon is actually unstable: the slightest perturbation might convert it to a final singularity. Whether or not this is the case  In this colloquium, I will explain the issue with pictures and discuss 
2 December 2020  Professor Jose Carrillo (Oxford)  AggregationDiffusion and Kinetic PDEs for collective behaviour: I will present a survey of micro, meso and macroscopic models where 
11 November 2020  Professor Poul Hjorth (Technical University of Denmark)  Poetry in Motion The unfolding of some strictly syntax based poetry is viewed from the perspective of dynamical systems theory. I will provide examples of this, in particular of a 13th century lyrical style, the generalization of which gives rise to various mathematical questions. In the search for answers we find ourselves in 20th century mathematics of iterative maps, and chaotic dynamics. Joint work with A.R.Champneys, U. of Bristol. 
28 October 2020  Professor Gerda Claeskens (KU Leuven)  Correct inference after model selection Postselection inference is a rather recent methodology that In this talk we shall mainly focus on the wellknown Akaike information criterion (AIC) for model selection and the effects on the construction of confidence intervals after its use. By unravelling the selection method, it is possible to incorporate the uncertainty about the selected model and to obtain confidence intervals that have the correct coverage. As a result, the postAIC selection confidence intervals are wider than 
29 April 2020  Professor Mathias Gaberdiel (ETH Zürich)  Postponed due to coronavirus 
18 March 2020  Professor Nick Higham FRS (Manchester)  Postponed due to coronavirus 
2018/19
School of Mathematics Colloquium Talks 20182019
27 September 2018 Dr Peter Hintz (MIT) : Stability of black holes
More than a hundred years ago, Schwarzschild first wrote down the mathematical description of a black hole; on a technical level, black holes are certain types of solutions of Einstein's equations of general relativity. While they have since become part of popular culture, many fundamental questions about them remain unanswered: for example, it is not yet known mathematically if they are stable! I will explain what that means and outline a recent proof of full nonlinear stability (obtained in joint work with A. Vasy) in the case that the cosmological constant is positive, a condition consistent with current cosmological models of the universe.
The talk is intended as a nontechnical introduction to the subject, with a focus on the central role played by modern microlocal and spectral theoretical techniques.
14 November 2018 Professor Vladimir Dotsenko (Trinity College Dublin) : Old and new aspects of the PoincaréBirkhoffWitt theorem
The PoincaréBirkhoffWitt theorem on universal enveloping algebras of Lie algebras is one of the fundamental results in many areas of mathematics: from differential geometry and representation theory to homological algebra and deformation quantisation. I shall give a short overview of that result and some of its proofs that emerged in about 120 years since Poincaré published a paper about it, and outline a new proof which perhaps captures its categorytheoretic essence in the best way possible. The talk is partly based on a joint work with Pedro Tamaroff.
12 December 2018 Professor Barbara Niethammer (Bonn) : Smoluchowski's classical coagulation model
In 1916 Smoluchowski derived a meanfield model for mass aggregation in order to develop a mathematical theory for coagulation processes. Since Smoluchowski's groundbreaking work his model has been used in a diverse range of applications such as aerosol physics, polymerization, population dynamics, or astrophysics. After reviewing some basic properties of the model I will address the fundamental question of dynamic scaling, that is whether solutions develop a universal selfsimilar form for large times.
This issue is only understood for some exactly solvable cases, while in the general case most questions are still completely open. I will give an overview of the main results in the past decades and explain why we believe that in general the scaling hypothesis is not true
30 January 2019 Professor Ivar Ekeland (ParisDauphine) : Inverse function theorems, soft and hard
6 March 2019 Professor Kathryn Hess (Lausanne) : Topology meets neuroscience
I will present an overview of the wide variety of applications of topology to neuroscience that my group has worked on over the past few years, including classification of neuron morphologies and structural and functional connectomics and network plasticity. This work has been carried out in collaboration with the Blue Brain Project at the EPFL.
13 March 2019 Professor Thomas Mikosch (Copenhagen) : Testing independence of random elements with the distance covariance
This is joint work with Herold Dehling (Bochum), Muneya Matsui (Nagoya), Gennady Samorodnitsky (Cornell) and Laleh Tafakori (Melbourne). Distance covariance was introduced by Székely, Rizzo and Bakirov (2007) as a measure of dependence between vectors of possibly distinct dimensions. Since then it has attracted attention in various fields of statistics and applied probability. The distance covariance of two random vectors X, Y is a weighted L2 distance between the joint characteristic function of (X, Y) and the product of the characteristic functions of X and Y. It has the desirable property that it is zero if and only if X, Y are independent. This is in contrast to classical measures of dependence such as the correlation between two random variables: zero correlation corresponds to the absence of linear dependence but does not give any information about other kinds of dependencies. We consider the distance covariance for stochastic processes X, Y defined on some interval and having square integrable paths, including Lévy processes, fractional Brownian, diffusions, stable processes, and many more. Since distance covariance is defined for vectors we consider discrete approximations to X, Y. We show that sample versions of the discretized distance covariance converge to zero if and only if X, Y are independent. The sample distance covariance is a degenerate Vstatistic and, therefore, has rate of convergence which is much faster than the classical √nrates. This fact also shows nicely in simulation studies for independent X, Y in contrast to dependent X, Y.
18 June 2019 Professor Graeme Milton (University of Utah) : Exact relations for Greens functions in linear partial differential equations and boundary field equalities: a generalisation of conservation laws
2017/18
8 November 2017 Speaker: Professor Constantin Teleman (Oxford) Title: Gauge theory, Mirror symmetry and Lagrangians
Abstract: A basic invariant of an isolated hypersurface singularity $f(mathbf{x})=0$ is its Jacobian ring, $mathbf{C}[mathbf{x}]/langle partialf/partial x_i angle$. It is known to have a emph{Frobenius algebra} structure, if a volume form is chosen in the ambient space. Frobenius algebras appear in connection to $2$dimensional topological field theories; the extended versions, posited by Segal, Kontsevich and others, involve a Frobenius category. Physicists understood that emph{matrix factorisations} provided a ``categorification'’ of the Jacobian ring, completing the structure of an extended TQFT. In this talk, I will discuss a possible generalisation of matrix factorisations, conjectured by Kapustin and Rozansky, and illustrate it with an example which provides a character calculus for $2$dimensional topological gauge theories, which is relevant to quantum cohomology and GromovWitten theory.
22 November 2017 Speaker: Professor Andrew Stuart (Caltech) Title: Blending Mathematical Models and Data
Abstract: A central research challenge for the mathematical sciences in the 21st century is the development of principled methodologies for the seamless integration of (often vast) data sets with (often sophisticated) mathematical models. Such data sets are becoming routinely available in almost all areas of engineering, science and technology, whilst mathematical models describing phenomena of interest are often built on decades, or even centuries, of human knowledge creation. Ignoring either the data or the models is clearly unwise and so the issue of combining them is of paramount importance. In this talk we will give a historical perspective on the subject, highlight some of the current research directions that it leads to, and describe some of the underlying mathematical frameworks being deployed and developed. The ideas will be illustrated by problems arising in the geophysical, biomedical and social sciences.
7 February 2018 Speaker: Professor Martin Hairer KBE FRS (Imperial) Title: Noisy rubber bands
Abstract: A "rubber band" constrained to remain on a manifold evolves by trying to shorten its length, eventually settling on a closed geodesic, or collapsing entirely. It is natural to try to consider a noisy version of such a model where each segment of the band gets pulled in random directions. Trying to build such a model turns out to be surprisingly difficult and generates a number of nice geometric insights, as well as some beautiful algebraic and analytical objects. We will survey some of the main results obtained on the way to this construction.
2016/17
12 October 2016 Speaker: Prof. Felix Otto (MPIMS, Leipzig) Title: Effective behavior of random media: From an error analysis to regularity theory
Abstract: Heterogeneous media are often naturally described in statistical terms, reflecting a lack of knowledge of details. How to extract their effective behavior on large scales, like the effective conductivity $a_{hom}=const$, from the statistical specifications, which are encoded in a stationary probability measure or ensemble $langlecdot angle$ on the space of microscopic conductivities $a=a(x)$? A practioneers numerical approach is to sample the medium according the $langlecdot angle$ and to determine $a_{hom}$ in the Cartesian directions by imposing simple boundary conditions. What is the error made in terms of the size of this ``representative volume element''? Our interest in what is called ``stochastic homogenization'' grew out of this error analysis, and now also includes a characterization of the leadingorder fluctuations.
In the course of developing such an error analysis, connections with the classical regularity theory of elliptic equations have emerged. More precisely, stochastic homogenization sheds a new light on a {it generic} largescale behavior of $a$harmonic functions  which is more regular than suggested by the classical counterexamples. This might be rephrased in geometric terms: How flat at infinity does a metric $a$ have to be such that the space of harmonic functions of a given polynomial growth rate has exactly the same dimension as in the Euclidean case $a_{hom}$. We give a sufficient criterion that is almost surely satisfied for the type of probability measures $langlecdot angle$ on metrics $a$ considered in stochastic homogenization.
26 October 2016 Speaker: Dr. James Maynard (Oxford) Title: Primes with restricted digits
Abstract: Many of the most important questions about prime numbers can be phrased as 'given some set A of integers, how many primes are in A?'. Unfortunately, even simple versions of such questions are often well beyond current techniques, and this is especially difficult if A is a 'thin' set of integers.
I will talk about recent work which shows that there are infinitely many prime numbers with no 7's in their decimal expansion, giving an example of a thin set where we do get a satisfactory answer. Ideas from probability (such as Markov chains), diophantine geometry (lattice point counting and rational approximation) and combinatorics all turn out to be important ingredients, alongside traditional analytic number theory.
2 November 2016 Speaker: Prof. Robert Weismantel (ETH, Zurich) Title: Integer Polynomial Optimization.
1 February 2017 Speaker: Dr. Nina Golyandina (St Petersburg) Amended Colloquium time: 16:00  17:00, E/0.15 Title: Singular spectrum analysis as a universal approach for finding structure in time series and digital images.
Abstract: Singular spectrum analysis (SSA) is an effective method for processing different objects such as time series and digital images, finding their structures and then using the found structure for trend and periodicity extraction, smoothing, parameter estimation, forecasting, gap imputations. SSA is known as a nonparametric tool, which is able to analyse time series without apriori assumptions about the object model. The method success is based on a specific construction of an adaptive decomposition, which is generated by the object itself. It is surprising how such a modelfree method can solve problems which are conventional for parametric methods. We discuss this kind of paradox and demonstrate the method abilities as well as the mathematics underlying the SSA approach.
15 March 2017 Speaker: Prof. Caroline Series FRS (Warwick) Title: The cover of the December AMS Notices
Abstract: The cover of the December 2016 AMS Notices shows an eyelike region picked out by blue and red dots and surrounded by green rays. The picture, drawn by Yasushi Yamashita, illustrates Gaven Martin’s search for the smallest volume hyperbolic orbifold. It represents a family of two generator groups of isometries of hyperbolic 3space which was recently studied, for quite different reasons, by myself, Yamashita and Ser Peow Tan. After explaining the coloured dots and their role in Martin’s search, we concentrate on the green rays. These are KeenSeries pleating rays which are used to locate spaces of discrete groups. We also introduce some rather mysterious `fake’ pleating rays which partially fill the space of nondiscrete groups and relate to a condition of Bowditch, mentioned but not explained in the Notices.
22 March 2017 Speaker: Prof. Frances Kirwan FRS (Oxford) Title: Moduli spaces of unstable curves.
Abstract: The construction of the moduli spaces of stable curves of fixed genus is one of the classical applications of Mumford's geometric invariant theory (GIT), developed in the 1960s. Here a projective curve is stable if it has only nodes as singularities and its automorphism group is finite. The aim of this talk is to describe these moduli spaces and outline their GIT construction, and then to explain how recent methods from nonreductive GIT can help us to classify the singularities of unstable curves in such a way that we can construct moduli spaces of unstable curves (of fixed singularity type).
2015/16
Date  Speaker  Abstract 

13 February 2024  Frank Sottile (Texas A&M University)  Galois groups in enumerative geometry and applications. In 1870 Jordan explained how Galois theory can be applied to problems from enumerative geometry, with the Galois group encoding intrinsic structure of the problem. Earlier, Hermite showed the equivalence of Galois groups with geometric monodromy groups, and in 1979 Harris initiated the modern study of Galois groups of enumerative problems. He posited that a Galois group should be “as large as possible”, in that it will be the largest group preserving internal symmetry in the geometric problem. I will describe this background and discuss some work in a longterm project to compute, study and use Galois groups of geometric problems, including those that arise in applications of algebraic geometry. A main focus is to understand Galois groups in the Schubert calculus, a wellunderstood class of geometric problems that has long served as a laboratory for testing new ideas in enumerative geometry. 
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