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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 up-to-date programme, please see our calendar of events

Previous talks

DateSpeakerAbstract

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 long-term 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 well-understood 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 top-down 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 bottom-up 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 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 = (ai+j)i,j≥0 of real numbers is totally positive if and only if the underlying sequence (an)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 atn can be expanded as a Stieltjes-type 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 (an)n≥0 of polynomials will be called coefficientwise Hankel-totally positive if the Hankel matrix H = (ai+j)i,j≥0 associated to (an)n≥0 is coefficientwise totally positive.  It turns out that many sequences of polynomials arising naturally in enumerative combinatorics are (empirically) coefficientwise Hankel-totally 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 decision-making 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 decision-maker 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 worst-case 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

  
DateSpeakerAbstract
19 April 2023Gesine Reinert (University of Oxford)

Assessing goodness of fit for network models

Networks are often used to represent complex dependencies in data, and network models can aid the understanding of such dependencies.  These models can be parametric, but they could also be implicit, such as the output of an automated synthetic data generator.  For assessing the goodness of fit of a model, independent replicas are often assumed.  However, when the data are given in the form of a network, usually there is only one network available.  Classical likelihood ratio methods may fail even in parametric models such as exponential random graph models, as due to an intractable normalising constant, the likelihood cannot be calculated explicitly.  This talk will present some network models.  We shall introduce a kernelized goodness of fit test (which is based on Stein's method), give performance guarantees, and illustrate its use.  The talk is based on joint work with Nathan Ross and with Wenkai Xu.

22 March 2023Volker Mehrmann (Technische Universität Berlin)

Energy-based mathematical modelling, simulation and control of real-world systems

Most real-world dynamical systems consist of subsystems from different physical domains, modelled by partial differential equations, ordinary differential equations and algebraic equations, combined with input and output connections.  To deal with such complex systems, in recent years the class of dissipative port-Hamiltonian (pH) descriptor systems has emerged as a very successful modelling methodology.  The main reasons are that the network-based interconnection of pH systems is again pH, Galerkin projection in PDE discretization and model reduction preserve the pH structure, and the physical properties are encoded in the geometric properties of the flow as well as the algebraic properties of the equations.  Furthermore, dissipative pH systems form a very robust representation under structured perturbations and directly indicate Lyapunov functions for stability analysis.  Yet another advantage of energy-based modelling via pH systems is that each separate model of a physical system can be an entire model catalogue from which models can be chosen in an adaptive way within simulation and optimization methods.  We will discuss the class of constrained pH systems and show that many classical real-world mathematical models can be formulated in this class.  We will illustrate the results with some real-world examples from gas transport and district heating systems, and point out emerging mathematical challenges.real-world systems

8 February 2023Nira Chamberlain OBE (Loughborough University and SNC-Lavalin)

The mathematics that can stop an AI apocalypse.

Abstract: How do you prevent artificial intelligence (AI) from taking over the world?  In this lecture, Professor Chamberlain will discuss how mathematics is providing crucial answers.  Mathematical modelling is the most creative side of applied mathematics, which itself connects pure mathematics with science and technology.  Mathematical models look into the real world, translate it into mathematics, solve that mathematics, and then apply the solution back into the real world. Professor Chamberlain will examine a mathematical model of the complexities of human behaviour that caused the 2008 Financial Crisis, and he’ll then go on to show that the same model can be used to investigate how to minimize the chances of an AI takeover.  The late great Professor Stephen Hawking once said, “The development of full artificial intelligence could spell the end of the human race”.  He also stated that he advocated research into precautionary measures to ensure that future super-intelligent machines remain under human control.  However, an AI apocalypse is not necessarily robots marching down the street, as there are several examples subtler than this.  So, what is the risk of an AI apocalypse, and can we calculate this probability?

DateSpeakerAbstract
20 October 2021Alain Goriely (Oxford)

Modelling dementia

Neurodegenerative diseases such as Alzheimer’s or Parkinson’s are
devastating conditions with poorly understood mechanisms and no known cure.

Yet a striking feature of these conditions is the characteristic pattern of invasion throughout the brain, leading to well-codified 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 2021Richard Thomas FRS (Imperial

Counting things:
enumerative algebraic geometry from string theory

For centuries mathematicians have generalised statements like “there is a unique line through any 2 points”, but with increasing technical
difficulties. It was not until the late 1990s that new ideas from
mathematics and physics allowed rigorous definitions to be made of these “curve counting problems”.

I will outline two different ways to count curves, assuming only a bit
of undergraduate complex analysis. The “MNOP conjecture” states that the two definitions give equivalent information. Its recent proof by
Pandharipande and Pixton has enabled the solution of various counting problems, such as the “KKV conjecture” from string theory, expressing all cuTBArve counting problems on “K3 surfaces” in terms of modular forms.

8 December 2021Xue-Mei Li (Imperial), TBA
4 May 2022Karen Vogtmann FRS (Warwick))TBA
21 April 2021Sylvia 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 ”mean-field” 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 2021Christoph Schweigert (Hamburg)

Quantum groups and antipodes

In the definition of a group in a basic algebra course, one does not
impose the axiom that (g^-1)^-1 =g for all group elements; one rather
derives this identity from the axioms.

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
linked to many classical structures in algebra.

24 February 2021Stefan Hollands (Leipzig)

(In)determinism Inside Black Holes

In classical General Relativity, the values of fields (e.g.
electromagnetic, fluids, ...) on spacetime are uniquely determined by
their initial values within the "domain of dependence". However, it may occur that the spacetime under consideration extends beyond this domain, and fields, therefore, are not entirely determined by their initial values. In fact, such a failure of determinism occurs inside all
physically relevant black holes.

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 --
and thus whether there is a real problem with determinism inside black holes -- has been put into question recently.

In this colloquium, I will explain the issue with pictures and discuss
whether quantum effects will come to the rescue of strong cosmic censorship.

2 December 2020Professor Jose Carrillo (Oxford)

Aggregation-Diffusion and Kinetic PDEs for collective behaviour:
applications in the sciences

I will present a survey of micro, meso and macroscopic models where
repulsion and attraction effects are included through pairwise
potentials. I will discuss their interesting mathematical features and
applications in mathematical biology and engineering. Qualitative
properties of local minimizers of the interaction energies are crucial
in order to understand these complex behaviours. I will showcase the
breadth of possible applications with three different phenomena in
applications: segregation, phase transitions and consensus.

11 November 2020Professor 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 2020Professor Gerda Claeskens (KU Leuven)

Correct inference after model selection

Post-selection inference is a rather recent methodology that
incorporates the extra variability added by model selection to perform
valid inference. When a model is not given, but is the result of a model search endeavour, the uncertainty about the model that is used for inference has consequences for hypothesis testing and for the
construction of confidence intervals for the model parameters of
interest. Ignoring this uncertainty leads to overoptimistic results,
p-values that are too small and confidence intervals that are too narrow.

In this talk we shall mainly focus on the well-known 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 post-AIC selection confidence intervals are wider than
the classic naïve confidence intervals, reaffirming that ignoring model
selection leads to invalid inference.

29 April 2020

Professor Mathias Gaberdiel (ETH Zürich)

Postponed due to coronavirus
pandemic

18 March 2020

Professor Nick Higham FRS (Manchester)

Postponed due to coronavirus
pandemic

School of Mathematics Colloquium Talks 2018-2019

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 non-technical 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 PoincaréBirkhoff-Witt theorem

The Poincaré-Birkhoff-Witt 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 category-theoretic 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 mean-field 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 self-similar 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 (Paris-Dauphine) : 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 V-statistic and, therefore, has rate of convergence which is much faster than the classical √n-rates. 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

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 Gromov-Witten 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.

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 leading-order 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} large-scale behavior of $a$-harmonic functions --- which is more regular than suggested by the classical counter-examples. 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 a-priori 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 eye-like 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 3-space 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 Keen-Series 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 non-discrete 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 non-reductive 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).

DateSpeakerAbstract

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 long-term 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 well-understood class of geometric problems that has long served as a laboratory for testing new ideas in enumerative geometry.

  

Find out more

For more information please contact:

Professor Roger Behrend

Professor Roger Behrend

Professor

Email
behrendr@cardiff.ac.uk
Telephone
+44 (0)29 2087 5543