Maths Colloquia

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 shear-thinning (e.g., in quicksand) or shear-thickening (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

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 Sub-Panel, 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 research-facing 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 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.

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 tⁿ 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 (a_n)_n≥0 of polynomials will be called coefficientwise Hankel-totally 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 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

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

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.

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

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?

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.

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.

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.

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.

(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.

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.

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

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.