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05 October 2018: Agelos Georgakopoulos (Warwick) From mafia expansion to analytic functions in percolation theory
I will present a (finite) random graph model that admits various definitions, one of which is via a percolation model on an infinite group. This will lead us to an excursion into classical results and open problems in percolation theory.

12 October 2018: Robert Gilman (Stevens Institute) Generating the symmetric group
It is well known that two randomly chosen permutations generate the alternating or symmetric group of degree n with probability tending to 1 as n goes to infinity. This is bad news if you would like to sample 2-generator permutation groups. We study a sampling method which provably avoids this problem, and we discuss a practical heuristic variation.

19 October 2018: Joel Hamkins (Oxford) The rearrangement number: how many rearrangements of a series suffice to validate absolute convergence?
The Riemann rearrangement theorem asserts that a series $\sum_n a_n$ is absolutely convergent if and only if every rearrangement $\sum_n a_{p(n)}$ of it is convergent, and furthermore, any conditionally convergent series can be rearranged so as to converge to any desired extended real value. How many rearrangements $p$ suffice to test for absolute convergence in this way? The rearrangement number, a new cardinal characteristic of the continuum, is the smallest size of a family of permutations, such that whenever the convergence and value of a convergent series is invariant by all these permutations, then it is absolutely convergent. The exact value of the rearrangement number turns out to be independent of the axioms of set theory. In this talk, I shall place the rearrangement number into a discussion of cardinal characteristics of the continuum, including an elementary introduction to the continuum hypothesis and an account of Freiling's axiom of symmetry.

26 October 2018: Andrzej Zuk (Paris 7 and Imperial) From PDEs to groups
We present a construction which associates to a KdV equation the lamplighter group. In order to establish this relation we use automata and random walks on ultra discrete limits. It is also related to the L2 Betti numbers introduced by Atiyah which are homotopy invariants of closed manifolds.

2 November 2018: Dorothy Buck (Bath) Topological Effects -- and Treatments -- of Confined Biopolymers
DNA, like any other long piece of string packed into a small space, would become highly knotted and tangled if there were no mechanisms to both keep it organised, and to untangle any knots that do arise. Every living organism has developed mechanisms to control DNA knotting in its cells, and many antibiotics and chemotherapeutics act by disturbing this control. This talk will give an overview of some of the topological methods that we utilise to model DNA knotting in open chains, closed circles and spatial graphs, and how the answers from these models aid experimentalists. (No prior biological knowledge needed.)

9 November 2018: Ed Brambley (Warwick) Mathematical modelling of aeroacoustics and metal forming
Aeroacoustics is the study of sound generated by airflow, typically in an aircraft engine but with applications to car noise and wind turbine noise, among others. This part of this talk will concentrate on models for acoustic linings that absorb sound. We will see how the model for the last 40 years is provably wrong (ill-posed), what we can do about it, and how even today we do not have a good predictive model for sound absorption by a surface in an airflow.

Metal Forming is bending bits of metal into the right shape. It is how most things from coke cans to cars are made. Finite element analysis is the tool of choice for engineers, but it is not fast enough for real-time control of metal forming processes; so these processes are not controlled. With a controlled metal forming process, we could make parts we can't today; we could reduce the energy needed to make parts; we could use more recycled metal; and, possibly, we could build machines that can make more than one part. The missing ingredient is a sufficiently accurate, sufficiently quick model of how metal forms, which is where mathematical modelling can help. This part of this talk will look at some recent attempts by my group at mathematical modelling of some simple metal forming processes.

16 November 2018: Nathalie Wahl (Copenhagen and Newton Institute) Homological stability: what is that for?
Homological stability is a topological property shared by many configuration spaces and groups of matrices, or, more generally, groups of symmetries. In recent years, it has turned into a powerful computational tool. The talk will give an overview of the subject.

23 November 2018: Impact colloquium - coordinated by Colm Connaughton

Join us for 3 short talks on research impact

1 - Sarah Hall - Why worry about making a difference? The impact agenda in higher education
Warwick’s new Head of Research Impact, Sarah Hall, will give a short presentation exploring various drivers for the impact agenda. This talk examines the question “Why should we be making a difference with research?” from national, institutional and personal perspectives.

2 - Colm Connaughton - Why buying a train ticket is NP-hard: ticket splitting and other mathematical problems in transportation systems
I will discuss some problems which have arisen from collaborations between the MathSys CDT and Thales UK on data-driven modelling and analysis of the UK’s transportation networks.

3 - Kat Rock - Modelling for elimination: integrating mathematical predictions with African sleeping sickness policy
I will outline how the BMGF-funded HAT Modelling & Economic Predictions for Policy project based at Warwick is working in conjunction with programme directors and strategy implementers to eliminate this deadly disease.

30 November 2018: Francesco Mezzadri (Bristol) Random Matrices, spectral moments and hypergeometric orthogonal polynomials
Spectral moments of random matrices have been extensively studied for almost forty years in many branches of mathematics and physics, like QCD, Quantum Transport, Number theory and Combinatorics. They continue to reveal an incredibly rich and varied mathematical structure. I will give an overview of the progress in the field over the past few years. In particular, I will discuss a fascinating recent discovery: the spectral moments of the classical random matrix ensembles are hypergeometric orthogonal polynomials belonging to the Askey scheme. This unifies different strands of current research and provides powerful analytical tools to study these quantities.

07 December 2018: Alejandro Adem (British Columbia and Newton Institute) The Topology of Commuting Matrices
In this talk we will discuss the structure of spaces of commuting elements in a compact Lie group. Their connected components and other basic topological properties will be discussed. We will also explain how they can be assembled to produce a space which classifies certain bundles and represents an interesting cohomology theory. A number of explicit examples will be provided for orthogonal, unitary and projective unitary groups.

11 January 2019: Kathryn Hess (Ecole Polytechnique de Lausanne) Topological adventures in neuroscience
Over the past decade, and particularly over the past five years, research at the interface of topology and neuroscience has grown remarkably fast. Topology has, for example, been successfully applied to objective classification of neuron morphologies and to automatic detection of network dynamics. In this talk I will focus on the algebraic topology of brain structure and function, describing results obtained in collaboration with the Blue Brain Project on digitally reconstructed microcircuits of neurons in the rat cortex. I will also outline our on-going work on the topology of synaptic plasticity. The talk will include an overview of the Blue Brain Project and a brief introduction to the topological tools that we use.

18 January 2019: Desmond J Higham (University of Strathclyde) A Nonlinear Spectral Method for Network Core-Periphery Detection

An important problem in network science is to identify core-periphery structure. Given a network, our task is to assign each node to either the core or periphery. Core nodes should be strongly connected across the whole network whereas peripheral nodes should be strongly connected only to core nodes. More generally, we may wish to assign a non-negative value to each node, with a larger value indicating greater "coreness." This type of problem is related to, but distinct from, commumnity detection (finding clusters) and centrality assignment (finding key players), and it arises naturally in the study of networks in social science and finance.

We derive and analyse a new iterative algorithm for detecting network core-periphery structure. Using techniques in nonlinear Perron-Frobenius theory, we prove global convergence to the unique solution of a relaxed version of a natural discrete optimization problem. On sparse networks, the cost of each iteration scales linearly with the number of nodes, making the algorithm feasible for large-scale problems. We give an alternative interpretation of the algorithm from the perspective of maximum likelihood reordering of a new logistic core--periphery random graph model. This viewpoint also gives a new basis for quantitatively judging a core--periphery detection algorithm. We illustrate the algorithm on a range of synthetic and real networks, and show that it offers advantages over the current state-of-the-art.

This is joint work with Francesco Tudisco (Strathclyde).

25 January 2019: Stavros Garoufalidis (Georgia Tech) A brief history of quantum topology
Mathematics is a centuries old subject, but topology is a 20th century subject, and quantum topology is a very recent one. We will give an overview of quantum topology and its connections with other parts of mathematics and physics.

1 February 2019: Apala Majumdar (Bath) The Myriad Hues of Nematic Liquid Crystals across Mathematics, Physics and Applications
Nematic liquid crystals are classical examples of mesophases that have physical properties intermediate between those of conventional solids and liquids. Nematics are anisotropic liquids with long-range orientational order, featured by the existence of distinguished directions. Consequently, nematics have directional properties, making them the working material of choice for several opto-electronic devices and the multi-billion dollar display industry. In this colloquium, we review the hierarchy of continuum theories for nematics - the Oseen-Frank, the Ericksen and the Landau-de Gennes theories, the relationships between these theories and adjacent theories in materials science such as nonlinear elasticity and the Ginzburg-Landau theories for superconductivity and new non-classical approaches that remedy some of the limitations of these conventional continuum theories. We illustrate the predictive powers of these continuum approaches by reviewing recent work on pattern formation in square wells filled with nematic liquid crystals, driven by geometrical, topological and energetic considerations. We discuss pattern formation in different asymptotic limits, with emphasis on multistability and the stabilizing/de-stabilizing effects of nematic defects and how this can lead to new applications with promising optical and mechanical properties.

8 February 2019: Gernot Akemann (Bielefeld) Products of Random Matrices: Universality and Applications
Random matrices continue to be a popular topic among mathematicians and physicists. After introducing the concept of an ensemble of a single random matrix and its universal predictions I will review some recent progress on products of random matrices. They may serve as an example of a toy model for chaotic dynamical systems, where the Lyapunov exponents that characterise its stability can be explicitly calculated. In a particular limit of large matrix size and a large number of factors point processes are found that interpolate between the known universal statistics of a single random matrix and a deterministic Lyapunov spectrum. The interpolating kernels can be identified with that of Dyson’s Brownian Motion with fixed initial conditions.
This is joint work with Z. Burda and M. Kieburg.

15 February 2019: Filip Rindler (Warwick) Singularities in measures

Over the last years it has been discovered that surprisingly many different problems of Analysis naturally lead to questions about singularities in (vector) measures (that is, parts of the measure that are not absolutely continuous with respect to Lebesgue measure). These problems come from both "pure" Analysis, such as the question for which measures Rademacher's Theorem on the differentiability of Lipschitz functions holds, as well as "applied" Analysis, such as the question to determine the fine structure of slip lines in elasto-plasticity.

It's a remarkable fact that many of the (vector) measures that naturally occur in the wild, satisfy an (under-determined) PDE and thus the task arises to analyse these PDEs and in particular their possible singularities. This leads to restrictions on the shape of these singularities, which yields several interesting results in various areas of Analysis. The essential difficulty for the analysis of these measure-PDEs is that many standard methods (such as harmonic analysis) are much weaker in an L^1-context and thus new strategies need to be developed.

In this talk, which is aimed at a general mathematical audience, I will survey recent (and ongoing) joint work with A Arroyo-Rabasa, G. De Philippis, J. Hirsch, and A. Marchese on several of these questions and point out some open problems.

22 February 2019: Richard Sharp (Warwick) Spectrum and growth covers and the Banach-Tarski paradox - [Room B3.03]

Two natural numerical invariants that can be associated to a Riemannian manifold are the bottom of the spectrum of the Laplacian operator and, if the manifold are negatively curved, the exponential growth rate of closed geodesics. Suppose we have a regular cover of a compact manifold. Then, for each of these quantities, we might ask under what circumstances we have equality between the number associated to the cover and the number associated to the base. This question becomes non-trivial questions once the cover is infinite. It turns out that the question has a common answer in the two cases and this depends only on the covering group as an abstract group. For the Laplacian, this result was obtained by Robert Brooks in the 1980s, and, for closed geodesics, it is a combination of work of Thomas Roblin (2005) and Rhiannon Dougall and myself (2016). I will discuss this problem, relating it to the Banach-Tarski Paradox. If time permits, I will discuss more recent results with Dougall, where closed geodesics are replaced with periodic orbits of certain flows. The talk will be aimed at a general mathematical audience.

01 March 2019: Sarah Rees (Newcastle) Building more automatic structures for groups
I'm talking about some new composition theorems for automatic groups, joint work with Hermiller, Holt and Susse, specifically relating to HNN extensions,
amalgamated products, and more generally graphs of groups that are (coset) automatic relative to appropriate subgroups. The concept of automaticity for a group was introduced by Thurston in the late 1980's, based on properties of the groups of compact hyperbolic 3-manifolds that had been identified by Cannon, which in particular facilitated computation with these groups. Automatic groups are finitely presented, with recognisable normal forms and word problem soluble in quadratic time, and if biautomatic they have soluble conjugacy problem. A number of properties of closure and composition for this class of groups were proved almost immediately after its definition; hence in particular the fundamental group of most (but certainly not all) compact 3-manifolds could be proved automatic.
I'll provide some background on the subject of automatic groups, providing some motivation and summarising what is known, what is open, and what cannot be true of groups in this class. I'll define Holt and Hurt's related concept of coset automaticity. And I'll provide some details of the methods we used to provide our recent results, and describe a few groups for which our results give automatic structures, where none were previously known.

08 March 2019: Greg Smith (Queen’s, Canada) Nonnegative Certificates and Sums of Squares
A multivariate real polynomial is nonnegative if its value at any real point is a real number greater than or equal to zero. These special polynomials play a central role many branches of mathematics including algebraic geometry, optimization theory, and dynamical systems. However, deciding whether a given polynomial is nonnegative is a hard. In this talk, we will review some general methods for certifying that a polynomial is nonnegative. We will then present optimal bounds for certificates in some important cases. This talk is based on joint work with Grigoriy Blekherman and Mauricio Velasco.

15 March 2019: Tobias Grafke (Warwick) Hydrodynamic instantons and the universal route to rogue waves
In stochastic systems, extreme events are known to be described by "instantons", saddle point configurations of the action of the associated stochastic field theory. In this talk, I will present experimental evidence of a hydrodynamic instanton in a real world fluid system: A 270m wave channel experiment in Norway. The experiment attempts to model conditions on the ocean in order to observe so-called rogue waves, realisations of extreme ocean surface elevation out of relatively calm surroundings. These rogue waves are also observed in the ocean, where they are rare and hard to predict but pose significant danger to naval vessels. We show that the instanton approach, which is rigorously grounded in large deviation theory, offers a unified description of rogue waves in the water tank, covering the entire range of parameters for deep water waves in the ocean. In particular, this approach allows for a unified description of both the predominantly linear and the highly nonlinear regimes, and is able to explain the experimental data in the tank regardless of the strength of the nonlinearity.

26 April 2019: Joel Ouaknine (Oxford / Max Planck) Decision Problems for Linear Dynamical Systems
Dynamical systems, both discrete and continuous, permeate vast areas of mathematics, physics, engineering, and computer science. In this talk, we consider a selection of natural decision problems for linear dynamical systems, such as reachability of a given hyperplane. Such questions have applications in a wide array of scientific areas, ranging from theoretical biology and software verification to quantum computing and statistical physics. Perhaps surprisingly, the study of decision problems for linear dynamical systems involves techniques from a variety of mathematical fields, including analytic and algebraic number theory, Diophantine geometry, and algebraic geometry. I will survey some of the known results as well as recent advances and open problems.

This is joint work with James Worrell.

3 May 2019: András Máthé (Warwick) Story of circle squaring
Soon after the famous Banach--Tarski paradox appeared (which only works in three dimensions and higher), in 1925 Tarski posed a similar problem in the plane. Is it possible to partition a disc in the plane into finitely many pieces and rearrange these to obtain a square (of the same area)? This became known as Tarski's circle squaring problem and was unsolved until 1990 when Laczkovich showed that this partitioning/rearranging was indeed possible.

I will survey this problem and tell about how graph theory and descriptive set theory have recently influenced nicer solutions to "circle squaring", that is, with pieces that are Lebesgue/Borel measurable.

10 May 2019: Kevin Buzzard (Imperial College) What is a proof?
In mathematics departments, we teach the 1st year undergraduates what a proof of a theorem is -- it is a rigorous sequence of logical steps, deducing the conclusion from the hypotheses, using the rules of our system and other results we've proved already.

This is not what the proofs in the mathematical literature look like *at all*. In the first half of my talk we will take a look at some human proofs which do not fit this mould, and yet are accepted by the community. I will talk about the reasons they are accepted, the risks that we are taking by not documenting the proofs carefully, and what is being done about it.

I have spent the last two years of my life learning about how computer scientists think about mathematics, and I am just about getting to the point where I think I understand the tools they have made and what can be done with them. In the second half of my talk I will explain an experiment we are doing with the undergraduates at Imperial College London, where we are learning how to use computer proof checkers. This might be of particular interest to the community at Warwick, because my impression is that your undergraduates and ours are of a similar high standard. I will finish by giving an explanation of the main goals and objectives of Tom Hales' "Formal Abstracts" project. I believe that the way humans do mathematics is going to change.
The talk will be accessible to mathematics undergraduates.

17 May 2019: Shin Nayatani (Nagoya) Metrics maximizing the first eigenvalue of the Laplacian on a closed surface
I will focus on recent progress on metrics that maximize the first eigenvalue of the Laplacian (under area normalization) on a closed surface. I start with Hersch-Yang-Yau inequality which gives an explicit upper bound for the invariant. I then overview the recent progress on the existence problem for maximizing metrics, including the affirmative resolution of Jacobson-Levitin-Nadirashvili-Nigam-Polterovich's conjecture, which explicitly predicts maximizing metrics in the case of genus two, by Toshihiro Shoda and myself.
I then discuss the beautiful relation between maximizing metrics and minimal surfaces in the sphere. Finally I’ll outline our proof of the JLNNP conjecture.

24 May 2019: Peter Bürgisser (TU Berlin) Efficient algorithms for moment polytopes and the null cone problem from invariant theory
Suppose a reductive group G acts linearly on a finite dimensional complex vector space V. The corresponding null cone, which may be thought of the set of "singular objects'', consists of those vectors that cannot be distinguished from the zero vector by means of polynomial invariants. The null cone was introduced by Hilbert in his seminal work on invariant theory around 1900.

Quite surprisingly, the computational problem of testing membership to the null cone turned out to be of relevance for geometric complexity theory, quantum information theory, and other areas. Notably, a thorough study of the null cone for the simultanous left/right action on tuples of matrices was crucial for finding a deterministic polynomial time algorithm for verifying algebraic identies.

Despite the algebraic nature of the problem, numerical optimization algorithms seem to be the most efficient general methods for solving the null cone problem. This also applies to the related problem of testing membership to moment polytopes, which e.g., generalizes Horn's problem on the eigenvalues of sums of matrices.

The goal of the talk is to provide an overview on these developments.

31 May 2019: Joel Tropp (CalTech) Applied random matrix theory
Random matrices now play a role in many areas of theoretical, applied, and computational mathematics. Therefore, it is desirable to have tools for studying random matrices that are flexible, easy to use, and powerful. Over the last fifteen years, researchers have developed a remarkable family of results, called matrix concentration inequalities, that balance these criteria. This talk offers an invitation to the field of matrix concentration inequalities and their applications. This talk is designed for a general mathematical audience.

7 June 2019: Nina Snaith (Bristol) Random matrix theory and number theory: Zeros, moments and determinants
For 20 years we have known that average values of characteristic polynomials of random unitary matrices provide a good model for moments of the Riemann zeta function. Now we consider mixed moments of characteristic polynomials and their derivatives, calculations which are motivated by questions on the distribution of zeros of the derivative of the Riemann zeta function.