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Mathematics Colloquium 2015-16 Abstracts

09 October 2015 Gregory Falkovich (Weizmann Institute of Science) Pipes and optic fibers, long things of physics
Abstract:
How systems loose coherence upon the increase of system size or power? Water pipes and fibers with normal dispersion both have their coherent (laminar) state linearly stable at all parameters. Understanding coherence loss and turbulence onset in such systems requires a new paradigm of probabilistic phase transition, which I describe in this talk. I also briefly describe an emerging theory of turbulent state in such systems.

23 October 2015 Marta Mazzocco (Loughborough) Painlevé’s equations, cluster algebras and quantisation
Abstract:
The famous Greek astronomer Ptolemy created his well-known table of chords in order to aid his astronomical observations. This table was based on the renowned relation between the four sides and the two diagonals of a quadrilateral whose vertices lie on a common circle.

In 2002, the mathematicians Fomin and Zelevinsky generalised this relation to introduce a new structure called cluster algebra. This is a set of clusters, each cluster made of n numbers called cluster variables. All clusters are obtained from some initial cluster by a sequence of transformations called mutations. Cluster algebras appear in a variety of topics, including total positivity, number theory, Teichmüller theory and computer graphics. A quantisation procedure for cluster algebras was proposed by Berenstein and Zelevinsky in 2005.

After introducing the basics about cluster algebras, in this talk we will link cluster algebras to the theory of Painlevé equations - very famous differential equations the solutions of which play the role of non-linear special functions. This link will provide the foundations to introduce a new class of cluster algebras of geometric type. We will show that the quantisation of these new cluster algebras provide a geometric setting for the Berenstein–Zelevinsky construction.

30 October 2015 Daniel Wise (McGil and IHP) Counting cycles in graphs
Abstract:
A "W-cycle" in a labelled digraph Γ is a closed path whose label is the word W. I will describe a simple result about bounding the number of W-cycles in a deterministically labelled connected digraph. The result appears to be very simple but we do not know of an elementary proof. Perhaps someone from the audience will have some ideas. This is joint work with Joseph Helfer, and has been proven independently by Lars Louder and Henry Wilton.

06 November 2015 David Firth (Warwick Statistics) Journal rankings: Why (and how) should we care?
Abstract:
The status of an academic journal has for many years been indicated (to librarians and others) by its "impact factor". The value of research, and even of individual researchers, is sometimes assessed (by university administrators and others) through the impact factors of journals in which work is published. The impact-factor measure itself has seen strong criticism in the academic literature, but its use persists.

Statisticians have had relatively little involvement in the "bibliometrics" (or "scientometrics") field so far. The work reported in this talk aims to highlight the importance of more principled approaches to journal ranking, with emphasis on what is measured and how accurately.

As a case study, and to help catch the minds of research statisticians who might contribute in this area, ranking of the main journals of Statistics is considered in some detail. The methods used are clearly more widely applicable.

This talk is based on the paper "Statistical Modelling of Citation Exchange Between Statistics Journals" (joint with C Varin and M Cattelan) which was discussed at an RSS Ordinary Meeting in May 2015. The paper, along with data and R-code to facilitate replication or further development of the work, are available via http://warwick.ac.uk/dfirth.

13 November 2015 Ian Stewart (Warwick) Thermoregulation in Possums and the Wedderburn-Malcev Theorem
Abstract:
Not a combination you expect to see in a title, but there’s a common theme:
homeostasis, and one way to interpret it formally in a dynamical system.

Homeostasis occurs in a biological or chemical system when some output variable remains approximately constant as an input parameter varies. For example, mammals keep their body temperature almost constant despite large changes of temperature in their environment. (This is the possum bit.)

Marty Golubitsky and I have reformulated homeostasis in the context of singularity theory by replacing ‘approximately constant over an interval’ by ‘zero derivative with respect to the input at a point’. To make mathematical sense, this condition should be invariant under suitable coordinate changes. General coordinate changes don’t work, but in network dynamics there are several natural classes of coordinate changes that do. Characterising them leads to problems in linear algebra, which we solve by a mixture of Lie theory and associative algebra structure theorems. (This is the Wedderburn-Malcev bit.)

20 November 2015 Christian Böhning (Hamburg) Birational automorphism groups and dynamical degrees
Abstract:
We will discuss rationality properties of algebraic varieties, in particular various obstructions to rationality; we will also show how dynamical spectra might be used to better understand the structure of birational automorphism groups and birational types of algebraic varieties, and present some results and computational tools for dynamical degrees.

We will try to make the larger part of the talk accessible to non-experts and develop as much as possible from scratch with minimum prerequisites. In particular, basic concepts from algebraic geometry, group cohomology or dynamical systems will be briefly recalled, in a way that will hopefully make apparent their intuitive meaning and the basic ideas they encapsulate without going into all the technicalities.

27 November 2015 Andrew Fowler (Oxford and Limerick) Predicting the unpredictable: how to explain statistical distributions in 'deterministic' systems
Abstract:
Real experimental data often produces regular distributions in various metrics, and we illustrate this with four examples: the size of drumlins, the distribution of roundworm infections in humans, the grain size distribution of exploding volcanic rocks, and the area-frequency distribution for landslides. We then offer explanations for the observed distributions in the first three of these, and solicit suggestions for a mechanism to explain the fourth.

04 December 2015 José Antonio Carrillo de la Plata (Imperial) Collective Behavior Models: Mathematical Perspectives
Abstract:
I will give an overview of the different levels of description of collective behavior models highlighting some of the interesting mathematical open problems in the subject. Calculus of variations, dynamical systems, mean-field limits for PDEs, kinetic and aggregation-diffusion equations naturally show up as necessary tools to solve some of these questions.

22 January 2016 Walter Dean (Warwick−Philosophy) Mathematical existence and the arithmetized completeness theorem
Abstract:
This talk will explore a traditional question in the philosophy of mathematics — i.e. whether (or in what sense) the consistency of a mathematical theory entails the existence of a mathematical structure satisfying its axioms? It is sometimes argued that an answer to this question was provided by Gödel’s (1929) proof of the Completeness Theorem for first-order logic which states that every proof-theoretically consistent set of sentences in a first-order language possesses a first-order model. In order to assess this claim, I will examine Gödel’s result in the context of Hilbert’s work on the construction of arithmetical models of theories of geometry and analysis as well as more contemporary work on computable model theory.

29 January 2016 Dirk Kreimer (Humboldt) Cutkosky and the cubical chain complex
Abstract:
Feynman graphs are ubiquitous in physics but also in mathematics by now. Still, the analytic structure of a graphs contribution to a physics amplitude remained a vexing problem in complex analysis. It turns out that considering pairs of a graph and its spanning forests, together with a study of the graphs Kirchhoff polynomials, provide data to illuminate the situation.

05 February 2016 Julia Wolf (Bristol) Quadratic Fourier analysis: why and when linear phases are not enough
Abstract:
Quadratic Fourier analysis has its origins in quantitative approaches to Szemerédi‘s theorem, which states that any sufficiently dense subset of the first N integers contains a k-term arithmetic progression. It has also found numerous other applications to long-standing problems in combinatorics, number theory, analysis, and theoretical computer science.

In applications of classical Fourier analysis, we often decompose a function into a structured part given by the frequencies supporting large Fourier coefficients, plus an error term that we hope will be negligible. In certain applications, however, the small Fourier coefficients cannot be ignored. Quadratic Fourier analysis yields decompositions with a suitably negligible error term. The price we pay is that the structured part of the function can no longer be expressed as a simple linear combination of linear exponentials. Instead, input from additive combinatorics allows us to infer that it has quadratic structure.

This talk will be an introduction to this fascinating subject and aims to highlight some recent applications and open problems.

12 February 2016 Marie-Therese Wolfram (Warwick) Collective dynamics: modeling, analysis and simulations
Abstract:
Collective dynamics refers to the coordinated behavior of large groups of individuals. It can be observed in animal groups, pedestrian crowds or opinion formation processes. In all these situations the synchronization of individual characteristics emerges from complex interactions of individuals with themselves as well as their environment. Two factors are mainly responsible for these complex phenomena - social interactions and external stimuli. In this talk we present different mathematical models to describe collective dynamics – ranging from biological applications to pedestrian motion. We start with the microscopic description and present the corresponding macroscopic equations. A main focus of the talk lies on segregation and alignment phenomena. We discuss the existence of such solutions on the PDE level and illustrate the behavior of the models with numerical simulations.

26 February 2016 Arnaud Chéritat (Toulouse) Straightening the square
Abstract:
One interesting and very useful way of creating deformations in the plane is by "straightening ellipse fields". An ellipse field records how a function distorts angles at small scales; a straightening is a map from the plane to the plane solves a simple partial differential equation: namely, all ellipses in the field must be mapped to circles. When it exists, the straightening map is essentially unique and can be used to solve various problems in geometry and topology.

We will look at a deceptively simple situation: outside a given square the ellipse field consists of circles and inside the square the ellipses are all parallel to the vertical side, with fixed ratio K = major axis / minor axis. In this case, there is a straightening map defined on the plane and the image of the square under the map is quite interesting! Now, here is a puzzle: what happens to this shape as K tends to infinity?

04 March 2016 Juliette Kennedy (Helsinki / Newton Institute) Turing, Gödel and some new Inner Models of Set Theory
Abstract:
In his 1946 Princeton Bicentennial Lecture Goedel suggested the problem of finding a notion of definability for set theory which is "formalism free" in a sense similar to the notion of computable function --- a notion which is very robust with respect to its various associated formalisms. One way to interpret this suggestion is to consider standard notions of definability in set theory, which are usually built over first order logic, and change the underlying logic. We show that constructibility is not very sensitive to the underlying logic, and the same goes for hereditary ordinal definability (or HOD). This is joint work with Menachem Magidor and Jouko Vaananen.

11 March 2016 Zemer Kosloff (Warwick) Symmetric Birkhoff sums in infinite ergodic theory
Abstract:
Aaronson showed in 1977 that there is no ergodic theorem in infinite measure space and the failure is very drastic in the sense that normalised Birkhoff sums of positive integrable function either tend to 0 almost surely or to infinity almost surely. Recently it was observed that this is not necessarily true for normalised symmetric Birkhoff sums where the summation is along a symmetric time interval as there are examples of infinite, ergodic systems for which the absolutely normalised symmetric Birkhoff sums of positive integrable functions may be almost surely bounded away from zero and infinity. In this talk I will explain this new phenomena and discuss the main result that although the failure is less drastic, there is no ergodic theorem for symmetric Birkhoff sums in infinite measure space and there exists a universal divergence statement.
The contents of this talk are a combination of 2 papers, one of them is joint with Benjamin Weiss (Hebrew University) and Jon Aaronson (Tel Aviv).

18 March 2016 Gernot Akemann (Bielefeld) Random matrices - an overview and recent developments
Abstract:
Why are random matrices as popular and exciting after more than 50 years of research? In this talk I will try to answer this question. First, the most basic random matrix models will be introduced, solved and the typical questions explained that are raised, including that of universality. One of the popular tools, the theory of orthogonal polynomials, will be used as an example that has benefitted a lot from the solution of more sophisticated and realistic random matrix models in recent years. Examples for various applications from physics to finance will be used throughout as illustrations.

29 April 2016 David Epstein (Warwick) Careless Inferiors
Abstract:
I will define two new objects in hyperbolic geometry, a "superior" and an "inferior". Peversely, it is the inferiors that are most interesting, and these can be further sub-divided into "careful" and "careless" inferiors. These objects can be uniquely labelled with finite strings of symbols, and the "careless inferiors" are the ones with remarkable properties under permutation of symbols. The talk requires no previous knowledge of hyperbolic geometry (or much other mathematics). It will particularly suit people who enjoy champagne and like anagrams.

13 May 2016 Eleanor Robson (UCL) Babylonian Village Numeracy, c 1600 BC
Abstract:
For the past few years I have been the epigrapher on the first British excavation in southern Iraq since the start of the UN sanctions regime in 1990, which banned international research in the country. The Ur Regional Archaeology Project, headed by archeologists from the University of Manchester, is exploring a village occupied until the mid-second millennium BC a few miles from the famous ancient city of Ur. It is shedding important new light on a previously little-known period of history, about 150 years after the reign of famous king Hammurabi, thanks to an archive of cuneiform tablets which it is my task to decipher and interpret. Some of these objects are only now emerging from the ground; I make an annual trip to study the new discoveries each February.

My talk will focus on two aspects of the finds which are important for the history of numeracy: the accounting practices in the archive, and the entirely unexpected evidence for formal scribal schooling, of a type hitherto found only in wealthy urban contexts.

20 May 2016 Phil Boyland (University of Florida) When Topology Forces Dynamics
Abstract:

Sometimes topology data about a dynamical system can dictate alower bound on its complexity. Because the topological data required is often very coarse, such results are robust and lend themselves to physical applications.

After covering the basics we present several theorems of this type including one on the growth of vorticity in certain periodically stirred two-dimensional Euler flows and another on the dynamics of time-periodic Lagrangian systems on hyperbolic manifolds. Applications will include viscous fluid mixing and the dynamics of mechanical linkages. A main theme in the talk is the analogy, pointed out by MacKay in 1988, between Morse's classical theorem on geodesics on higher genus surfaces and the isotopy stability of the dynamics of Thurston's pseudo Anosov maps.

27 May 2016 Gavin Brown (Warwick) Moebius strips and flops and flips
Abstract:
The central axis of a Moebius strip has normal bundle -1: if you try to move it, it will break in 1 place, as the usual construction shows at once. You can also contract the central axis to a point without much drama: the less-usual knitted construction shows this when you pull the middle fibre tight, or you could stare at the spiral staircase in the library from above if you don't have any wool. This example appears everywhere in 2-dimensional complex geometry and its role has been well understood since the millennium before last.

In three (complex) dimensions there are similarly immovable, contractible curves. In fact there are many, not just a single key case, and broad features of their classification have been known for decades. But even though they can be made by sellotaping a couple of bits of complex 3-space together with a twist, we don't know how to write them down explicitly, by equations for example. (The best result of this nature to date is by Miles, around the time I took my 11+ exam.) I'll talk about how Michael Wemyss and I approach this.

03 June 2016 Miklos Abert (Alfréd Rényi) Groups and graph convergence
Abstract:
The story of local sampling convergence of sparse graphs started with the beautiful result of Benjamini and Schramm, that a large planar graph looks recurrent, when observed from a uniform random vertex. Since then, the notion received a lot of attention and by now, it has established ties to group theory, geometry, statistical physics, stochastic processes and algebraic topology.

A couple of recent applications connected to group theory, from various authors are: 1) Ramanujan graphs have few short cycles; 2) an injective cellular automata over a sofic group must be surjective; 3) higher rank locally symmetric spaces have large injectivity radius almost everywhere; 4) if two Bernoulli shifts over a sofic group are isomorphic then they have the same base entropy. I will discuss these results as well as some open problems.



10 June 2016 Tony Lelievre (ENPC Paris) Metastability: a journey from stochastic processes to semiclassical analysis
Abstract:
A stochastic process is metastable if it stays for a very long period of time in a region of the phase space (called a metastable region) before going to another metastable region, where it again remains trapped. Such processes naturally appear in many applications, metastability being related to a two time scale mechanism: the small time scale corresponds to the vibration period within the metastable regions and the large time scale is associated with the transitions between metastable states. For example, in molecular dynamics, the metastable regions are typically associated with the atomic conformations of a molecule (or an ensemble of molecules), and one is actually interested in simulating and studying the transitions between these conformations.

In this talk, I will explain how the exit events from a metastable state can be studied using an eigenvalue problem. This point of view is useful to build very efficient algorithms to simulate metastable stochastic processes (using in particular parallel architectures). It also gives a new way to prove the Eyring-Kramers laws and to justify the parametrization of an underlying Markov chain (Markov state model), using techniques form semiclassical analysis.