Morning Speakers

Lara Alcock

Loughborough University

Conditional Inference in Undergraduate Mathematics Students

Conditional inference has been studied for decades in cognitive psychology - we know a lot about what inferences people do and do not accept from statements of the form ‘If A then B’. We know much less about conditional inference in mathematics students, despite the obvious importance of this type of reasoning for understanding theorems and proofs. This talk will give a speedy overview of relevant research in cognitive psychology, then present findings from a sequence of studies of conditional inference in mathematics undergraduates. These studies investigate inferences from abstract, everyday and mathematical conditionals, and include evidence on believability effects and some striking individual differences. I will invite discussion of what the findings mean for teaching and learning, particularly at the transition to proof-based undergraduate mathematics.

Jennifer Balakrishnan

Boston University

Quadratic Chabauty for modular curves

By Faltings' theorem, the set of rational points on a curve of genus 2 or more is finite. We describe how p-adic heights ("quadratic Chabauty") can be used to determine this set for certain curves of genus 2 or more, in the spirit of Kim's nonabelian Chabauty program. In particular, we discuss what aspects of the quadratic Chabauty method can be made practical for certain modular curves and highlight several examples. This is based on joint work with  Alexander Betts, Netan Dogra, Daniel Hast, Aashraya Jha, Steffen Mueller, Jan Tuitman, and Jan Vonk.

Tim Burness

University of Bristol

Simple groups, fixed point ratios and applications

Let G be a group acting on a finite set X. The fixed point ratio (FPR) of an element g in G is simply the proportion of points in X fixed by g. Calculating, or bounding, FPRs has been a central problem in permutation group theory for many decades, finding numerous applications. In this talk, I will survey some of the main FPR results in the special case where G is a simple group, which is an area where there has been several major advances in recent years. I will also highlight the diverse range of applications, which includes powerful new results on bases for permutation  groups, the connectivity of generating graphs and the commuting probability of finite groups.

Robert Gray

University of East Anglia

The geometry of the word problem for groups and inverse monoids

The most fundamental algorithmic problem in algebra is the word problem, which asks whether there is an algorithm that takes two expressions over a set of generators and decides whether they represent the same element. Despite the huge advances that have been made in this area over the past century, there are still many basic questions about the word problem, and related algorithmic problems, for finitely presented groups and monoids that remain open.

Important work of Ivanov, Margolis, Meakin, and Stephen in the 1990s and 2000s shows how algorithmic problems for groups and monoids can be related to corresponding questions about finitely presented inverse monoids. This is a natural class that lies between groups and monoids, and corresponds to the abstract study of partial symmetries. There is a powerful range of geometric methods for studying inverse monoids such as the Scheiblich/Munn description of free inverse monoids and Stephen's procedure for constructing Schutzenberger graphs.

In this talk I'll explain these connections, and discuss recent advances in our understanding of the behaviour of the geometry of Cayley graphs of inverse monoids, and how this has been used to prove new and unexpected results about their algorithmic and algebraic properties.

Philipp Habegger

University of Basel

Torsion Points on Families of Abelian Varieties

Points of finite order are sparse on a subvariety of an abelian variety, except for algebraic subgroups and their components. This is the Manin-Mumford Conjecture, proved by Raynaud in the 1980s, when sparse is interpreted as not Zariski dense. Later, Pink formulated a relative variant of the conjecture in a family of abelian varieties. Masser and Zannier obtained the first results for a curve in a family of abelian varieties which were later extended to surfaces together with Corvaja and Tsimerman. I will speak about joint work with Ziyang Gao in higher dimensions. Our work relies on the Pila-Zannier counting strategy. We rely on definability results of Peterzil-Starchenko in o-minimal geometry which built upon earlier work of Wilkie, van den Dries, and Miller. Additionally, we use new height bounds from joint work with Dimitrov and Gao.

Michael Magee

University of Durham

Optimal spectral gaps

I'll discuss spectral gaps in the following contexts:

Of particular interest are spectral gaps that are in some a priori sense optimal or almost optimal. The discussion will centre on recent developments giving new examples of (almost) optimal spectral gaps.

Photo: LMS

Beatrice Pelloni

Heriot-Watt University

Dispersive revivals, or the Talbot effect revisited

I will discuss a surprising phenomenon, first observed experimentally in linear optics and quantum wave transmission, and known variously as the Talbot effect, (quantum) revivals, or dispersive quantisation. This phenomenon, and its dual phenomenon of fractalisation,  is clearly manifested in the solution of periodic dispersive equation, starting from a discontinuous initial condition. Then, at times that are rational multiple of the spatial period (“rational” times), the solution is discontinuous, indeed it is built from translated copies of the initial condition, while at all other times, the solution is wildly oscillatory, with positive fractal dimension, but it is continuous. I will discuss the mathematical description of this phenomenon, the regularity of solutions, and the effect on it of nonlinearity and of non-periodic boundary conditions, in particular the surprising appearance of logarithmic cusps when the boundary conditions interfere with the periodic symmetry.

Kobi Peterzil

University of Haifa

Closure and Hausdorff limits, in tori and nilmanifolds, via o-minimality

In a series of papers, joint with Sergei Starchenko, we began with the following question:  Given a subset X of R^n definable in an o minimal structure, and given a lattice L in R^n, with π from R^n to R^n / L, what can be said about the closure of π(X) in the torus? A similar question is asked for definable families of sets, and the Hausdorff limits of their image, in tori and also in nilmanifolds.

Despite the fact that lattices are inherently non-definable in o-minimal structures, the "flattness of X at infinity" allows us to answer the question using linear spaces associated to the set X.

The goal of the talk is to explain the problem and the answers, using many examples, without a need for model theoretic background.

Carola Schönlieb

University of Cambridge

A whistle stop tour from nonlinear PDEs and Riemannian calculus to deep neural networks for scientific imaging

In this talk we will discuss mathematical problems that arise in scientific imaging, ranging from variational models and PDEs for image analysis and inverse imaging problems to discrete geodesic calculus for shape matching, as well as recent advances where such mathematical models are complemented and replaced by deep neural networks. We will particularly focus on the current interplay and tension between mathematical modelling and modern AI approaches, first and foremost deep neural networks.

The talk is furnished with applications to low-dose computed tomography for cancer screening, electron microscopy for modelling protein dynamics, and fast magnetic resonance tomography for imaging fluid flow.

Pablo Shmerkin

University of British Columbia

Counting intersections of fuzzy lines

Given a finite set of lines in the plane, among which no two are parallel, how many points lie in at least two of the lines? If all the lines go through a single point, then the answer is "just one". But what if not all lines go through a point? In 1983, J. Beck gave an essentially sharp answer. For example, he showed that if no point lies in half of the lines, then the  number of line intersections is close to the square of the number of lines, which is the maximum possible value. This can be thought of as a dichotomy: either "many" of the lines meet at a point, or there is a "nearly maximal" number of line intersections. After reviewing Beck's Theorem, I will discuss a "fuzzy" variant of this problem in which lines are replaced by thin tubes, and we identify points that are close to each other. This innocent-looking question turns out to be closely related to deep problems in harmonic analysis, combinatorial geometry, and geometric measure theory. The result I will present can be seen as a fuzzy analog of Beck's theorem, and was obtained in collaboration with Tuomas Orponen (Jyväskylä) and Hong Wang (Courant). The talk will be accessible to a wide audience; no specialized background will be assumed.

Caroline Terry

Ohio State University

Measuring combinatorial complexity via regularity lemmas

Many tools have been developed in combinatorics to study global structure in finite graphs.  One such tool is called Szemerédi’s regularity lemma, which gives a structural decomposition for any large finite graph.  Beginning with work of Alon-Fischer-Newman, Lovász-Szegedy, and Malliaris-Shelah, it has been shown over the last 15 years that regularity lemmas can be used to detect structural dichotomies in graphs, and that these dichotomies have deep connections to model theory.  One striking example is a dichotomy in the size of regular partitions, first observed by Alon-Fox-Zhao.  Specifically, if a hereditary graph property ℋ has finite VC-dimension, then results of Alon-Fischer-Newman and Lovász-Szegedy imply all graphs in ℋ have regular partitions of size polynomial is 1/ε. On the other hand, if ℋ has infinite VC-dimension, then results of Gowers and Fox-Lovász show there are graphs in ℋ whose smallest 1/ε-regular partition has size at least an exponential tower of height polynomial in 1/ε.  In this talk, I present several analogous dichotomies in the setting of hereditary properties of 3-uniform hypergraphs.

Matthew Tointon

University of Bristol

Structure, expansion and probability in transitive graphs

Celebrated theorems of Gromov, Trofimov and Coulhon—Saloff-Coste combine to give a remarkable dichotomy for vertex-transitive graphs: such graphs must either resemble highly structured Cayley graphs, or must exhibit “expansion” in a certain sense. This in turn has had a number of striking applications, particularly to probability, such as Varopoulos’s characterisation of those transitive graphs on which the random walk is recurrent, and Duminil-Copin, Goswami, Raoufi, Severo and Yadin’s recent proof that a transitive graph has a non-trivial percolation phase unless it is “one-dimensional” in a precise sense. Over the past 15 years or so there have been a number of quantitative, finitary refinements of these results, notably due to Shalom—Tao, Breuillard—Green—Tao, and Tessera and me. I will survey these results and describe some recent and forthcoming probabilistic applications.