Past

Abstract: We use groupoids to describe a geometric framework which can host a generalisation of Hairer's regularity structures to manifolds. In this setup, Hairer's re-expansionmap (usually denoted \Gamma) is a (direct) connection on a gauge groupoid and can therefore be viewed as a groupoid counterpart of a (local) gauge field. This definitions enables us to make the link between re-expansion maps (direct connections), principal connections and path connections, to understand the flatness of the direct connection in terms of that of the manifold and, finally, to easily build a polynomial regularity structure which we compare to the one given by Driver, Diehl and Dahlquist. (Join work with Sara Azzali, Alessandra Frabetti and Sylvie Paycha).

Quantum Yang-Mills theory is an important part of the Standard model built by physicists to describe elementary particles and their interactions. One approach to the mathematical substance of this theory consists in constructing a probability measure on an infinite-dimensional space of connections on a principal bundle over space-time. However, in the physically realistic 4-dimensional situation, the construction of this measure is still an open mathematical problem. The subject of this talk will be the physically less realistic 2-dimensional situation, in which the construction of the measure is possible, and fairly well understood.

In probabilistic terms, the 2-dimensional Yang-Mills measure is the distribution of a stochastic process with values in a compact Lie group (for example the unitary group U(N)) indexed by the set of continuous closed curves with finite length on a compact surface (for example a disk, a sphere or a torus) on which one can measure areas. It can be seen as a Brownian motion (or a Brownian bridge) on the chosen compact Lie group indexed by closed curves, the role of time being played in a sense by area.

In this talk, I will describe the physical context in which the Yang-Mills measure is constructed, and describe it without assuming any prior familiarity with the subject. I will then present a set of results obtained in the last few years by Antoine Dahlqvist, Bruce Driver, Franck Gabriel, Brian Hall, Todd Kemp, James Norris and myself concerning the limit as N tends to infinity of the Yang-Mills measure constructed with the unitary group U(N). 

 

After discussing a general excision technique for constructing canonical orientations for moduli spaces that derive from an elliptic equation, I shall
explain how to carry out this program in the case of G2-instantons and the 7-dimensional real Dirac operator. In many ways our approach can
be regarded as a categorification of the Atiyah-Singer index theorem. (Based on joint work with Dominic Joyce.)

The E-string theory is usually considered as the simplest among 6D (1,0) superconformal field theories. Nonetheless, we still have little information about its spectrum of operators. In this talk, I'm going to describe our recent geometric approach using F-theory compactification on an elliptic Calabi-Yau threefold. The elliptic fibration is non-flat, which means that there are complex surface components in the fiber direction. From the geometry of non-flat fiber, we read out an infinite tower of particle states in the E-string theory. I will also discuss its relevance to 4D standard model building, which is a main motivation of this work.
 

Teaching ethics to the mathematicians who need it most
For the last 20 years it has become increasingly obvious, and increasingly pressing, that mathematicians should be taught some ethical awareness so as to realise the impact of their work. This extends even to those more highly trained, like graduate students and postdocs. But which mathematicians should we be teaching this to, what should we be teaching them, and how should we do it? In this talk I’ll explore the idea that all mathematicians will, at some stage, be faced with ethical challenges stemming from their work, and yet few are ever told beforehand.
 

Mathematics is both the language and the instrument that connects our abstract understanding with the physical world, thus knowledge of mathematics quickly translates to substantial knowledge and influence on the way the world works.  But those who have the greatest ability to understand and manipulate the world hold the greatest capacity to do damage and inflict harm.  In this talk I'll explain that yes, there is ethics in mathematics, and that it is up to us as mathematicians to make good ethical choices in order to prevent our work from becoming harmful.

In this work we will attempt, via virtual models, to interpret how structure and body positioning impact upon the outcomes of Multi-Breath-Washout tests. 

By extrapolating data from CT images, a virtual reduced dimensional airway/vascualr network will be constructed. Using this network both airway and blood flow profiles will be calculated. These profiles will then be used to model gas transport within the lungs. The models will allow us to investigate the role of airway restriction, body position during testing and washout gas choice have on MBW measures. 
 

Many scientists, and in particular mathematicians, report difficulty in understanding thermodynamics. So why is thermodynamics so difficult? To attempt an answer, we begin by looking at the components in an exposition of a scientific theory. These include a mathematical core, a motivation for the choice of variables and equations, some historical remarks, some examples and a discussion of how variables, parameters, and functions (such as equations of state) can be inferred from experiments. There are other components too, such as an account of how a theory relates to other theories in the subject.

It will be suggested that theories of thermodynamics are hard to understand because (i) many expositions appear to argue from the particular to the general (ii) there are several different thermodynamic theories that have no obvious logical or mathematical equivalence (iii) each theory really is subtle and requires intense study (iv) in most expositions different theories are mixed up, and the different components of a scientific exposition are also mixed up. So, by presenting one theory at a time, and by making clear which component is being discussed, we might reduce the difficulty in understanding any individual thermodynamic theory. The key is perhaps separation of the mathematical core from the physical motivation. It is also useful to realise that a motivation is not generally the same as a proof, and that no theory is actually true.

By way of illustration we will attempt expositions of two of the simplest thermodynamic theories – reversible and then irreversible thermodynamics of homogeneous materials – where the mathematical core and the motivation are discussed separately. In conclusion we’ll relate these two simple theories to other, foundational and generalised, thermodynamic theories.

In the past decade, deep learning methods have achieved unprecedented performance on a broad range of problems in various fields from computer vision to speech recognition. So far research has mainly focused on developing deep learning methods for Euclidean-structured data. However, many important applications have to deal with non-Euclidean structured data, such as graphs and manifolds. Such data are becoming increasingly important in computer graphics and 3D vision, sensor networks, drug design, biomedicine, high energy physics, recommendation systems, and social media analysis. The adoption of deep learning in these fields has been lagging behind until recently, primarily since the non-Euclidean nature of objects dealt with makes the very definition of basic operations used in deep networks rather elusive. In this talk, I will introduce the emerging field of geometric deep learning on graphs and manifolds, overview existing solutions and outline the key difficulties and future research directions. As examples of applications, I will show problems from the domains of computer vision, graphics, high-energy physics, and fake news detection. 

Clustering is a very important task in data analytics and is usually addressed using (i) statistical tools based on maximum likelihood estimators for mixture models, (ii) techniques based on network models such as the stochastic block model, or (iii) relaxations of the K-means approach based on semi-definite programming (or even simpler spectral approaches). Statistical approaches of type (i) often suffer from not being solvable with sufficient guarantees, because of the non-convexity of the underlying cost function to optimise. The other two approaches (ii) and (iii) are amenable to convex programming but do not usually scale to large datasets. In the big data setting, one usually needs to resort to data subsampling, a preprocessing stage also known as "coreset selection". We will present this last approach and the problem of selecting a coreset for the special cases of K-means and spectral-type relaxations.

Seminal work of Hida tells us that if a modular eigenform is ordinary at p then we can always find other eigenforms, of different weights, that are congruent to our given form. Even better, it says that we can find q-expansions with coefficients in p-adic analytic function of the weight variable k that when evaluated at positive integers give the q-expansion of classical eigenforms. His construction of these families uses mainly the geometry of the modular curve and its ordinary locus.
In a joint work with Marc-Hubert Nicole, we obtained similar results for Drinfeld modular forms over function fields. After an extensive introduction to Drinfeld modules, their moduli spaces, and Drinfeld modular forms, we shall explain how to construct Hida families for ordinary Drinfeld modular forms.

 One of the most defining features of modern financial networks is their inherent complex and intertwined structure. In particular the often observed core-periphery structure plays a prominent role. Here we study and quantify the impact that the complexity of networks has on contagion effects and system stability, and our focus is on the channel of default contagion that describes the spread of initial distress via direct balance sheet exposures. We present a general approach describing the financial network by a random graph, where we distinguish vertices (institutions) of different types - for example core/periphery - and let edge probabilities and weights (exposures) depend on the types of both the receiving and the sending vertex. Our main result allows to compute explicitly the systemic damage caused by some initial local shock event, and we derive a complete characterization of resilient respectively non-resilient financial systems in terms of their global statistical characteristics. Due to the random graphs approach these results bear a considerable robustness to local uncertainties and small changes of the network structure over time. Applications of our theory demonstrate that indeed the features captured by our model can have significant impact on system stability; we derive resilience conditions for the global network based on subnetwork conditions only. 

The Strominger-Yau-Zaslow (SYZ) conjecture postulates that mirror dual Calabi-Yau manifolds carry dual special Lagrangian fibrations. Within the study of Mirror Symmetry the SYZ conjecture has provided a particularly fruitful point of convergence of ideas from Riemannian, Symplectic, Tropical, and Algebraic geometry over the last twenty years. I will attempt to provide a brief overview of this aspect of Mirror Symmetry.

Blisters form when a thin surface layer of a solid body separates/delaminates from the underlying bulk material over a finite, bounded region. It is ubiquitous in a range of industrial applications, e.g. blister test is applied to assess the strength of adhesion between thin elastic films and their solid substrates, and during natural processes, such as formation and spreading of laccoliths or retinal detachment.

We study a special case of blistering, in which a thin elastic membrane is adhered to the substrate by a thin layer of viscous fluid. In this scenario, the expansion of the newly formed blister by fluid injection occurs via a displacement flow, which peels apart the adhered surfaces through a two-way interaction between flow and deformation. If the injected fluid is less viscous than the fluid already occupying the gap, patterns of short and stubby fingers form on the propagating fluid interface in a radial geometry. This process is regulated by membrane compliance, which if increased delays the onset of fingering to higher flow rates and reduces finger amplitude. We find that the morphological features of the fingers are selected in a simple way by the local geometry of the compliant cell. In contrast, the local geometry itself is determined from a complex fluid–solid interaction, particularly in the case of rectangular blisters. Furthermore, changes to the geometry of the channel cross-section in the latter case lead to a rich variety of possible interfacial patterns. Our experiments provide a link between studies of airway reopening, Saffman-Taylor fingering and printer’s instability.   

We discuss the key role that bespoke linear algebra plays in modelling PDEs with random coefficients using stochastic Galerkin approximation methods. As a specific example, we consider nearly incompressible linear elasticity problems with an uncertain spatially varying Young's modulus. The uncertainty is modelled with a finite set of parameters with prescribed probability distribution.  We introduce a novel three-field mixed variational formulation of the PDE model and and  assess the stability with respect to a weighted norm. The main focus will be  the efficient solution of the associated high-dimensional indefinite linear system of equations. Eigenvalue bounds for the preconditioned system can be  established and shown to be independent of the discretisation parameters and the Poisson ratio.  We also  discuss an associated a posteriori error estimation strategy and assess proxies for the error reduction associated with selected enrichments of the approximation spaces.  We will show by example that these proxies enable the design of efficient  adaptive solution algorithms that terminate when the estimated error falls below a user-prescribed tolerance.

This is joint work with Arbaz Khan and Catherine Powell

Tanut Treetanthiploet
---------------------
Exploration vs Exploitation under Statistical Uncertainty

The exploration vs Exploitation trade-off can be quantified and studied through the notion of statistical uncertainty using the theory of nonlinear expectations. The dynamic allocation problem of multi-armed bandits will be discussed. In the case of a finite state space in discrete time, we can describe the value function in terms of the solution to a discrete BSDE and obtain a similar notion to the Bellman equation. We also give an approximation scheme to evaluate decisions in the simple setting.


Julien Vaes
-----------
Optimal Execution Strategy Under Price and Volume Uncertainty

In the seminal paper on optimal execution of portfolio transactions, Almgren and Chriss define the optimal trading strategy to liquidate a fixed volume of a single security under price uncertainty. Yet there exist situations, such as in the power market, in which the volume to be traded can only be estimated and becomes more accurate when approaching a specified delivery time. To meet the need of efficient strategies in these situations, we have developed  a model that accounts for volume uncertainty and show that a risk-averse trader has benefit in delaying their trades. We show that the optimal strategy is a trade-off between early and late trades to balance risk associated to both price and volume. With the incorporation of a risk term for the volume to trade, the static optimal strategies obtained with our model avoid the explosion in the algorithmic complexity associated to dynamic programming solutions while yielding to competitive performance.
 

In this talk I will discuss recent work with P. Daskalopoulos on sufficient conditions to prove uniqueness of complete graphs evolving by mean curvature flow. It is interesting to remark that the behaviour of solutions to mean curvature flow differs from the heat equation, where non-uniqueness may occur even for smooth initial conditions if the behaviour at infinity is not prescribed for all times. 

NSOP1 is a class of first order theories containing simple theories, which contains many natural examples that somehow slip-out of the simple context.

As in simple theories, NSOP1 theories admit a natural notion of independence dubbed Kim-independence, which generalizes non-forking in simple theories and satisfies many of its properties.

In this talk I will explain all these notions, and in particular talk about recent progress (joint with Nick Ramsey) in the study of Kim-independence, showing transitivity and several consequences.

I will give a description of a method introduced by N. Ivanov to study the abstract commensurator of a group by using a rigid action of that group on a graph. We will sketch Ivanov's theorem regarding the abstract commensurator of a mapping class group. Time permitting, I will describe how these methods are used in some of my recent work with Horbez on outer automorphism groups of free groups.

We survey recent research related to the Extension Property of Partial Isomorhisms (EPPA, also known as Hrushovski property) and, perhaps surprisingly, relate it to structural Ramsey theory.   This is based on a joint work with David Evans, Jan Hubicka and Matej Konecny.
 

Not much is known about the Chow rings  of moduli spaces of abelian varieties or more general Shimura varieties. The tautological ring of a Shimura variety of Hodge type is a subring of its Chow ring containing many "interesting" classes. I will talk about joint work with Torsten Wedhorn on this ring as well as its characteristic p variant. The later is strongly related to the question of understanding the cycle classes of Ekedahl-Oort strata in the Chow ring.

Cluster Adjacency is a geometric principle which defines a subclass of multiple polylogarithms with analytic properties compatible with that of scattering amplitudes and Feynman loop integrals. We use this principle to a priori remove the redundances in the perturbative bootstrap approach and efficiently compute the four-loop NMHV heptagon. Moreover, cluster adjacency is naturally applied to the space of $A_n$ polylogarithms and generates numerous structures therein to be explored further.

A common strategy to solve shape optimization problems is to select an initial domain and to update it iteratively until it satisfies certain optimality crietria. In the presence of PDE-constraints, computing these updates requires solving a boundary value problem on a domain that changes at every iteration. We explain how to use isoparametric finite elements to tackle this issue. We also show how finite elements allow computing these updates without deriving shape derivative formulas by hand.

There was major progress on perfect graphs in the early 2000's: Chudnovsky, Robertson, Thomas and I proved the ``strong perfect graph theorem'' that a graph is perfect if and only if it has no odd hole or odd antihole; and Chudnovsky, Cornuejols, Liu, Vuscovic and I found a polynomial-time algorithm to test whether a graph has an odd hole or odd antihole, and thereby test if it is perfect. (A ``hole'' is an induced cycle of length at least four, and an ``antihole'' is a hole in the complement graph.)

What we couldn't do then was test whether a graph has an odd hole, and this has stayed open for the last fifteen years, despite some intensive effort. I am happy to report that in fact it can be done in poly-time (in time O(|G|^{12}) at the last count), and in this talk we explain how.

Joint work with Maria Chudnovsky, Alex Scott, and Sophie Spirkl.

The classical Dirac operator is part of an osp(1|2) realisation inside the Weyl-Clifford algebra which is Pin-invariant. This leads to a multiplicity-free decomposition of the space of spinor-valued polynomials in irreducible modules for this Howe dual pair. In this talk we review an abstract generalisation A of the Weyl algebra that retains a realisation of osp(1|2) and we determine its centraliser algebra explicitly. For the special case where A is a rational Cherednik algebra, the centralizer algebra provides a refinement of the previous decomposition whose analogue was no longer irreducible in general. As an example, for the  group S3 in specific, we will examine the finite-dimensional irreducible modules of the centraliser algebra.

In this talk, we consider solving nonlinear systems of equations and the unconstrained minimization problem using Newton’s method methods from the Halley class. The methods in this class have in general local and third order rate of convergence while Newton’s method has quadratic convergence. In the unconstrained optimization case, the Halley methods will require the second and third derivative. Third-order methods will, in most cases, use fewer iterations than a second-order method to reach the same accuracy. However, the number of arithmetic operations per iteration is higher for third-order methods than for a second-order method. We will demonstrate that for a large class of problems, the ratio of the number of arithmetic operations of Halley’s method and Newton’s method is constant per iteration (independent of the number of unknowns).

We say that the sparsity pattern of the third derivative (or tensor) is induced by the sparsity pattern of the Hessian matrix. We will discuss some datastructures for matrices where the indices of nonzero elements of the tensor can be computed. Historical notes will be merged into the talk.

Coastal vegetation has a well-known effect of attenuating waves; however, quantifiable measures of attenuation for general wave and vegetation scenarios are not well known. On the plant scale, there are extensive studies in predicting the dynamics of a single plant in an oscillatory flow. On the coastal scale however, there are yet to be compact models which capture the dynamics of both the flow and vegetation, when the latter exists in the form of a dense canopy along the bed. In this talk, we will discuss the open questions in the field and the modelling approaches involved. In particular, we investigate how micro-scale effects can be homogenised in space and how periodic motions can be averaged in time.

I will review the fishnet model, which is an integrable scalar QFT, obtained by an extreme gamma deformation of N=4 super Yang-Mills. The theory has a peculiar perturbative expansion in which many quantities at a fixed loop order are given by a single Feynman diagram. This feature allows the reinterpretation of Feynman loop integrals as integrable systems.

We examine the relationship between social structure and sentiment through the analysis of a large collection of tweets about the Irish Marriage Referendum of 2015. We obtain the sentiment of every tweet with the hashtags #marref and #marriageref that was posted in the days leading to the referendum, and construct networks to aggregate sentiment and use it to study the interactions among users. Our analysis shows that the sentiment of outgoing mention tweets is correlated with the sentiment of incoming mentions, and there are significantly more connections between users with similar sentiment scores than among users with opposite scores in the mention and follower networks. We combine the community structure of the follower and mention networks with the activity level of the users and sentiment scores to find groups that support voting ‘yes’ or ‘no’ in the referendum. There were numerous conversations between users on opposing sides of the debate in the absence of follower connections, which suggests that there were efforts by some users to establish dialogue and debate across ideological divisions. Our analysis shows that social structure can be integrated successfully with sentiment to analyse and understand the disposition of social media users around controversial or polarizing issues. These results have potential applications in the integration of data and metadata to study opinion dynamics, public opinion modelling and polling.

The Euler equation describing motion of ideal fluids goes back to 1755. 
The analysis of the equation is challenging since it is nonlinear and nonlocal. Its solutions are often unstable and spontaneously generate small scales. The fundamental question of global regularity vs finite time singularity formation 
remains open for the Euler equation in three spatial dimensions. In this lecture, I will review the history of this question and its connection with the arguably greatest unsolved problem of classical physics, turbulence. Recent results on small scale and singularity formation in two dimensions and for a number of related models will also be presented.

I will describe an extremely easy construction with formal group laws, and a 
slightly more subtle argument to show that it can be done in a coordinate-free
way with formal groups.  I will then describe connections with a range of other
phenomena in stable homotopy theory, although I still have many more 
questions than answers about these.  In particular, this should illuminate the
relationship between the Lambda algebra and the Dyer-Lashof algebra at the
prime 2, and possibly suggest better ways to think about related things at 
odd primes.  The Morava K-theory of symmetric groups is well-understood
if we quotient out by transfers, but somewhat mysterious if we do not pass
to that quotient; there are some suggestions that dilation will again be a key
ingredient in resolving this.  The ring $MU_*(\Omega^2S^3)$ is another
object for which we have quite a lot of information but it seems likely that 
important ideas are missing; dilation may also be relevant here.
 

This talk is based on joint work with Dominic Joyce and Markus Upmeier. Issues we'd like to talk about are a) the orientability of moduli spaces that
appear in various gauge-theoretic problems; and b) how to orient those moduli spaces if they are orientable. We begin with briefly mentioning backgrounds and motivation, and recall basics in gauge theory such as the Atiyah-Hitchin-Singer complex and the Kuranishi model by taking the anti-self-dual instanton moduli space as an example. We then describe the orientability and canonical orientations of the anti-self-dual instanton moduli space, and other
gauge-theoretic moduli spaces which turn up in current research interests.

 Author of Alan Turing: The Enigma, sharing his academic path and experience as activist for LGBTQ+ rights

In this talk, we show that a range of non-trivial SU(3) structures can be constructed on large classes of Calabi-Yau three-folds. Among the possible SU(3) structures we find Strominger-Hull systems, suitable for heterotic or type II string compactifications. These SU(3) structures of Strominger-Hull type have a non-vanishing and non-closed three-form flux which needs to be supported by source terms in the associated Bianchi identity. We discuss the possibility of finding such source terms and present first steps towards their explicit construction. Provided suitable sources exist, our methods lead to Calabi-Yau compactifications of string theory with a non Ricci-flat, physical metric which can be written down explicitly and in analytic form. The talk is based on the paper 1805.08499.

Thomas Prince The double life of the number 24.

The number 24 appears in a somewhat surprising result in the study of polyhedra with integer lattice points. In a different setting, the number 24 is the Euler characteristic of a K3 surface: a four (real) dimensional object which plays a central role in algebraic geometry. We will hint at why both instances of 24 are in fact the same, and suggest that integral affine geometry can be used to interpolate between the realm of integral polytopes and the world of complex algebraic geometry.

Mohit Dalwadi A multiscale mathematical model of bacterial nutrient uptake

In mathematical models that include nutrient delivery to bacteria, it is prohibitively expensive to include many small bacterial regions acting as volumetric nutrient sinks. To combat this problem, such models often impose an effective uptake instead. However, it is not immediately clear how to relate properties on the bacterial scale with this effective result. For example, one may intuitively expect the effective uptake to scale with bacterial volume for weak first-order uptake, and with bacterial surface area for strong first-order uptake. I will present a general model for bacterial nutrient uptake, and upscale the system using homogenization theory to determine how the effective uptake depends on the microscale bacterial properties. This will show us when the intuitive volume and surface area scalings are each valid, as well as the correct form of the effective uptake when neither of these scalings is appropriate.
 

Are you keen to share your love of maths with non-mathematicians, but aren’t sure where to start? Whether you're keen to get involved in outreach activities at Oxford, or you'd just like to explain to your friends and family what you do all term, there's something for everyone in our interactive hour of workshop activities, and lots of laughs along the way. Just bring plenty of enthusiasm, and come prepared with a bit of mathematics you particularly like. 

This session is open to all, and no prior outreach experience is necessary.

In this talk I will show the nature, the properties and the features of the Pareto Optimality in a diverse set of phenomena modeled as complex networks.
I will present a composite design methodology for multi-objective modeling and optimization of complex networks.  The method is based on the synergy of different algorithms and computational techniques for the analysis and modeling of natural systems (e.g., metabolic pathways in prokaryotic and eukaryotic cells) and artificial systems (e.g., traffic networks, analog circuits and solar cells).

“Pareto Optimality in Multilayer Network Growth”
G. Nicosia et al, Phys. Rev. Lett., 2018

Multiple zeta values are generalizations of the special values of Riemann's zeta function at positive integers. They satisfy a large number of algebraic relations some of which were already known to Euler. More recently, the interpretation of multiple zeta values as periods of mixed Tate motives has led to important new results. However, this interpretation seems insufficient to explain the occurrence of several phenomena related to modular forms.

The aim of this talk is to describe an analog of multiple zeta values for complex elliptic curves introduced by Enriquez. We will see that these define holomorphic functions on the upper half-plane which degenerate to multiple zeta values at cusps. If time permits, we will explain how some of the rather mysterious modular phenomena pertaining to multiple zeta values can be interpreted directly via the algebraic structure of their elliptic analogs.

We analyse the consequences of conservative portfolio compression, i.e., netting cycles in financial networks, on systemic risk.  We show that the recovery rate in case of default plays a significant role in determining whether portfolio compression is potentially beneficial.  If recovery rates of defaulting nodes are zero then compression weakly reduces systemic risk. We also provide a necessary condition under which compression strongly reduces systemic risk.  If recovery rates are positive we show that whether compression is potentially beneficial or harmful for individual institutions does not just depend on the network itself but on quantities outside the network as well. In particular we show that  portfolio compression can have negative effects both for institutions that are part of the compression cycle and for those that are not. Furthermore, we show that while a given conservative compression might be beneficial for some shocks it might be detrimental for others. In particular, the distribution of the shock over the network matters and not just its size.  

Accurate simulation of electromagnetic scattering by ice crystals in clouds is an important problem in atmospheric physics, with single scattering results feeding directly into the radiative transfer models used to predict long-term climate behaviour. The problem is challenging for numerical simulation methods because the ice crystals in a given cloud can be extremely varied in size and shape, sometimes exhibiting fractal-like geometrical characteristics and sometimes being many hundreds or thousands of wavelengths in diameter. In this talk I will focus on the latter "high-frequency" issue, describing a hybrid numerical-asymptotic boundary element method for the simplified problem of acoustic scattering by penetrable convex polygons, where high frequency asymptotic information is used to build a numerical approximation space capable of achieving fixed accuracy of approximation with frequency-independent computational cost. 

It is shown by Kashiwara and Schapira (1980s) that for every constructible sheaf on a smooth manifold, one can construct a closed conic Lagrangian subset of its cotangent bundle, called the microsupport of the sheaf. This eventually led to the equivalence of the category of constructible sheaves on a manifold and the Fukaya category of its cotangent bundle by the work of Nadler and Zaslow (2006), and Ganatra, Pardon, and Shende (2018) for partially wrapped Fukaya categories. One can try to generalise this and conjecture that Fukaya category of a Weinstein manifold can be given by constructible (microlocal) sheaves associated to its skeleton. In this talk, I will explain these concepts and confirm the conjecture for a family of Weinstein manifolds which are certain quotients of A_n-Milnor fibres. I will outline the computation of their wrapped Fukaya categories and microlocal sheaves on their skeleta, called pinwheels.

Just like optimization needs derivatives, shape optimization needs shape derivatives. Their definition and computation is a classical subject, at least concerning first order shape derivatives. Second derivatives have been studied as well, but some aspects of their theory still remains a bit mysterious for practitioners. As a result, most algorithms for shape optimization are first order methods.

To understand this situation better and in a general way, we consider first and second order shape sensitivities of integrals on smooth submanifolds using a variant of shape differentiation. Instead of computing the second derivative as the derivative of the first derivative, we choose a one-parameter family of perturbations  and compute first and second derivatives with respect to that parameter. The result is a  quadratic form in terms of a perturbation vector field that yields a second order quadratic model of the perturbed functional, which can be used as the basis of a second order shape optimization algorithm. We discuss the structure of this derivative, derive domain expressions and Hadamard forms in a general geometric framework, and give a detailed geometric interpretation of the arising terms.

Finally, we use our results to construct a second order SQP-algorithm for shape optimization that exhibits indeed local fast convergence.

In this talk I will illustrate how solutions of nonlinear nonlocal Fokker-Planck equations in a bounded domain with no-flux boundary conditions can be approximated by Cauchy problems with increasingly strong confining potentials defined outside such domain. Two different approaches are analysed, making crucial use of uniform estimates for energy and entropy functionals respectively. In both cases we prove that the problem in a bounded domain can be seen as a limit problem in the whole space involving a suitably chosen sequence of large confining potentials.
This is joint work with Maria Bruna and José Antonio Carrillo.
 

Traditionally, logic was thought of as `principles of right reason'. Early twentieth century philosophy of mathematics focused on the problem of a general foundation for all mathematics. In contrast, the last 70 years have seen model theory develop as the study and comparison of formal theories for studying specific areas of mathematics. While this shift began in work of Tarski, Robinson, Henkin, Vaught, and Morley, the decisive step came with Shelah's stability theory. After this paradigm shift there is a systematic search for a short set of syntactic conditions which divide first order theories into disjoint classes such that models of different theories in the same class have similar mathematical properties. This classification of theories makes more precise the idea of a `tame structure'. Thus, logic (specifically model theory) becomes a tool for organizing and doing mathematics with consequences for combinatorics, diophantine geometry, differential equations and other fields. I will present an account of the last 70 years in model theory that illustrates this shift. This reports material in my recent book published by Cambridge: Formalization without Foundationalism: Model Theory and the Philosophy of Mathematical Practice.

Euler sums and p-adic L-functions

Abstract:  It is a central question in the theory of infinite relational structures as to which structures carry analogues of Ramsey’s Theorem.  This question, of interest for several decades, has gained recent momentum as it was brought into focus by Kechris, Pestov, and Todorcevic, when they proved a deep correspondence between Ramsey theory and topological dynamics.  

In this talk, we provide background on the Ramsey theory of the Rado graph, solved by Sauer.  A longstanding open question was whether Henson graphs, the k-clique-free analogues of the Rado graph, have similar features.  We present the speaker’s recent work solving the Ramsey theory of the Henson graphs.  The techniques developed open new lines of investigation for other relational structures with forbidden configurations.  As a byproduct of these methods, we may obtain Ramsey properties for Borel colorings on copies of the Rado graph, with respect to a certain topology.

Abstract:  It is a central question in the theory of infinite relational structures as to which structures carry analogues of Ramsey’s Theorem.  This question, of interest for several decades, has gained recent momentum as it was brought into focus by Kechris, Pestov, and Todorcevic, when they proved a deep correspondence between Ramsey theory and topological dynamics.  

In this talk, we provide background on the Ramsey theory of the Rado graph, solved by Sauer.  A longstanding open question was whether Henson graphs, the k-clique-free analogues of the Rado graph, have similar features.  We present the speaker’s recent work solving the Ramsey theory of the Henson graphs.  The techniques developed open new lines of investigation for other relational structures with forbidden configurations.  As a byproduct of these methods, we may obtain Ramsey properties for Borel colorings on copies of the Rado graph, with respect to a certain topology.