Past

ABSTRACT

We approximate the solution of the stationary Stokes equations with various conforming and nonconforming inf-sup stable pairs of finite element spaces on simplicial meshes. Based on each pair, we design a discretization that is quasi-optimal and pressure robust, in the sense that the velocity H^1-error is proportional to the best H^1-error to the analytical velocity. This shows that such a property can be achieved without using conforming and divergence-free pairs. We bound also the pressure L^2-error, only in terms of the best approximation errors to the analytical velocity and the analytical pressure. Our construction can be summarized as follows. First, a linear operator acts on discrete velocity test functions, before the application of the load functional, and maps the discrete kernel into the analytical one.

Second, in order to enforce consistency, we  possibly employ a new augmented Lagrangian formulation, inspired by Discontinuous Galerkin methods.

Theerawat Bhudisaksang
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Adaptive robust control with statistical learning

We extend the adaptive robust methodology introduced in Bielecki et al. and propose a continuous-time version of their approach. Bielecki et al. consider a model in which the distribution of the underlying (observable) process depends on unknown parameters and the agent uses observations of the process to estimate the parameter values. The model is made robust to misspecification because the agent employs a set of ambiguity measures that contains measures where the parameter are inside a confidence region of their estimator. In our extension, we construct the set of ambiguity measures such that each probability measure in the set has a semimartingale characterisation lies in a restricted set. Finally, we prove the dynamic programming principle of the adaptive robust control in continuous time problem using measurable selection theorems, and we show that the value function can be characterised as the solution of a non-linear partial differential equation.

Yufei Zhang
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A neural network based policy iteration algorithm with global convergence of values and controls for stochastic games on domains

In this talk, we propose a class of neural network based numerical schemes for solving semi-linear Hamilton-Jacobi-Bellman-Isaacs (HJBI) boundary value problems which arise naturally from exit time problems of diffusion processes with controlled drift. We exploit a policy iteration to reduce the semilinear problem into a sequence of linear Dirichlet problems, which are subsequently approximated by a multilayer feedforward neural network ansatz. We establish that the numerical solutions converge globally in the H^2-norm, and further demonstrate that this convergence is superlinear, by interpreting the algorithm as an inexact Newton iteration for the HJBI equation. Moreover, we construct the optimal feedback controls from the numerical value functions and deduce convergence. The numerical schemes and convergence results are then extended to HJBI boundary value problems corresponding to controlled diffusion processes with oblique boundary reflection. Numerical experiments on the stochastic Zermelo navigation problem are presented to illustrate the theoretical results and to demonstrate the effectiveness of the method. 
 

The finite element exterior calculus is a powerful approach to study many problems under the same lens. The canonical finite element spaces (see Arnold, Falk and Winther) are tied together with an exact sequence and have the required smoothness to define the exterior derivatives weakly. However, some applications require spaces that are more smooth (e.g. plate bending problems, incompressible flows). In this talk we will discuss some recent results in developing finite element spaceson simplicial triangulations with more smoothness, that also fit in an exact sequence. This is joint work with Guosheng Fu, Anna Lischke and Michael Neilan.

KW states that every finite abelian extension of the rationals is contained in a cyclotomic extension. In a previous talk, this was reduced to considering cyclic extensions of the local fields Q_p of prime power order l^r. When l\neq p, general theory is sufficient, however for l=p, more specific (although not necessarily more abstruse) descriptions become necessary.
I will focus on the simple structure of Q_p's extensions to obstruct the remaining obstructions to KW (and hopefully provoke some interest in local fields in those less familiar). Time-permitting, I will talk about this theorem in the context of class field theory and/or Hilbert's 12th problem.

Many toric degenerations and integrable systems of the Grassmannians Gr(2, n) are described by trees, or equivalently subdivisions of polygons. These degenerations can also be seen to arise from the cones of the tropicalisation of the Grassmannian. In this talk, I focus on particular combinatorial types of cones in tropical Grassmannians Gr(k,n) and prove a necessary condition for such an initial degeneration to be toric. I will present several combinatorial conjectures and computational challenges around this problem.  This is based on joint works with Kristin Shaw and with Oliver Clarke.

Fireshape is a shape optimization library based on finite elements. In this talk I will describe how Florian Wechsung and I developed Fireshape and will share my reflections on lessons learned through the process.

We derive sharp bounds for the accuracy of approximate eigenvectors (Ritz vectors) obtained by the Rayleigh-Ritz process for symmetric eigenvalue problems. Using information that is available or easy to estimate, our bounds improve the classical Davis-Kahan sin-theta theorem by a factor that can be arbitrarily large, and can give nontrivial information even when the sin-theta theorem suggests that a Ritz vector might have no accuracy at all. We also present extensions in three directions, deriving error bounds for invariant subspaces, singular vectors and subspaces computed by a (Petrov-Galerkin) projection SVD method, and eigenvectors of self-adjoint operators on a Hilbert space.

Commuting concerns people’s spatial behaviour resulting from the geographic separation of home and workplace and is connected with their willingness to seek economic opportunities outside their place of residence (Rouwendal J., Nijkamp P., 2004). Such opportunities are usually found in the urban areas, so this phenomenon is often a subject of urban studies or research focusing on city centres (Drejerska N., Chrzanowska M., 2014). In literature, commuting patterns are used to determine the boundaries of local and regional labour markets. Furthermore, labour market is one of the most important features for the delimitation of functional regions, as commuting involves not only working outside one’s place of residence but also, among other things, using various services offered there, from shopping to health or cultural services. Taking this into account, it can be stated that commuting is an important characteristic of relations between territories, and these relations form complex networks.

People decide to commute to work for various reasons. Most commuters travel from a small town, village or rural area to a city or town where they have a wider range of employment opportunities. However, people differ in their attitudes toward commuting. While some people find it troublesome, others enjoy their daily travel. There are also people who regard commuting as the necessary condition for supporting themselves and their families. Therefore, commuting is an important factor that should be taken into account in the research on the quality of life and quality of work.

The main goals of this presentation is to identify and analyse relations between communities (municipalities) from the perspective of labour market, especially commuting in the vicinity of Warsaw, Data on the number of commuters come from the Central Statistical Office of Poland and cover the year 2011.

 Bibliography

Drejerska N., Chrzanowska M., 2014: Commuting in the Warsaw suburban area from a spatial perspective – an example of empirical research, Acta Universitatis Lodziensis. Folia Oeconomica 2014, Vol. 6, no 309, pp. 87-96.

Rouwendal J., Nijkamp P., 2004: Living in Two Worlds: A Review of Home-to-Work Decisions, Growth and Change, Volume 35, Issue 3, p. 287.

The classical theory of integration concern integrals of differential forms over domains of integration. In geometric terms, this corresponds to a canonical pairing between de Rham cohomology and singular homology. For varieties defined over the reals, one can make use of complex conjugation to define a real-valued pairing between de Rham cohomology and its dual, de Rham homology. The corresponding theory of integration, that we call single-valued integration, pairs a differential form with a `dual differential form’. We will explain how single-valued periods are computed and give an application to superstring amplitudes in genus zero. This is joint work with Francis Brown.
 

I plan to present a brief introduction to optimal control theory (no background knowledge assumed), and discuss a fascinating and oft-forgotten family of problems where the optimal control behaves very strangely; it changes state infinitely often in finite time. This causes havoc in practice, and even more so in the literature.
 

"We study the hydrodynamic limit for the isothermal dynamics of an anharmonic chain under hyperbolic space-time scaling under varying tension. The temperature is kept constant by a contact with a heat bath, realised via a stochastic momentum-preserving noise added to the dynamics. The noise is designed to be large at the microscopic level, but vanishing in the macroscopic scale. Boundary conditions are also considered: one end of the chain is kept fixed, while a time-varying tension is applied to the other end. We show that the volume stretch and momentum converge to a weak solution of the isothermal Euler equations in Lagrangian coordinates with boundary conditions."

We will discuss a variant of Taubes’s Seiberg-Witten to Gromov theorem in the context of a 4-manifold with cylindrical ends, equipped with a nontrivial harmonic 2-form. This harmonic 2-form is allowed to be asymptotic to 0 on some (but not all) of its ends, and may have nondegenerate zeros along 1-submanifolds. Corollaries include various positivity results; some simple special cases of these constitute a key ingredient in Kutluhan-Lee-Taubes’s proof of HM = HF (Monopole Floer homology equals Heegaard Floer homology). The aforementioned general theorem is motivated by (potential) extensions of the HM = HF and Lee-Taubes’s HM = PFH (Periodic Floer homology) theorems.

What actually happens when you submit an article to a journal? How does refereeing work in practice? How can you keep editors happy as an author or referee? How does one become a referee or editor? What does 'publication' mean with the internet and arXiv?

In this panel we'll discuss what happens between finishing writing a mathematical paper and its final (?) publication, looking at the various roles that people play and how they work best.

Featuring Helen Byrne, Rama Cont and Jonathan Pila.

 

This talk will cover the connections of persistence with the topology of random structures. This includes an overview of various results from stochastic topology as well as the role persistence ideas  play in the analysis. This will include results on the maximally persistent classes and minimum spanning acycles/generalised trees.

This year sees the 100th anniversary of Emmy Noether receiving her Habilitation and thus becoming the first women to be granted the right to teach and lecture at a university in Prussia (now Germany).  Noether shaped modern algebra and her influence was felt in many other fields including topology.

We will start by exploring what algebraic topology is, how the subject was shaped by algebra (under the influence of Noether), before considering some current challenges and applications.

Despite their tiny size, microorganisms play a huge role in many biological, medical, and engineering phenomena. For example, massive plankton blooms are an integral part of the oceanic ecosystem. Algal cells incorporate carbon dioxide, which affects global warming. In industry, microorganisms are used in bioreactors to produce food and medicines and to treat sewage. The human body hosts hundreds of microorganism species, and the number of microorganisms in the human body is roughly double the number of cells in the body. In the intestine, approximately 1 kg of enterobacteria form a unique ecosystem, called the gut flora, which plays important roles in digestion and in relation to infection. Because of the considerable influence that microorganisms have on human life, the study of their behavior and function is important.

Recent research has demonstrated the importance of biomechanics in understanding the behavior and functions of microorganisms. For example, red tides can be induced by the interplay between the background flow and swimming cells. A dense suspension of bacteria can generate a coherent structure, which strongly enhances mass transport in a suspension. These phenomena show that the physical environments around cells alter their behavior and biological functions. Such a biomechanical understanding is still lacking in microbiology, and we believe that biomechanics can provide new perspectives on future microbiology.

In this talk, we first introduce some of our studies of the behavior of individual swimming microorganisms near surfaces. We show that hydrodynamic forces can trap cells at liquid–air or liquid–solid interfaces. We then introduce interactions between a pair of swimming microorganisms, because a two-body interaction is the simplest many-body interaction. We show that our mathematical models can describe the interactions between two nearby swimming microorganisms. Collective motions formed by a group of swimming microorganisms are also introduced. We show that some collective motions of microorganisms, such as coherent structures of bacterial suspensions, can be understood in terms of fluid mechanics. We then discuss how cellular-level phenomena can change the rheological and diffusion properties of a suspension. The macroscopic properties of a suspension are strongly affected by mesoscale flow structures, which in turn are strongly affected by the interactions between cells. Hence, a bottom-up strategy, i.e., from a cellular level to a continuum suspension level, represents a natural approach to the study of a suspension of swimming microorganisms. Finally, we discuss whether our understanding of biological functions can be strengthened by the application of biomechanics, and how we can contribute to the future of microbiology.

Berry's random wave conjecture is a heuristic that the eigenfunctions of a classically ergodic system ought to display Gaussian random behaviour, as though they were random waves, in the large eigenvalue limit. We discuss two manifestations of this conjecture for eigenfunctions of the Laplacian on the modular surface: Planck scale mass equidistribution, and an asymptotic for the fourth moment. We will highlight how the resolution of these two problems in this number-theoretic setting involves a delicate understanding of the behaviour of certain families of L-functions.

In the first part of the talk, we study risk-sharing equilibria where heterogenous agents trade subject to quadratic transaction costs. The corresponding equilibrium asset prices and trading strategies are characterised by a system of nonlinear, fully-coupled forward-backward stochastic differential equations. We show that a unique solution generally exists provided that the agents’ preferences are sufficiently similar. In a benchmark specification, the illiquidity discounts and liquidity premia observed empirically correspond to a positive relationship between transaction costs and volatility.
In the second part of the talk, we discuss how the model can be calibrated to time series of prices and the corresponding trading volume, and explain how extensions of the model with general transaction costs, for example, can be solved numerically using the deep learning approach of Han, Jentzen, and E (2018).
 (Based on joint works with Martin Herdegen and Dylan Possamai, as well as with Lukas Gonon and Xiaofei Shi)

Multiplicative preprojective algebras (MPAs) were originally defined by Crawley-Boevey and Shaw to encode solutions of the Deligne-Simpson problem as irreducible representations. 
MPAs have recently appeared in the literature from different perspectives including Fukaya categories of plumbed cotangent bundles (Etgü and Lekili) and, similarly, microlocal sheaves 
on rational curves (Bezrukavnikov and Kapronov.) After some motivation, I'll suggest a purely algebraic approach to study these algebras. Namely, I'll outline a proof that MPAs are 
2-Calabi-Yau if Q contains a cycle and an inductive argument to reduce to the case of the cycle itself.

Cracks in many soft solids behave very differently to the classical picture of fracture, where cracks are long and thin, with damage localised to a crack tip. In particular, small cracks in soft solids become highly rounded — almost circular — before they start to extend. However, despite being commonplace, this is still not well understood. We use a phase-separation technique in soft, stretched solids to controllably nucleate and grow small, nascent cracks. These give insight into the soft failure process. In particular, our results suggest fracture occurs in two regimes. When a crack is large, it obeys classical linear-elastic fracture mechanics, but when it is small it grows in a new, scale-free way at a constant driving stress.

Stochastic control problems are closely related to free boundary problems, where both the underlying fully nonlinear PDEs and the boundaries separating the action and waiting regions are integral parts of the problems. In this talk, we will propose a class of stochastic N-player games and show how the free boundary problems involve moving boundaries due to the additional game nature. We will provide explicit Nash equilibria by solving a sequence of Skorokhod problems. For the special cases of resource allocation problems, we will show how players change their strategies based on different network structures between players and resources. We will also talk about the insights from a sharing economy perspective. This talk is based on a joint work with Xin Guo (UC Berkeley) and Wenpin Tang (UCLA).

We present some regularity results for the gradient of solutions to very degenerate equations, which exhibit a great lack of ellipticity.
In particular we show that local weak solutions of the orthotropic p−harmonic equation are locally Lipschitz, for every $p\geq 2$ and in every dimension.
The results presented in this talk have been obtained in collaboration with Pierre Bousquet (Toulouse), Lorenzo Brasco (Ferrara) and Anna Verde (Napoli).
 

The Poincare Conjecture was first formulated over a century ago and states that there is only one closed simply connected 3-manifold, hinting at a link between 3-manifolds and their fundamental groups. This seemingly basic fact went unproven until the early 2000s when Perelman proved Thurston's much more powerful Geometrisation Conjecture, providing us with a powerful structure theorem for understanding all closed 3-manifolds.
In this talk I will introduce the results developed throughout the 20th century that lead to Thurston and Perelman's work. Then, using Geometrisation as a black box, I will present a proof of the Poincare Conjecture. Throughout we shall follow the crucial role that the fundamental group plays and hopefully demonstrate the geometric and group theoretical nature of much of the modern study of 3-manifolds.
As the title suggests, no prior understanding of 3-manifolds will be expected.
 

The Kronecker-Weber theorem states that every finite abelian extension of the rationals is contained in some cyclotomic field. I will present a proof that emphasizes the standard local-global philosophy by first proving it for the p-adics and then deducing the global case.

Many time-dependent PDEs -- KdV, Burgers, Gray-Scott, Allen-Cahn, Navier-Stokes and many others -- combine a higher-order linear term with a lower-order nonlinear term.  This talk will review the method of exponential integrators for solving such problems with better than 2nd-order accuracy in time.

A defect-based a posteriori error estimate for Krylov subspace approximations to the matrix exponential is introduced. This error estimate constitutes an upper norm bound on the error and can be computed during the construction of the Krylov subspace with nearly no computational effort. The matrix exponential function itself can be understood as a time propagation with restarts. In practice, we are interested in finding time steps for which the error of the Krylov subspace approximation is smaller than a given tolerance. Finding correct time steps is a simple task with our error estimate. Apart from step size control, the upper error bound can be used on the fly to test if the dimension of the Krylov subspace is already sufficiently large to solve the problem in a single time step with the required accuracy.

I will review the idea that entanglement must ultimately be understood in terms of modes, rather than in terms of particles. The most striking instance of mode entanglement is a single particle entangled state, which I will discuss both in the case of bosons as well as in the case of fermions. I then proceed to show that the Aharonov-Bohm effect can be understood by using a single electron entangled state. Finally, I will argue that this demonstrates beyond doubt that the Aharonov-Bohm effect is non non-local, contrary to what is frequently claimed in the literature.

The advent of social media and the resulting ability to instantaneously communicate ideas and messages to connections worldwide is one of the great consequences arising from the telecommunications revolution over the last century. Individuals do not, however, communicate only upon a single platform; instead there exists a plethora of options available to users, many of whom are active on a number of such media. While each platform offers some unique selling point to attract users, e.g., keeping up to date with friends through messaging and statuses (Facebook), photo sharing (Instagram), seeing information from friends, celebrities and numerous other outlets (Twitter) or keeping track of the career paths of friends and past colleagues (Linkedin), the platforms are all based upon the fundamental mechanisms of connecting with other users and transmitting information to them as a result of this link.

In this talk a model for the spreading of online information or “memes" on multiplex networks is introduced and analyzed using branching-process methods. The model generalizes that of [Gleeson et al., Phys. Rev. X., 2016] in two ways. First, even for a monoplex (single-layer) network, the model is defined for any specific network defined by its adjacency matrix, instead of being restricted to an ensemble of random networks. Second, a multiplex version of the model is introduced to capture the behavior of users who post information from one social media platform to another. In both cases the branching process analysis demonstrates that the dynamical system is, in the limit of low innovation, poised near a critical point, which is known to lead to heavy-tailed distributions of meme popularity similar to those observed in empirical data.

[1] J. P. Gleeson et al. “Effects of network structure, competition and memory time on social spreading phenomena”. Physical Review X 6.2 (2016), p. 021019.

[2] J. D. O’Brien et al. "Spreading of memes on multiplex networks." New Journal of Physics 21.2 (2019): 025001.

For measuring families of curves, or, more generally, of measures, $M_p$-modulus is traditionally used. More recent studies use so-called plans on measures. In their fundamental paper Ambrosio, Di Marino and Savare proved that these two approaches are in some sense equivalent within $1<p<\infty$. We consider the limiting case $p=1$ and show that the $AM$-modulus can be obtained alternatively by the plan approach. On the way, we demonstrate unexpected behavior of the $AM$-modulus in comparison with usual capacities.

This is a joint work with Vendula Honzlov\'a Exnerov\'a, Ond\v{r}ej F.K. Kalenda and Olli Martio. Partially supported by the grant GA\,\v{C}R P201/18-07996S of the Czech Science Foundation.

X.S. Lin defined an invariant of knots in S^3 by counting represenations 
of the knot group into SU(2) with fixed meridinal holonomy. Lin's 
invariant was subsequently shown to coincide with the Levine-Tristam 
signature. I'll define an analogous total Lin invariant which counts 
repesentations into both SU(2) and SL_2(R). Unlike the SU(2) version, this 
invariant does not (as far as I know) coincide with other known 
invariants. I'll describe some applications to left-orderability of Dehn 
fillings and branched covers, as well as a curious connection with the 
Alexander polynomial. This is joint work with Nathan Dunfield.

We observe a noisy version of a large rank-one matrix. Depending on the strength of the noise, can we recover non-trivial information on the matrix? This problem, interesting on its own, will be motivated by its link with a "spin glass" model, which is a model of statistical mechanics where a large number of variables interact with one another, with random interactions that can be positive or negative. The resolution of the initial question will involve a Hamilton-Jacobi equation

I will discuss new results for the gradient field models with uniformly convex potential (also known as the Ginzburg-Landau field). A connection between the scaling limits of the field and elliptic homogenization was introduced by Naddaf and Spencer in 1997. We quantify the existing central limit theorems in light of recent advances in quantitative homogenization; and positively settle a conjecture of Funaki and Spohn about the surface tension. Joint work with Scott Armstrong. 

I will describe joint work with Bruno Premoselli which gives a new existence theorem for negatively curved Einstein 4-manifolds, which are obtained by smoothing the singularities of hyperbolic cone metrics. Let (M_k) be a sequence of compact 4-manifolds and let g_k be a hyperbolic cone metric on M_k with cone angle \alpha (independent of k) along a smooth surface S_k. We make the following assumptions:

1. The injectivity radius i(k) of M_k tends to infinity (where in defining injectivity radius we ignore those geodesics which hit the cone singularity)

2. The normal injectivity radius of S_k is at least i(k)/2.

3. The area of the singular locii satisfy A(S_k)\leq C \exp(5 i(k)/2) for some C independent of k.

When these assumptions hold, we prove that for all large k, M_k carries a smooth Einstein metric of negative curvature. The proof involves a gluing theorem and a parameter dependent implicit function theorem (where k is the parameter). As I will explain, negative curvature plays an essential role in the proof. (For those who may be aware of our arxiv preprint, https://arxiv.org/abs/1802.00608 [arxiv.org], the work
I will describe has a new feature, namely we now treat all cone angles, and not just those which are greater than 2\pi. This gives lots more examples of Einstein 4-manifolds.)

How do you create a self-sustaining, flourishing academic community in a developing country? What kind of challenges need to be overcome to ensure that quality education becomes available? What can we do to help make it happen? In this talk, we will describe our experience visiting the University of Yangon in Myanmar. During the visit, we delivered a course to the academic staff, and discussed future collaborations between Oxford and Yangon, as well as further directions for Mathematical education in Myanmar, all the while marvelling at the wonders of the Burmese culture.

I'll talk about smooth solutions of Euler equations with compactly supported velocities, and applications to other equations.

In this talk I want to outline the proofs our of main results, i.e. the localisation theorem and the identification of the homotopy type of Grothendieck-Witt theory in terms of K- and L-theory.
Finally, as a small application I want to present a refinement and extension of certain maps relating certain Madsen-Tillmann spectra and orthogonal/symplectic algebraic K-theory spectra of the integers.

All original material is joint work with B.Calmès, E.Dotto, Y.Harpaz, M.Land, K.Moi, D.Nardin, T.Nikolaus and W.Steimle.
 

I will start by briefly reviewing the Tate construction and in particular, the Tate diagonal. Using these I will then illustrate Lurie’s notion of Poincaré categories by considering Poincaré structures on module categories over a ring (spectrum) in detail. In particular, I will describe the somewhat subtle genuine Poincaré structure on the category of perfect complexes of an ordinary ring, which conjecturally links the classical notion of the Grothendieck-Witt spectrum to our derived version. Finally, I will compute its associated L-groups.

I will explain how Lurie‘s approach to L-theory via Poincaré categories can be extended to yield cobordism categories of Poincaré objects à la Ranicki. These categories can be delooped by an iterated Q-construction and the resulting spectrum is a derived version of Grothendieck-Witt-theory.  Its homotopy type can be described in terms of K- and L-theory as conjectured by Hesselholt-Madsen. Furthermore, it has a clean universal property analogous to that of K-theory, localisation sequences in much greater generality than classical Grothendieck-Witt theory, gives a cycle description of Weiss-Williams‘ LA-theory and allows for maps from the geometric cobordism category, refining and unifying various known invariants.

All original material is joint work with B.Calmès, E.Dotto, Y.Harpaz, M.Land, K.Moi, D.Nardin, T.Nikolaus and W.Steimle.