Past

Graph sampling with noise is a fundamental problem in graph signal processing (GSP). A popular biased scheme using graph Laplacian regularization (GLR) solves a system of linear equations for its reconstruction. Assuming this GLR-based reconstruction scheme, we propose a fast sampling strategy to maximize the numerical stability of the linear system--i.e., minimize the condition number of the coefficient matrix. Specifically, we maximize the eigenvalue lower bounds of the matrix that are left-ends of Gershgorin discs of the coefficient matrix, without eigen-decomposition. We propose an iterative algorithm to traverse the graph nodes via Breadth First Search (BFS) and align the left-ends of all corresponding Gershgorin discs at lower-bound threshold T using two basic operations: disc shifting and scaling. We then perform binary search to maximize T given a sample budget K. Experiments on real graph data show that the proposed algorithm can effectively promote large eigenvalue lower bounds, and the reconstruction MSE is the same or smaller than existing sampling methods for different budget K at much lower complexity.

Integrable models provide simplified examples of quantum field theories with self-interaction. As often in relativistic quantum theory, their local observables are difficult to control mathematically. One either tries to construct pointlike local quantum fields, leading to possibly divergent series expansions, or one defines the local observables indirectly via wedge-local quantities, losing control over their explicit form.

We propose a new, hybrid approach: We aim to describe local quantum fields; but rather than exhibiting their n-point functions and verifying the Wightman axioms, we establish them as closed operators affiliated with a net of von Neumann algebras. This is shown to work at least in the Ising model.

George Soules [1] introduced a set of vectors $r_1,...,r_N$ with the remarkable property that for any set of ordered numbers $\lambda_1\geq\dots\geq\lambda_N$, the matrix $\sum_n \lambda_nr_nr_n^T$ has nonnegative off-diagonal entries. Later, it was found [2] that there exists a whole class of such vectors - Soules vectors - which are intimately connected to binary rooted trees. In this talk I will describe the construction of Soules vectors starting from a binary rooted tree, and introduce some basic properties. I will also cover a number of applications: the inverse eigenvalue problem, equitable partitions in Laplacian matrices and the eigendecomposition of the Clauset-Moore-Newman hierarchical random graph model.

[1] Soules (1983), Constructing Symmetric Nonnegative Matrices
[2] Elsner, Nabben and Neumann (1998), Orthogonal bases that lead to symmetric nonnegative matrices

In joint work with Peter Topping we introduce pyramid Ricci flows, defined throughout uniform regions of spacetime that are not simply parabolic cylinders, and enjoying curvature estimates that are not required to remain spatially constant throughout the domain of definition. This weakened notion of Ricci flow may be run in situations ill-suited to the classical theory. As an application of pyramid Ricci flows, we obtain global regularity results for three-dimensional Ricci limit spaces (extending results of Miles Simon and Peter Topping) and for higher dimensional PIC1 limit spaces (extending not only the results of Richard Bamler, Esther Cabezas-Rivas and Burkhard Wilking, but also the subsequent refinements by Yi Lai).
 

Many singular stochastic PDEs are expected to be universal objects that govern a wide range of microscopic models in different universality classes. Two notable examples are KPZ and \Phi^4_3. In these cases, one usually finds a parameter in the system, and tunes according to the space-time scale in such a way that the system rescales to the SPDE in the large-scale limit. We justify this belief for a large class of continuous microscopic growth models (for KPZ) and phase co-existence models (for Phi^4_3), allowing microscopic nonlinear mechanisms far beyond polynomials. Aside from the framework of regularity structures, the main new ingredient is a moment bound for general nonlinear functionals of Gaussians. This essentially allows one to reduce the problem of a general function to that of a polynomial. Based on a joint work with Martin Hairer, and another joint work in progress with Chenjie Fan and Jiawei Li. 

In a recent paper, Basterra, Bobkova, Ponto, Tillmann and Yeakel defined
topological operads with homological stability (OHS) and proved that the
group completion of an algebra over an OHS is weakly equivalent to an
infinite loop space.

In this talk, I shall outline a construction which to an algebra A over
an OHS associates a new infinite loop space. Under mild conditions on
the operad, this space is equivalent as an infinite loop space to the
group completion of A. This generalises a result of Wahl on the
equivalence of the two infinite loop space structures constructed by
Tillmann on the classifying space of the stable mapping class group. I
shall also talk about an application of this construction to stable
moduli spaces of high-dimensional manifolds in thesense of Galatius and
Randal-Williams.

We will start by presenting two basic probabilistic effects for questions concerning the regularity of functions and nonlinear operations on functions. We will then overview well-posedenss results for the nonlinear wave equation, the nonlinear Schr\"odinger equation and the nonlinear heat equation, in the presence of singular randomness.

In this talk we discuss new effective methods to test pairwise containment of arbitrary (possibly singular) subvarieties of any smooth projective toric variety and to determine algebraic multiplicity without working in local rings. These methods may be implemented without using Gröbner bases; in particular any algorithm to compute the number of solutions of a zero-dimensional polynomial system may be used. The methods arise from techniques developed to compute the Segre class s(X,Y) of X in Y for X and Y arbitrary subschemes of some smooth projective toric variety T. In particular, this work also gives an explicit method to compute these Segre classes and other associated objects such as the Fulton-MacPherson intersection product of projective varieties.
These algorithms are implemented in Macaulay2 and have been found to be effective on a variety of examples. This is joint work with Corey Harris (University of Oslo).

A virtue of the special geometry underlying the string theory moduli space of  Calabi--Yau manifolds is the existence of a canonical choice of moduli space coordinates. In heterotic theories, as much as we would desire it, there is no obvious choice of coordinates and so we should be covariant. I will discuss some issues in doing this.

Aura Raulo (Ecological and Evolutionary Dynamics) and Marie-Claire Koschowitz (Vertebrate Palaeobiology) discuss their work and its mathematical challenges.

Aura Raulo

" Aura Raulo is a graduate student in Zoology Department working on transmission of symbiotic bacteria in the social networks of their animal hosts"
Title: Heaps in networks - How we share our microbiota through kisses
Abstract: Humans, like all vertebrates have a microbiome, a diverse community of symbiotic bacteria that live in and on us and are crucial for our functioning. These bacteria help us digest food, regulate our mood and function as a key part of our immune system. Intriguingly, while they are part of us, they are, unlike our other cells, in constant flux between us, challenging the traditional definition of a biological individual. Many of these bacteria need intimate social contact to be transmitted from human to human, making social network analysis tools handy in explaining their community dynamics.What then is a recipe for a ``good microbiome”? Theories and evidence implies that the most healthy and immunologically robust microbiome composition is both diverse, semi-stable and somewhat synchronized among closely interacting individuals, but little is known about what kind of transmission landscapes determine these bacterial cocktails. In my talk, I will present humanmicrobiome as a network trait: a metacommunity of cells shaped by an equilibrium of isolation and contact among their hosts. I propose that we do notnecessarily need to think of levels of life (e.g. cells, individuals, populations) as being neatly nested inside of each other. Rather, aggregations of cooperating cells (both bacteria and human cells) can be considered as mere tighter clusters in their interaction network, dynamically creating de novo defined units of life. I will present a few game theoretical evolutionary dilemmas following from this perspective and highlight outstanding questions in mapping how network position of the host translates into community composition of bacteria in flux.

Marie Koschowitz
“Marie Koschowitz is a PhD student in the Department of Zoology and the Department of Earth Sciences, working on comparative physiology and large scale evolutionary patterns in reptiles such as crocodiles, birds and dinosaurs."
Title: Putting the maths into dinosaurs – A zoologist's perspective
Abstract: Contemporary palaeontology is a subject area that often deals with sparse data.Therefore, palaeontologists became rather inventive in pursuit of getting the most out of what is available. If we find a dinosaur’s skull that shows prominent, but puzzling, bony ridges without any apparent function, how can we make meaningful interpretations of its purpose in the living animal that was? If we are confronted with a variety of partially preserved bones from animals looking anatomically similar, but not quite alike, how can we infer relationships in the absence of genetic data?Some methods that resolve these questions, such as finite element analysis, were borrowed from engineering. Others, like comparative phylogenetics or MCMC generalised mixed effects models, are even more directly based on mathematical computations. All of these approaches help us to calculate things like a raptors bite-force and understand the ins and outsof their skulls anatomy, or why pterosaurs and plesiosaurs aren’t exactly dinosaurs. This talk aims to presents a selection of current approaches to applied mathematics which have been inspired by interdisciplinary research – and to foster awareness of all the ways how mathematicians can get involved in “dinosaur research”, if they feel inclined to do so.


 

The class of sheaf Laplacians can be characterized as the convex closure of a certain set of sparse semidefinite matrices. From this viewpoint, the study of sheaf Laplacians becomes a question of linear algebra on sparse matrices. I will discuss the applications of this perspective to the problems of approximating, sparsifying, and learning sheaves.

This workshop will focus on the main causes of exam stress, anxiety and panic and look at practical strategies to manage and overcome these issues. We will also review strategies to best support exam preparation.

Dr Ruth Collins is a Chartered Psychologist who specialises in the management of anxiety and panic. She is also a trained mindfulness teacher and an associate of the Oxford Mindfulness Centre.

Inertia-gravity waves (IGWs) are ubiquitous in the ocean and the atmosphere. Once generated (by tides, topography, convection and other processes), they propagate and scatter in the large-scale, geostrophically-balanced background flow. I will discuss models of this scattering which represent the background flow as a random field with known statistics. Without assumption of spatial scale separation between waves and flow, the scattering is described by a kinetic equation involving a scattering cross section determined by the energy spectrum of the flow. In the limit of small-scale waves, this equation reduces to a diffusion equation in wavenumber space. This predicts, in particular, IGW energy spectra scaling as k^{-2}, consistent with observations in the atmosphere and ocean, lending some support to recent claims that (sub)mesoscale spectra can be attributed to almost linear IGWs.  The theoretical predictions are checked against numerical simulations of the three-dimensional Boussinesq equations.
(Joint work with Miles Savva and Hossein Kafiabad.)

The discrete fundamental groups of a metric space can be thought of as fundamental groups that `ignore' closed loops up to some specified size R. As the parameter R grows, these groups have been used to produce interesting invariants of coarse geometry. On the other hand, as R gets smaller one would expect to retrieve the usual fundamental group as a limit. In this talk I will try to briefly illustrate both these aspects.

The problem of decomposing a given dataset as a superposition of basic motifs arises in a wide range of application areas, including neural spike sorting and the analysis of astrophysical and microscopy data. Motivated by these problems, we study a "short-and-sparse" deconvolution problem, in which the goal is to recover a short motif a from its convolution with a random spike train $x$. We formulate this problem as optimization over the sphere. We analyze the geometry of this (nonconvex) optimization problem, and argue that when the target spike train is sufficiently sparse, on a region of the sphere, every local minimum is equivalent to the ground truth, up to symmetry (here a signed shift). This characterization obtains, e.g., for generic kernels of length $k$, when the sparsity rate of the spike train is proportional to $k^{-2/3}$ (i.e., roughly $k^{1/3}$ spikes in each length-$k$ window). This geometric characterization implies that efficient methods obtain the ground truth under the same conditions. 

Our analysis highlights the key roles of symmetry and negative curvature in the behavior of efficient methods -- in particular, the role of a "dispersive" structure in promoting efficient convergence to global optimizers without the need to explicitly leverage second-order information. We sketch connections to broader families of benign nonconvex problems in machine learning and signal processing, in which efficient methods obtain global optima independent of initialization. These problems include variants of sparse dictionary learning, tensor decomposition, and phase recovery.

Joint work with Yuqian Zhang, Yenson Lau, Han-Wen Kuo, Dar Gilboa, Sky Cheung, Abhay Pasupathy

Atom Probe Tomography is a powerful 3D mass spectrometry technique. By pulsing the sample apex with an electric field, surface atoms are ionised and collected by a detector. A 3D image of estimated initial ion positions is constructed via an image reconstruction protocol. Current protocols assume ion trajectories follow a stereographic projection. However, this method assumes a hemispherical sample apex that fails to account for varying material ionisation rates and introduces severe distortions into atomic distributions for complex material systems.

We aim to develop continuum models and use this to derive a time-dependent mapping describing how ion initial positions on the sample surface correspond to final impact positions on the detector. When correctly calibrated with experiment, such a mapping could be used for performing reconstruction.

Currently we track the sample surface using a level set method, while the electric field is solved via BEM or a FEM-BEM coupling. These field calculations must remain accurate close to the boundary. Calibrating unknown evaporation parameters with experiment requires an ensemble of models per experiment. Therefore, we are also looking to maximise model efficiency via BEM compression methods i.e. fast multipole BEM. Efficiently constructing and reliably interpolating the non-bijective trajectory mapping, while accounting for ion trajectory overlap and instabilities (at sample surface corners), also presents intriguing problems.

This project is in collaboration with Cameca, the leading manufacturer of commercial atom probe instruments. If successful in minimising distortions such a technique could become valuable within the semiconductor industry.

PLEASE NOTE THAT THIS SEMINAR IS CANCELLED DUE TO UNFORSEEN CIRCUMSTANCES.

We introduce a new probabilistic model for primes, which we believe is a better predictor for large gaps than the models of Cramer and Granville. We also make strong connections between our model, prime k-tuple counts, large gaps and the "square-root sieve".  In particular, our model makes a prediction about large prime gaps that may contradict the models of Cramer and Granville, depending on the tightness of a certain sieve estimate. This is joint work with Bill Banks and Terence Tao.

Gauge-theoretic invariants such as Donaldson or Seiberg–Witten invariants of 4-manifolds, Casson invariants of 3-manifolds, Donaldson–Thomas invariants of Calabi–Yau 3- and 4-folds, and putative Donaldson–Segal invariants of G_2 manifolds are defined by constructing a moduli space of solutions to an elliptic PDE as a (derived) manifold and integrating the (virtual) fundamental class against cohomology classes. For a moduli space to have a (virtual) fundamental class it must be compact, oriented, and (quasi-)smooth. We first describe a general framework for addressing orientability of gauge-theoretic moduli spaces due to Joyce–Tanaka–Upmeier. We then show that the moduli stack of perfect complexes of coherent sheaves on a Calabi–Yau 4-fold X is a homotopy-theoretic group completion of the topological realisation of the moduli stack of algebraic vector bundles on X. This allows one to extend orientations on the locus of algebraic vector bundles to the boundary of the (compact) moduli space of coherent sheaves using the universal property of homotopy-theoretic group completions. This is a necessary step in constructing Donaldson–Thomas invariants of Calabi–Yau 4-folds. This is joint work with Yalong Cao and Dominic Joyce.

We present a consistent neural network based calibration method for a number of volatility models-including the rough volatility family-that performs the calibration task within a few milliseconds for the full implied volatility surface.
The aim of neural networks in this work is an off-line approximation of complex pricing functions, which are difficult to represent or time-consuming to evaluate by other means. We highlight how this perspective opens new horizons for quantitative modelling: The calibration bottleneck posed by a slow pricing of derivative contracts is lifted. This brings several model families (such as rough volatility models) within the scope of applicability in industry practice. As customary for machine learning, the form in which information from available data is extracted and stored is crucial for network performance. With this in mind we discuss how our approach addresses the usual challenges of machine learning solutions in a financial context (availability of training data, interpretability of results for regulators, control over generalisation errors). We present specific architectures for price approximation and calibration and optimize these with respect different objectives regarding accuracy, speed and robustness. We also find that including the intermediate step of learning pricing functions of (classical or rough) models before calibration significantly improves network performance compared to direct calibration to data.

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Abstract

The dynamical fields that emanate from self-similarly expanding ellipsoidal regions undergoing phase change (change in density, i.e., volume collapse, and change in moduli) under pre-stress, constitute the dynamic generalization of the seminal Eshelby inhomogeneity problem (as an equivalent inclusion problem), and they consist of pressure, shear, and M waves emitted by the surface of the expanding ellipsoid and yielding Rayleigh waves in the crack limit. They may constitute the model of Deep Focus Earthquakes (DFEs) occurring under very high pressures and due to phase change. Two fundamental theorems of physics govern the phenomenon, the Cauchy-Kowalewskaya theorem, which based on dimensional analysis and analytic properties alone, dictates that there is zero particle velocity in the interior, and Noether’s theorem that extremizes (minimizes for stability) the energy spent to move the boundary so that it does not become a sink (or source) of energy, and determines the self-similar shape (axes expansion speeds). The expression from Noether’s theorem indicates that the expanding region can be planar, thus breaking the symmetry of the input and the phenomenon manifests itself as a newly discovered one of a “dynamic collapse/ cavitation instability”, where very large strain energy condensed in the very thin region can escape out. In the presence of shear, the flattened very thin ellipsoid (or band) will be oriented in space so that the energy due to phase change under pre-stress is able to escape out at minimum loss condensed in the core of dislocations gliding out on the planes where the maximum configurational force (Peach-Koehler) is applied on them. Phase change occurring planarly produces in a flattened expanding ellipdoid a new defect present in the DFEs. The radiation patterns are obtained in terms of the equivalent to the phase change six eigenstrain components, which also contain effects due to planarity through the Dynamic Eshelby Tensor for the flattened ellipsoid. Some models in the literature of DFEs are evaluated and excluded on the basis of not having the energy to move the boundary of phase discontinuity. Noether’s theorem is valid in anisotropy and nonlinear elasticity, and the phenomenon is independent of scales, valid from the nano to the very large ones, and applicable in general to other dynamic phenomena of stress induced martensitic transformations, shear banding, and amorphization.

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
----------------------

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
-----------

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).