Past

The Weak Gravity Conjecture holds that in any consistent theory of quantum gravity, gravity must be the weakest force. This simple proposition has surprisingly nontrivial physical consequences, which in the case of supersymmetric string/M-theory compactifications lead to nontrivial geometric consequences for Calabi-Yau manifolds. In this talk we will describe these conjectured geometric consequences in detail and show how they are realized in concrete examples, deriving new results about 5d supersymmetric black holes in the process.

The classical Universal Approximation Theorem certifies that the universal approximation property holds for the class of neural networks of arbitrary width. Here we consider the natural `dual' theorem for width-bounded networks of arbitrary depth, for a broad class of activation functions. In particular we show that such a result holds for polynomial activation functions, making this genuinely different to the classical case. We will then discuss some natural extensions of this result, e.g. for nowhere differentiable activation functions, or for noncompact domains.
 

Tensors are higher dimensional analogues of matrices; they are used to record data with multiple changing variables. Interpreting tensor data requires finding low rank structure, and the structure depends on the application or context. Often tensors of interest define semi-algebraic sets, given by polynomial equations and inequalities. I'll give a characterization of the set of tensors of real rank two, and answer questions about statistical models using probability tensors and semi-algebraic statistics. I will also describe work on learning a path from its three-dimensional signature tensor. This talk is based on joint work with Guido Montúfar, Max Pfeffer, and Bernd Sturmfels.

We study nearly singular PDEs that arise in the solution of linear elasticity and incompressible flow. We will demonstrate, that due to the nearly singular nature, standard methods for the solution of the linear systems arising in a finite element discretisation for these problems fail. We motivate two key ingredients required for a robust multigrid scheme for these equations and construct robust relaxation and prolongation operators for a particular choice of discretisation.
 

Numerous researchers recently applied empirical spectral analysis to the study of modern deep learning classifiers. We identify and discuss an important formal class/cross-class structure and show how it lies at the origin of the many visually striking features observed in deepnet spectra, some of which were reported in recent articles and others unveiled here for the first time. These include spectral outliers and small but distinct bumps often seen beyond the edge of a "main bulk". The structure we identify organizes the coordinates of deepnet features and back-propagated errors, indexing them as an NxC or NxCxC array. Such arrays can be indexed by a two-tuple (i,c) or a three-tuple (i,c,c'), where i runs across the indices of the train set; c runs across the class indices and c' runs across the cross-class indices. This indexing naturally induces C class means, each obtained by averaging over the indices i and c' for a fixed class c. The same indexing also naturally defines C^2 cross-class means, each obtained by averaging over the index i for a fixed class c and a cross-class c'. We develop a formal process of spectral attribution, which is used to show the outliers are attributable to the C class means; the small bump next to the "main bulk" is attributable to between-cross-class covariance; and the "main bulk" is attributable to within-cross-class covariance. Formal theoretical results validate our attribution methodology.
We show how the effects of the class/cross-class structure permeate not only the spectra of deepnet features and backpropagated errors, but also the gradients, Fisher Information matrix and Hessian, whether these are considered in the context of an individual layer or the concatenation of them all. The Kronecker or Khatri-Rao product of the class means in the features and the class/cross-class means in the backpropagated errors approximates the class/cross-class means in the gradients. These means of gradients then create C and C^2 outliers in the spectrum of the Fisher Information matrix, which is the second moment of these gradients. The outliers in the Fisher Information matrix spectrum then create outliers in the Hessian spectrum. We explain the significance of this insight by proposing a correction to KFAC, a well known second-order optimization algorithm for training deepnets.

The relationship between the large-scale geometry of a group and its algebraic structure can be studied via three notions: a group's quasi-isometry class, a group's abstract commensurability class, and geometric actions on proper geodesic metric spaces. A common model geometry for groups G and G' is a proper geodesic metric space on which G and G' act geometrically. A group G is action rigid if every group G' that has a common model geometry with G is abstractly commensurable to G. For example, a closed hyperbolic n-manifold group is not action rigid for all n at least three. In contrast, we show that free products of closed hyperbolic manifold groups are action rigid. Consequently, we obtain the first examples of Gromov hyperbolic groups that are quasi-isometric but do not virtually have a common model geometry. This is joint work with Daniel Woodhouse.

In 1985, McDowell introduced a family of parabolically induced Whittaker modules over a complex semisimple Lie algebra, which includes both Verma modules and the nondegenerate Whittaker modules studied by Kostant. Many classical results for Verma modules and the Bernstein--Gelfand--Gelfand category O have been generalized to the category of Whittaker modules introduced by Milicic--Soergel, including the classification of irreducible objects and the Kazhdan--Lusztig conjectures. Contravariant forms on Verma modules are unique up to scaling and play a key role in the definition of the Jantzen filtration. In this talk I will discuss a classification of contravariant forms on parabolically induced Whittaker modules. In a recent result, joint with Anna Romanov, we show that the dimension of the space of contravariant forms on a parabolically induced Whittaker module is given by the cardinality of a Weyl group. This result illustrates a divergence from classical results for Verma modules, and gives insight to two significant open problems in the theory of Whittaker modules: the Jantzen conjecture and the absence of an algebraic definition of duality.

In an increasingly urbanized world, where cities are changing continuously, it is essential for policy makers to have access to regularly updated decision-making tools for an effective management of urban areas. An example of these tools is the delineation of cities into functional areas which provides knowledge on high spatial interaction zones and their socioeconomic composition. In this paper, we presented a method for the structural analysis of a city, specifically for the determination of its functional areas, based on communities detection in graphs. The nodes of the graph correspond to geographical units resulting from a cartographic division of the city according to the road network. The edges are weighted using a Gaussian distance-decay function and the amount of spatial interactions between nodes. Our approach optimize the modularity to ensure that the functional areas detected have strong interactions within their borders but lower interactions outside. Moreover, it leverages on POIs' entropy to maintain a good socioeconomic heterogeneity in the detected areas. We conducted experiments using taxi trips and POIs datasets from the city of Porto, as a study case. Trough those experiments, we demonstrate the ability of our method to portray functional areas while including spatial and socioeconomic dynamics.
 

Here is the scientific program.

Keynote speakers:

Roger Kamm, Cecil and Ida Green Distinguished Professor of Biological and Mechanical Engineering, MIT

Alicia El Haj,  Interdisciplinary Chair of Cell Engineering, Healthcare Technology Institute, University of Birmingham

Invited speakers (confirmed to date):

Davide Ambrosi, Politecnico di Torino, Italy

Anthony Callanan, University of Edinburgh, UK

Ruth Cameron, University of Cambridge, UK

Sonia Contera, University of Oxford, UK

Linda Cummings, New Jersey Institute of Technology, USA

Mohit Dalwadi, University of Oxford, UK

John Dunlop, University of Salzburg, Austria

John King, Nottingham, UK

Nati Korin, Technion, Israel

Catriona Lally, Trinity College Dublin, Ireland

Sandra Loerakker, TU Eindhoven, Netherlands

Ivan Martin, University of Basel, Switzerland

Scott McCue, Queensland University of Technology, Australia

Pierre-Alexis Mouthuy, University of Oxford, UK

Tom Mullin,  University of Oxford, UK

Ramin Nasehi, Politecnico di Milano, Italy

Reuben O'Dea, University of Nottingham, UK

James Oliver, University of Oxford, UK

Ioannis Papantoniou, KU Leuven, Belgium

Ansgar Petersen, Julius Wolf Institute Berlin, Germany

Luigi Preziosi, Politecnico di Torino, Italy

Rebecca Shipley, University College London, UK

Barbara Wagner, Weierstrass Institute for Applied Analysis and Stochastics, Berlin

Cathy Ye, Oxford University, UK

Feihu Zhao, TU Eindhoven, Netherlands

This meeting is being held in celebration of Prof Philip Maini's 60th birthday. Prof Maini has been an internationally leading researcher in mathematical biology for decades. He is currently the Director of the Wolfson Centre for Mathematical Biology, a position he has held since 1998. In the past 20 years he has grown the group significantly. He has established countless interdisciplinary collaborations, has over 400 publications in numerous areas of mathematical biology, with major contributions in mathematical modelling of tumours, wound healing and embryonic pattern formation. He has been elected Fellow of the Royal Society (FRS), Fellow of the Academy of Medical Sciences (FMedSci), and Foreign Fellow of the Indian National Science Academy (FNA). He has served or is serving on editorial board of a large number of journals, and was Editor-in-Chief of the Bulletin of Mathematical Biology [2002-15]. And yet his service to the community cannot be captured just by numbers and titles. Anyone who has met him and worked with him cannot but notice and be touched by his unfailing generosity and the many sacrifices he has made and continues to make day in and day out to help students, early career researchers, and fellow faculty alike.

This meeting provides an opportunity to celebrate Prof Maini's many accomplishments; to thank him for all of his sacrifices; and to bring together the large number of researchers – mathematicians, biologists, physiologists, and clinicians – that he has worked with and interacted with over the years. More broadly, the meeting provides a unique opportunity to reflect on mathematical biology, to provide perspectives on the trajectory of a field that was scarcely recognised and had very few dedicated researchers in the days of Prof Maini's own DPhil; yet a field that has grown tremendously since then. Much of this growth is attributable to the work of Prof Maini, so that today the value of mathematics in biology is increasingly recognized by biologists and clinicians, and with theoretical predictions of mathematical models having cemented a role in advancing biological understanding. 

Speakers

David Sumpter, Uppsala University (Public lecture), Derek Moulton, University of Oxford, Hans Othmer, Minnesota University, Jen Flegg, University of Melbourne, Jim Murray, University of Washington, Jonathan Sherratt, Heriot-Watt University, Kevin Painter, Heriot-Watt University, Linus Schumacher, University of Edinburgh, Lucy Hutchinson, Roche, Mark Chaplain, University of St Andrews, Mark Lewis, University of Alberta, Mary Myerscough, University of Sydney, Natasha Martin, University of Bristol, Noemi Picco, Swansea University, Paul Kulesa, Stowers Institute for Medical Research, Ruth Baker, University of Oxford, Santiago Schnell, University of Michigan, Tim Pedley, University of Cambridge

Organising committee

Ruth Baker (University of Oxford)

Derek Moulton (University of Oxford)

Helen Byrne (University of Oxford)

Santiago Schnell (University of Michigan)

Mark Chaplain (University of St Andrews)

The analysis and characterization of human mobility using population-level mobility models is important for numerous applications, ranging from the estimation of commuter flows to modeling trade flows. However, almost all of these applications have focused on large spatial scales, typically from intra-city level to inter-country level. In this paper, we investigate population-level human mobility models on a much smaller spatial scale by using them to estimate customer mobility flow between supermarket zones. We use anonymized mobility data of customers in supermarkets to calibrate our models and apply variants of the gravity and intervening-opportunities models to fit this mobility flow and estimate the flow on unseen data. We find that a doubly-constrained gravity model can successfully estimate 65-70% of the flow inside supermarkets. We then investigate how to reduce congestion in supermarkets by combining mobility models with queueing networks. We use a simulated-annealing algorithm to find store layouts with lower congestion than the original layout. Our research gives insight both into how customers move in supermarkets and into how retailers can arrange stores to reduce congestion. It also provides a case study of human mobility on small spatial scales.

Please register at:

https://sites.google.com/view/geometryandstrings2019/home

 It has been long known that intersection theory on the moduli space of punctured Riemann surfaces encodes observables in two-dimensional quantum gravity. It is natural to ask whether interacting theories could also admit a similar description. In the genus-zero case we put forward a twisted version of intersection theory on the moduli space and propose that it computes tree-level scattering amplitudes in a range of quantum field theories with a finite spectrum of particles. The resulting intersection numbers exhibit two alternative kinds of localization formulae. The first one receives contributions only from boundaries of the moduli space, thus leading to a Feynman diagram expansion, while the second one localizes on critical points of a certain Morse function.

We investigate a nonlinear version of coevolving voter models, in which both node states and network structure update as a coupled stochastic dynamical process. Most prior work on coevolving voter models has focused on linear update rules with fixed rewiring and adopting probabilities. By contrast, in our nonlinear version, the probability that a node rewires or adopts is a function of how well it "fits in" within its neighborhood. To explore this idea, we incorporate a parameter σ that represents the fraction of neighbors of an updating node that share its opinion state. In an update, with probability σq (for some nonlinearity parameter q), the updating node rewires; with complementary probability 1−σq, the updating node adopts a new opinion state. We study this mechanism using three rewiring schemes: after an updating node deletes a discordant edge, it then either (1) "rewires-to-random" by choosing a new neighbor in a random process; (2) "rewires-to-same" by choosing a new neighbor in a random process from nodes that share its state; or (3) "rewires-to-none" by not rewiring at all (akin to "unfriending" on social media). We compare our nonlinear coevolving model to several existing linear models, and we find in our model that initial network topology can play a larger role in the dynamics, whereas the choice of rewiring mechanism plays a smaller role. A particularly interesting feature of our model is that, under certain conditions, the opinion state that is initially held by a minority of nodes can effectively spread to almost every node in a network if the minority nodes views themselves as the majority. In light of this observation, we relate our results to recent work on the majority illusion in social networks.

Reference: 

Kureh, Yacoub H., and Mason A. Porter. "Fitting In and Breaking Up: A Nonlinear Version of Coevolving Voter Models." arXiv preprint arXiv:1907.11608 (2019).

In the speaker's previous joint work with Hans-Joachim Hein, a mass formula for asymptotically locally Euclidean (ALE) Kaehler manifolds was proved, assuming only relatively weak fall-off conditions on the metric. However, the case of real dimension four presented technical difficulties that led us to require fall-off conditions in this special dimension that are stronger than the Chrusciel fall-off conditions that sufficed in higher dimensions. This talk will explain how a new proof of the 4-dimensional case, using ideas from symplectic geometry, shows that Chrusciel fall-off suffices to imply all our main results in any dimension. In particular, I will explain why our Penrose-type inequality for the mass of an asymptotically Euclidean Kaehler manifold always still holds, given only this very weak metric fall-off hypothesis.
 

The vulnerability of a host population to a specific disease measures how likely pathogen introduction will lead to an epidemic outbreak, and how hard it is to contain or eliminate an ongoing one. Predicting vulnerability is thus key to designing risk-reduction strategies that limit disease burden on public health and economic development. To do that, highly-resolved data tracking contacts and mobility of the host population need to integrate into detailed models of disease dynamics. This represents a twofold challenge. Firstly, we need theoretical frameworks that turn data feeds into predictors of epidemic risk, and can identify which of the structural features of the host population drive its vulnerability. Secondly, we need new ways to access, analyze, and share the relevant contact and mobility data: a necessary step to make our predictions realistic and reliable. In my talk, I will address both issues. I will show how to analytically derive the conditions that discriminate between epidemic regime and quick pathogen extinction, by representing empirically measured contacts as time-evolving complex networks. The analytical core of this theory leads to a broad range of applications. At the same time, its data-driven nature prompts context-specific predictions that can inform policymaking, as I will show in two case studies: reorganizing nurse scheduling to reduce the risk of spread of healthcare-associated infections; linking the features of livestock trade movements to the spatial spread of cattle diseases. The latter application is also an example of how limited access and incomplete data collection represent a big hurdle to predictive vulnerability analysis. To overcome this, I will present a collaborative platform for analyzing and comparing trade networks coming from several European countries. Using a bring code to the data approach, our platform surmounts the strict regulations preventing data sharing, and builds an algorithm that predicts vulnerability even in situations when limited data on cattle trade are available. The ultimate goal of all these theoretical and numerical developments is to inform strategies that reduce the vulnerability of the host population by restructuring its contacts. However, such restructuring may entail a feedback effect, acting as selective pressure on the pathogen itself. In the last part of my talk, I will extend the developed formalism to modeling evolutionary pathways that maximize the invasion potential of the pathogen, given the observed host population structure. Specifically, I will link the emergence of exotic replication behaviors in plant-infecting viruses to historical changes in plant distribution patterns.

We study the stability properties of the Eulerian mean flow generated by monochromatic surface-gravity waves propagating in a rotating frame. The wave averaged equations, also known as the Craik-Leibovich equations, govern the evolution of the mean flow. For propagating waves in a rotating frame these equations admit a steady depth-dependent base flow sometimes called the Ekman-Stokes spiral, because of its resemblance to the standard Ekman spiral. This base flow profile is controlled by two non-dimensional numbers, the Ekman number Ek and the Rossby number Ro. We show that this steady laminar velocity profile is linearly unstable above a critical Rossby number Roc(Ek). We determine the threshold Rossby number as a function of Ek using a numerical eigenvalue solver, before confirming the numerical results through asymptotic expansions in the large/low Ek limit. These were also confirmed by nonlinear simulations of the Craik-Leibovich equations. When the system is well above the linear instability threshold, Ro >> Roc, the resulting flow fluctuates chaotically. We will discuss the possible implications in an oceanographic context, as well as for laboratory experiments.

In recent years, much attention has been given to single-cell RNA sequencing techniques as they allow researchers to examine the functions and relationships of single cells inside a tissue. In this study, we combine single-cell RNA sequencing data with protein–protein interaction networks (PPINs) to detect active modules in cells of different transcriptional states. We achieve this by clustering single-cell RNA sequencing data, constructing node-weighted PPINs, and identifying the maximum-weight connected subgraphs with an exact Steiner-Tree approach. As a case study, we investigate RNA sequencing data from human liver spheroids but the techniques described here are applicable to other organisms and tissues. The benefits of our novel method are two-fold: First, it allows us to identify important proteins (e.g., receptors) which are not detected from a differential gene-expression analysis as they only interact with proteins that are transcribed in higher levels. Second, we find that different transcriptional states have different subnetworks of the PPIN significantly overexpressed. These subnetworks often reflect known biological pathways (e.g., lipid metabolism and stress response) and we obtain a nuanced picture of cellular function as we can associate them with a subset of all analysed cells.

We will talk about our recent results on the uniqueness of regular reflection solutions for the potential flow equation in a natural class of self-similar solutions. The approach is based on a nonlinear version of method of continuity. An important property of solutions for the proof of uniqueness is the convexity of the free boundary.

The stabilization of thermo-mechanical systems is a classical problem in thermodynamics and well

understood in a context of gases. The objective of this talk is to indicate the role of null-Lagrangians and

certain transport/stretching identities in stabilizing thermomechanical systems associated with general

thermoelastic free energies. This allows to prove various convergence results among thermomechan-

ical theories, and suggests a variational scheme for the approximation of the equations of adiabatic

thermoelasticity.

In this talk, I will talk about the existence of global weak solutions for the compressible Navier-Stokes equations, in particular, the viscosity coefficients depend on the density. Our main contribution is to further develop renormalized techniques so that the Mellet-Vasseur type inequality is not necessary for the compactness.  This provides existence of global solutions in time, for the barotropic compressible Navier-Stokes equations, for any $\gamma>1$, in three dimensional space, with large initial data, possibly vanishing on the vacuum. This is a joint work with D. Bresch, A. Vasseur.

Automatic structures are algebraic structures, such as graphs, groups
and partial orders, that can be presented by automata. By varying the 
classes of automata (e.g. finite automata, tree automata, omega-automata) 
one varies the classes of automatic structures. The class of all automatic 
structures is robust in the sense that it is closed under many natural
algebraic and model-theoretic operations.  
In this talk, we give formal definitions to 
automatic structures, motivate the study, present many examples, and
explain several fundamental theorems.  Some results in the area
are deeply connected  with algebra, additive combinatorics, set theory, 
and complexity theory. 
We then motivate and pose several important  unresolved questions in the
area.

It is unnecessary to emphasize important place of algorithms in computer science. Many efficient and convenient algorithms are designed by borrowing or revising ancient mathematical algorithms and methods. For example, recursive method, exhaustive search method, greedy method, “divide and conquer” method, dynamic programming method, reiteration algorithm, circulation algorithm, among others.

From the perspective of the history of computer science, it is necessary to study the history of algorithms used in the computer computations. The history of algorithms for computer science is naturally regarded as a sub-object of history of mathematics. But historians of mathematics, at least those who study history of mathematics in China, have not realized it is important in the history of mathematics. Historians of Chinese mathematics paid little attention to these studies, mainly having not considered from this research angle. Relevant research is therefore insufficient in the field of history of mathematics.

The mechanization thought and algorithmization characteristic of Chinese traditional (and therefore, East Asian) mathematics, however, are coincident with that of computer science. Traditional Chinese algorithms, therefore, show their importance historical significance in computer science. It is necessary and important to survey traditional algorithms again from the point of views of computer science. It is also another angle for understanding traditional Chinese mathematics.

There are many things in the field that need to be researched. For example, when and how were these algorithms designed? What was their mathematical background? How were they applied in ancient mathematical context? How are their complexity and efficiency of ancient algorithms?

In the present paper, we will study the circulation structure in traditional Chinese mathematical algorithms. Circulation structures have great importance in the computer science. Most algorithms are designed by means of one or more circulation structures. Ancient Chinese mathematicians were familiar them with the circulation structures and good at their applications. They designed a lot of circulation structures to obtain their desirable results in mathematical computations. Their circulation structures of dozen ancient algorithms will be analyzed. They are selected from mathematical and astronomical treatises, and also one from the Yijing (Book of Changes), the oldest of the Chinese classics.

In this talk I will describe how to make sense of the function $(1+t)^x$ over the integers. I will explain how different rings of analytic functions can be defined over the integers, and how this leads to global analytic geometry and global Hodge theory. If time permits I will also describe an analytic version of lambda-rings and how this can be used to define a cohomology theory for schemes over Z. This is joint work with Federico Bambozzi and Adam Topaz. 

 For a semisimple modular tensor category the Reshetikhin-Turaev construction yields an extended three-dimensional topological field theory and hence by restriction a modular functor. By work of Lyubachenko-Majid the construction of a modular functor from a modular tensor category remains possible in the non-semisimple case. We explain that the latter construction is the shadow of a derived modular functor featuring homotopy coherent mapping class group actions on chain complex valued conformal blocks and a version of factorization and self-sewing via homotopy coends. On the torus we find a derived version of the Verlinde algebra, an algebra over the little disk operad (or more generally a little bundles algebra in the case of equivariant field theories). The concepts will be illustrated for modules over the Drinfeld double of a finite group in finite characteristic. This is joint work with Christoph Schweigert (Hamburg).

The notion of a higher Segal object was introduces by Dyckerhoff and Kapranov as a general framework for studying (higher) associativity inherent
in a wide range of mathematical objects. Most of the examples are related to Hall algebra type constructions, which include quantum groups. We describe a construction that assigns to a simplicial object S a datum H(S)  which is naturally interpreted as a "d-lax A-infinity algebra” precisely when S is a (d+1)-Segal object. This extends the extensively studied d=2 case.

Aden Forrow
Optimal transport and cell differentiation

Abstract
Optimal transport is a rich theory for comparing distributions, with both deep mathematics and application ranging from 18th century fortification planning to computer graphics. I will tie its mathematical story to a biological one, on the differentiation of cells from pluripotency to specialized functional types. First the mathematics can support the biology: optimal transport is an apt tool for linking experimental samples across a developmental time course. Then the biology can inspire new mathematics: based on the branching structure expected in differentiation pathways, we can find a regularization method that dramatically improves the statistical performance of optimal transport.

Paul Ziegler
Geometry and Arithmetic

Abstract
For a family of polynomials in several variables with integral coefficients, the Weil conjectures give a surprising relationship between the geometry of the complex-valued roots of these polynomials and the number of roots of these polynomials "modulo p". I will give an introduction to this circle of results and try to explain how they are used in modern research.
 

We can see the simplest setting of persistence from a functional point of view: given a fixed finite simplicial complex, we have the barcode function which, given a filter function over this complex, returns the corresponding persistent diagram. The bottleneck distance induces a topology on the space of persistence diagrams, and makes the barcode function a continuous map: this is a consequence of the stability Theorem. In this presentation, I will present ongoing work that seeks to deepen our understanding of the analytic properties of the barcode function, in particular whether it can be said to be smooth. Namely, if we smoothly vary the filter function, do we get smooth changes in the resulting persistent diagram? I will introduce a notion of differentiability/smoothness for barcode valued maps, and then explain why the barcode function is smooth (but not everywhere) with respect to the choice of filter function. I will finally explain why these notions are of interest in practical optimisation/learning situations. 

The amount and complexity of biological data has increased rapidly in recent years with the availability of improved biological tools. When applying persistent homology to large data sets, many of the currently available algorithms however fail due to computational complexity preventing many interesting biological applications. De Silva and Carlsson (2004) introduced the so called Witness Complex that reduces computational complexity by building simplicial complexes on a small subset of landmark points selected from the original data set. The landmark points are chosen from the data either at random or using the so called maxmin algorithm. These approaches are not ideal as the random selection tends to favour dense areas of the point cloud while the maxmin algorithm often selects outliers as landmarks. Both of these problems need to be addressed in order to make the method more applicable to biological data. We study new ways of selecting landmarks from a large data set that are robust to outliers. We further examine the effects of the different subselection methods on the persistent homology of the data.

Mitral regurgitation is one of the most common valve diseases in the UK and contributes to 50% of the transcatheter mitral valve replacement (TMVR) procedures with bioprosthetic valves. TMVR is generally performed in frailer, older patients unlikely to tolerate open-heart surgery or further interventions. One of the side effects of implanting a bioprosthetic valve is a condition known as left ventricular outflow obstruction, whereby the implanted device can partially obstruct the outflow of blood from the left ventricle causing high flow resistance. The ventricle has then to pump more vigorously to provide adequate blood supply to the circulatory system and becomes hypertrophic. This ultimately results in poor contractility and heart failure.
We developed personalised image-based models to characterise the complex relationship between anatomy, blood flow, and ventricular function both before and after TMVR. The model prediction provides key information to match individual patient and device size, such as postoperative changes in intraventricular pressure gradients and blood residence time. Our pilot data from a cohort of 7 TMVR patients identified a correlation between the degree of outflow obstruction and the deterioration of ventricular function: when approximately one third of the outflow was obstructed as a result of the device implantation, significant increases in the flow resistance and the average time spent by the blood inside the ventricle were observed, which are in turn associated with hypertrophic ventricular remodelling and blood stagnation, respectively. Currently, preprocedural planning for TMVR relies largely on anecdotal experience and standard anatomical evaluations. The haemodynamic knowledge derived from the models has the potential to enhance significantly pre procedural planning and, in the long term, help develop a personalised risk scoring system specifically designed for TMVR patients.
 

This work aims at developing new approach for parameterizing mesoscale eddy effects for use in non-eddy-resolving ocean circulation models. These effects are often modelled as some diffusion process or a stochastic forcing, and the proposed approach is implicitly related to the latter category. The idea is to approximate transient eddy flux divergence in a simple way, to find its actual dynamical footprints by solving a simplified but dynamically relevant problem, and to relate the ensemble of footprints to the large-scale flow properties.

We report on a line of work initiated by Dan-Cohen and Wewers and continued by Dan-Cohen and the speaker to explicitly compute the zero loci arising in Kim's non-abelian Chabauty's method. We explain how this works, an important step of which is to compute bases of a certain motivic Hopf algebra in low degrees. We will summarize recent work by Dan-Cohen and the speaker, extending previous computations to $\mathbb{Z}[1/3]$ and proposing a general algorithm for solving the unit equation. Many of the methods in the more recent work are inspired by recent ideas of Francis Brown. Finally, we indicate future work, in which we hope to use elliptic motivic periods to explicitly compute points on punctured elliptic curves and beyond.

Arakelov geometry studies schemes X over ℤ, together with the Hermitian complex geometry of X(ℂ).
Most notably, it has been used to give a proof of Mordell's conjecture (Faltings's Theorem) by Paul Vojta; curves of genus greater than 1 have at most finitely many rational points.
In this talk, we'll introduce some of the ideas behind Arakelov theory, and show how many results in Araklev theory are analogous—with additional structure—to classic results such as intersection theory and Riemann Roch.

Our aim is to construct high order approximation schemes for general 
semigroups of linear operators $P_{t},t \ge 0$. In order to do it, we fix a time 
horizon $T$ and the discretization steps $h_{l}=\frac{T}{n^{l}},l\in N$ and we suppose
that we have at hand some short time approximation operators $Q_{l}$ such
that $P_{h_{l}}=Q_{l}+O(h_{l}^{1+\alpha })$ for some $\alpha >0$. Then, we
consider random time grids $\Pi (\omega )=\{t_0(\omega )=0<t_{1}(\omega 
)<...<t_{m}(\omega )=T\}$ such that for all $1\le k\le m$, $t_{k}(\omega 
)-t_{k-1}(\omega )=h_{l_{k}}$ for some $l_{k}\in N$, and we associate the approximation discrete 
semigroup $P_{T}^{\Pi (\omega )}=Q_{l_{n}}...Q_{l_{1}}.$ Our main result is the 
following: for any approximation order $\nu $, we can construct random grids $\Pi_{i}(\omega )$ and coefficients 
$c_{i}$, with $i=1,...,r$ such that $P_{t}f=\sum_{i=1}^{r}c_{i} E(P_{t}^{\Pi _{i}(\omega )}f(x))+O(n^{-\nu})$
with the expectation concerning the random grids $\Pi _{i}(\omega ).$ 
Besides, $Card(\Pi _{i}(\omega ))=O(n)$ and the complexity of the algorithm is of order $n$, for any order
of approximation $\nu$. The standard example concerns diffusion 
processes, using the Euler approximation for $Q_l$.
In this particular case and under suitable conditions, we are able to gather the terms in order to produce an estimator of $P_tf$ with 
finite variance.
However, an important feature of our approach is its universality in the sense that
it works for every general semigroup $P_{t}$ and approximations.  Besides, approximation schemes sharing the same $\alpha$ lead to
the same random grids $\Pi_{i}$ and coefficients $c_{i}$. Numerical illustrations are given for ordinary differential equations, piecewise 
deterministic Markov processes and diffusions.

When a liquid drop is deposited over a solid surface whose temperature is sufficiently above the boiling point of the liquid, the drop does not experience nucleate boiling but rather levitates over a thin layer of its own vapor. This is known as the Leidenfrost effect. Whilst highly undesirable in certain cooling applications, because of a drastic decrease of the energy transferred between the solid and the evaporating liquid due to poor heat conductivity of the vapor, this effect can be of great interest in many other processes profiting from this absence of contact with the surface that considerably reduces the friction and confers an extreme mobility on the drop. During this presentation, I hope to provide a good vision of some of the knowledge on this subject through some recent studies that we have done. First, I will present a simple fitting-parameter-free theory of the Leidenfrost effect, successfully validated with experiments, covering the full range of stable shapes, i.e., from small quasi-spherical droplets to larger puddles floating on a pocketlike vapor film. Then, I will discuss the end of life of these drops that appear either to explode or to take-off. Finally, I will show that the Leidenfrost effect can also be observed over hot baths of non-volatile liquids. The understanding of the latter situation, compare to the classical Leidenfrost effect on solid substrate, provides new insights on the phenomenon, whether it concerns levitation or its threshold.

Abstract:  As is well known, every rational representation of a finite group $G$ can be realized over $\mathbb{Z}$, that is, the corresponding $\mathbb{Q}G$-module $V$ admits a $\mathbb{Z}$-form. Although $\mathbb{Z}$-forms are usually far from being unique, the famous Jordan--Zassenhaus Theorem shows that there are only finitely many $\mathbb{Z}$-forms of any given $\mathbb{Q}G$-module, up to isomorphism. Determining the precise number of these isomorphism classes or even explicit representatives is, however, a hard task in general. In this talk we shall be concerned with the case where $G$ is the symmetric group $\mathfrak{S}_n$ and $V$ is a simple $\mathbb{Q}\mathfrak{S}_n$-module labelled by a hook partition. Building on work of Plesken and Craig we shall present some results as well as open problems concerning the construction of the
integral forms of these modules. This is joint work with Tommy Hofmann from Kaiserslautern.

Partial differential equations (PDEs) are among the most universal tools used in modelling problems in nature and man-made complex systems. For example, stochastic PDEs are a fundamental ingredient in models for nonlinear filtering problems in chemical engineering and weather forecasting, deterministic Schroedinger PDEs describe the wave function in a quantum physical system, deterministic Hamiltonian-Jacobi-Bellman PDEs are employed in operations research to describe optimal control problems where companys aim to minimise their costs, and deterministic Black-Scholes-type PDEs are highly employed in portfolio optimization models as well as in state-of-the-art pricing and hedging models for financial derivatives. The PDEs appearing in such models are often high-dimensional as the number of dimensions, roughly speaking, corresponds to the number of all involved interacting substances, particles, resources, agents, or assets in the model. For instance, in the case of the above mentioned financial engineering models the dimensionality of the PDE often corresponds to the number of financial assets in the involved hedging portfolio. Such PDEs can typically not be solved explicitly and it is one of the most challenging tasks in applied mathematics to develop approximation algorithms which are able to approximatively compute solutions of high-dimensional PDEs. Nearly all approximation algorithms for PDEs in the literature suffer from the so-called "curse of dimensionality" in the sense that the number of required computational operations of the approximation algorithm to achieve a given approximation accuracy grows exponentially in the dimension of the considered PDE. With such algorithms it is impossible to approximatively compute solutions of high-dimensional PDEs even when the fastest currently available computers are used. In the case of linear parabolic PDEs and approximations at a fixed space-time point, the curse of dimensionality can be overcome by means of Monte Carlo approximation algorithms and the Feynman-Kac formula. In this talk we introduce new nonlinear Monte Carlo algorithms for high-dimensional nonlinear PDEs. We prove that such algorithms do indeed overcome the curse of dimensionality in the case of a general class of semilinear parabolic PDEs and we thereby prove, for the first time, that a general semilinear parabolic PDE with a nonlinearity depending on the PDE solution can be solved approximatively without the curse of dimensionality.