Past

During development, cells take on specific fates to properly build tissues and organs. These cell fates are regulated by short and long range signalling mechanisms, as well as feedback on gene expression and protein activity. Despite the high conservation of these signalling pathways, we often see different cell fate outcomes in similar tissues or related species in response to similar perturbations. How these short and long range signals work to control patterning during development, and how the same network can lead to species specific responses to perturbations, is not yet understood. Exploiting the high conservation of developmental pathways, we theoretically and experimentally explore mechanisms of cell fate patterning during development of the egg laying structure (vulva) in nematode worms. We developed differential equation models of the main signalling networks (EGF/Ras, Notch and Wnt) responsible for vulval cell fate specification, and validated them using experimental data. A complex, biologically based model identified key network components for wild type patterning, and relationships that render the network more sensitive to perturbations. Analysis of a simplified model indicated that short and long range signalling play complementary roles in developmental patterning. The rich data sets produced by these models form the basis for further analysis and increase our understanding of cell fate regulation in development.

We construct a functor from arithmetic schemes (and dominant morphisms) to dynamical systems which allows to recover the Hasse-Weil zeta function of a scheme as a Ruelle type zeta function of the corresponding dynamical system. We state some further properties of this correspondence and explain the relation to the work of Kucharczyk and Scholze who realize the Galois groups of fields containing all roots of unity as (etale) fundamental groups of certain topological spaces. We also explain the main reason why our dynamical systems are not yet the right ones and in what regard they need to be refined.
 

Heterogeneity in Space and Time: Novel Dispersion Relations in Morphogenesis

Dr. Andrew Krause

Motivated by recent work with biologists, I will showcase some results on Turing instabilities in complex domains. This is scientifically related to understanding developmental tuning in the whiskers of mice, and in synthetic quorum-sensing patterning of bacteria. Such phenomena are typically modelled using reaction-diffusion systems of morphogens, and one is often interested in emergent spatial and spatiotemporal patterns resulting from instabilities of a homogeneous equilibrium. In comparison to the well-known effects of how advection or manifold structure impacts the modes which may become unstable in such systems, I will present results on instabilities in heterogeneous systems, reaction-diffusion systems on evolving manifolds, as well as layered reaction-diffusion systems. These contexts require novel formulations of classical dispersion relations, and may have applications beyond developmental biology, such as in understanding niche formation for populations of animals in heterogeneous environments. These approaches also help close the vast gap between the simplistic theory of instability-driven pattern formation, and the messy reality of biological development, though there is still much work to be done in concretely demonstrating such a theory's applicability in real biological systems.
 

Cavity flow characteristics and applications to kidney stone removal

Dr. Jessica Williams

Ureteroscopy is a minimally invasive surgical procedure for the removal of kidney stones. A ureteroscope, containing a hollow, cylindrical working channel, is inserted into the patient's kidney. The renal space proximal to the scope tip is irrigated, to clear stone particles and debris, with a saline solution that flows in through the working channel. We consider the fluid dynamics of irrigation fluid within the renal pelvis, resulting from the emerging jet through the working channel and return flow through an access sheath . Representing the renal pelvis as a two-dimensional rectangular cavity, we investigate the effects of flow rate and cavity size on flow structure and subsequent clearance time of debris. Fluid flow is modelled with the steady incompressible Navier-Stokes equations, with an imposed Poiseuille profile at the inlet boundary to model the jet of saline, and zero-stress conditions on the outlets. The resulting flow patterns in the cavity contain multiple vortical structures. We demonstrate the existence of multiple solutions dependent on the Reynolds number of the flow and the aspect ratio of the cavity using complementary numerical simulations and PIV experiments. The clearance of an initial debris cloud is simulated via solutions to an advection-diffusion equation and we characterise the effects of the initial position of the debris cloud within the vortical flow and the Péclet number on clearance time. With only weak diffusion, debris that initiates within closed streamlines can become trapped. We discuss a flow manipulation strategy to extract debris from vortices and decrease washout time.

 

In this talk we will discuss a data-driven approach based on neural networks (NN) for calibrating financial asset price models. Determining optimal values of the model parameters is formulated as training hidden neurons within a machine learning framework, based on available financial option prices. The framework consists of two parts: a forward pass in which we train the weights of the NN off-line, valuing options under many different asset model parameter settings; and a backward pass, in which we evaluate the trained NN-solver on-line, aiming to find the weights of the neurons in the input layer. We will show how the same data-driven approach can be used to estimate the Black-Scholes implied volatility and dividend yield for American options in a fast and robust way. We then discuss the complexity of the optimization problem through an analysis of the loss surface of the neural network. We finally will present some numerical examples which show that neural networks can be an efficient and reliable technique for the calibration of financial assets and the extraction of implied information.

Here, a connection is a algebraic structure that is weaker than an algebra and stronger than a module. I will define this structure and give examples. I will then define the quantum product and explain how connections capture important properties of this product. I will finish by stating a new result which describes how this extends to equivariant Floer cohomology. No knowledge of symplectic topology will be assumed in this talk.
 

If asked, we all say we aim to to good research and make sensible decisions. In mathematics, the choice of criteria to optimise is often explicit, and we know there is no complete ordering in more than one dimension.

Statisticians involved in multi-disciplinary research need to reflect on how their understanding of uncertainty and statistical methods can contribute to reliable and reproducible research. The ISI Declaration of Professional Ethics provides a framework for statisticians.  Judging what is "normal" and what is "best" requires an appreciation of the assumptions and guidelines of other disciplines.

I will briefly discuss the requirements for design and analysis in medical research, and relate this to debates on reproducible research and p-values in social science research. Issues arising from informed and uninformed consent will be outlined.

Examples might include medical research in developing countries, toxic tort or wrongful birth claims, big data and use of routine administrative or commercial data.

In this talk, we discuss Sobolev embeddings into rearrangement-invariant function spaces on (regular) domains in $\mathbb{R}^n$ endowed with measures whose decay on balls is dominated by a power $d$ of their radius, called $d$-Frostman measures. We show that these embeddings can be deduced from one-dimensional inequalities for an operator depending on $n$, $d$ and the order $m$ of the Sobolev space. We also point out an interesting feature of this theory - namely that the results take a substantially different form depending on whether the measure is decaying fast ($d\geq n-m$) or slowly ($d<n-m$). This is a
joint work with Andrea Cianchi and Lubos Pick.

 This is a longstanding problem. We will present candidate axioms that we believe form a complete axiom system satisfied by $F_p^{alg}((t))$.  A currently "unfinished" proof will be discussed.   This work is joint with Françoise Delon (Paris) and Arno Fehm (Dresden).

 Let phi be an outer automorphism of a free group. A topological representative of phi is a marked graph G along with a homotopy equivalence f: G → G which induces the outer automorphism phi on the fundamental group of G. For any given outer automorphism, the choice of topological representative is far from unique. Handel and Bestvina showed that sufficiently nice automorphisms admit a special type of topological representative called a train track map, whose dynamics can be well understood. 
In this talk I will outline the definition and motivation for train tracks, and give a sketch of Handel and Bestvina’s algorithm for finding them.
 

Hawkes processes are a class of point processes used to model self-excitation and cross-excitation between different types of events. They are characterized by the auto-regressive structure of their conditional intensity, and there exists several extensions to the original linear Hawkes model. In this talk, we start by defining Hawkes processes and give a brief overview of some of their basic properties. We then review some approaches to parametric and non-parametric estimation of Hawkes processes and discuss some applications to problems with large data sets in high frequency finance and social networks.

There is a classical mathematical theorem (due to Banach and Tarski) that implies the following shocking statement: An orange can be divided into finitely many pieces, these pieces can be rotated and rearranged in such a way to yield two oranges of the same size as the original one. In 1929 J.~von Neumann recognizes that one of the reasons underlying the Banach-Tarski paradox is the fact that on the unit ball there is an action of a discrete subgroup of isometries that fails to have the property of amenability ("Mittelbarkeit" in German).

In this talk we will address more recent developments in relation to the dichotomy amenability vs. existence of paradoxical decompositions in different mathematical situations like, e.g., for metric spaces, for algebras and operator algebras. We will present a result unifying all these approaches in terms of a class of C*-algebras, the so-called uniform Roe algebras.

P. Ara, K. Li, F. Lledó and J. Wu, Amenability of coarse spaces and K-algebras , Bulletin of Mathematical Sciences 8 (2018) 257-306;
P. Ara, K. Li, F. Lledó and J. Wu, Amenability and uniform Roe algebras, Journal of Mathematical Analysis and Applications 459 (2018) 686-716;

The ordinary braid group ${\mathrm Br}_n$ is a well-known algebraic structure which encodes configurations of $n$ non-touching strands (“braids”) up to continious transformations (“isotopies”). A classical result of Khovanov and Thomas states that there is a natural categorical action of ${\mathrm Br}_n$ on the derived category of the cotangent bundle of the variety of complete flags in ${\mathbb C}^n$. 

In this talk, I will introduce a new structure: the category ${\mathrm GBr}_n$ of generalised braids. These are the braids whose strands are allowed to touch in a certain way. They have multiple endpoint configurations and can be non-invertible, thus forming a category rather than a group. In the context of triangulated categories, it is natural to impose certain relations which result in the notion of a skein-triangulated representation of ${\mathrm GBr}_n$. A decade-old conjecture states that there is a skein-triangulated action of ${\mathrm GBr}_n$ on the cotangent bundles of the varieties of full and partial flags in ${\mathbb C}^n$. We prove this conjecture for $n = 3$. We also show that, in fact, any categorical action of ${\mathrm Br}_n$ can be lifted to a categorical action of ${\mathrm GBr}_n$, generalising a result of Ed Segal. This is a joint work with Rina Anno and Lorenzo De Biase.

The standard rounding procedure in floating-point computations is round to nearest (RN). In this talk we consider an alternative rounding strategy called stochastic rounding (SR) which has the appealing property of being exact (actually exact!) in expectation. In the first part of the talk we discuss recent developments in probabilistic rounding error analysis and we show how rounding errors grow at an O(\sqrt{n}) rate rather than O(n) when SR is employed. This shows that Wilkinson's rule of thumb provably holds for this type of rounding. In the second part of the talk we consider the application of SR to parabolic PDEs admitting a steady state solution. We show that when the heat equation is solved in half precision RN fails to compute an accurate solution, while SR successfully solves the problem to decent accuracy.
 

I will explain the basics of 2-representation theory and will explain an approach to classifying 'simple' 2-representations of the Hecke 2-category (aka Soergel bimodules) for finite Coxeter types.

Consider all planar graphs on n vertices. What is the smallest graph that contains them all as induced subgraphs? We provide an explicit construction of such a graph on $n^{4/3+o(1)}$ vertices, which improves upon the previous best upper bound of $n^{2+o(1)}$, obtained in 2007 by Gavoille and Labourel.

In this talk, we will gently introduce the audience to the notion of so-called universal graphs (graphs containing all graphs of a given family as induced subgraphs), and devote some time to a key lemma in the proof. That lemma comes from a recent breakthrough by Dujmovic et al. regarding the structure of planar graphs, and has already many interesting consequences - we hope the audience will be able to derive more. This is based on joint work with Cyril Gavoille and Michal Pilipczuk.

We consider the problem of dynamically pricing multiple products on a network of resources, such as that faced by an airline selling tickets on its route network. For computational reasons this inherently stochastic problem is often approximated deterministically, however even the deterministic dynamic pricing problem can be impractical to solve. For this reason we have derived a novel iterative Quadratic Programming approximation to the deterministic dynamic pricing problem that is not only very fast to solve in practice but also exhibits a provably linear rate of convergence. This is joint work with Saksham Jain and Raphael Hauser.
 

Spatial networks are often the most natural way to represent spatial information of different kinds. One of the outstanding problems in current spatial network research is to effectively quantify the heterogeneity of the discrete-valued spatial distributions underlying a spatial graph. In this talk we will presentsome recent alternative approaches to estimate heterogeneity in spatial networks based on simple dynamical processes running on them.

The parabolic Allen-Cahn equations is the gradient flow of phase transition energy and can be viewed as a diffused version of mean curvature flows of hypersurfaces. It has been known by the works of Ilmanen and Tonegawa that the energy densities of the Allen-Cahn flows converges to mean curvature flows in the sense of varifold and the limit varifold is integer rectifiable. It is not known in general whether the transition layers have higher regularity of convergence yet. In this talk, I will report on a joint work with Huy Nguyen that under the low entropy condition, the convergence of transition layers can be upgraded to C^{2,\alpha} sense. This is motivated by the work of Wang-Wei and Chodosh-Mantoulidis in elliptic case that under the condition of stability, one can upgrade the regularity of convergence.

We will discuss some problems around independence of l in compatible systems of Galois representations, mostly focusing on the independence of l of algebraic monodromy groups. We will explain how these problems fit into the context of the Langlands program, and present results both in characteristic zero and in positive characteristic settings.

We will first construct pairs of homotopic 2-spheres smoothly embedded in a 4-manifold that are smoothly equivalent (via an ambient diffeomorphism preserving homology) but not even topologically isotopic. Indeed, these examples show that Gabai's recent "4D Lightbulb Theorem" does not hold without the 2-torsion hypothesis. We will proceed to discuss two distinct ways of obstructing such an isotopy, as well as related invariants which can be used to obstruct an isotopy between pairs of properly embedded disks (rather than spheres) in a 4-manifold.

The deep neural network has achieved impressive results in various applications, and is involved in more and more branches of science. However, there are still few theories supporting its empirical success. In particular, we miss the mathematical tool to explain the advantage of certain structures of the network, and to have quantitive error bounds. In our recent work, we used a regularised relaxed control problem to model the deep neural network.  We managed to characterise its optimal control by the invariant measure of a mean-field Langevin system, which can be approximated by the marginal laws. Through this study we understand the importance of the pooling for the deep nets, and are capable of computing an exponential convergence rate for the (stochastic) gradient descent algorithm.