Past

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.