Past

Topological field theories give rise to a wealth of algebraic structures, extending
the E_n algebra expressing the "topological OPE of local operators". We may interpret these algebraic structures as defining (slightly noncommutative) algebraic varieties and stacks, called moduli stacks of vacua, and relations among them. I will discuss some examples of these structures coming from the geometric Langlands program and their applications. Based on joint work with Andy Neitzke and Sam Gunningham. 

We will discuss the percolation model on the hexagonal grid. In 2001 S. Smirnov proved conformal invariance of its scaling limit through the use of a tricky auxiliary combinatorial construction.

We present a more conceptual approach, implying that the construction in question can be thought of as geometrically natural one.

The main goal of the talk is to make it believable that not all nice and useful objects in the field have been already found.

No background is required.

We present sharp gradient estimates for the solution of the filtering equation and report on its applications in a high order cubature method for the nonlinear filtering problem.

A C^k-extension of a smooth and connected Lorentzian manifold (M,g) is an isometric embedding of M into a proper subset of a connected Lorentzian manifold (N,h) of the same dimension, where the Lorentzian metric h is C^k regular. If no such extension exists, then we say that (M,g) is C^k-inextendible. The study of low-regularity inextendibility criteria for Lorentzian manifolds is motivated by the strong cosmic censorship conjecture in general relativity.

The Schwarzschild spacetime is manifestly inextendible as a Lorentzian manifold with a C^2 regular metric. In this talk I will describe how one
proves the stronger statement that the maximal analytic Schwarzschild spacetime is inextendible as a Lorentzian manifold with a continuous metric.

In the last years complex networks tools contributed to provide insights on the structure of research, through the study of collaboration, citation and co-occurrence networks. The network approach focuses on pairwise relationships, often compressing multidimensional data structures and inevitably losing information. In this paper we propose for the first time a simplicial complex approach to word co-occurrences, providing a natural framework for the study of higher-order relations in the space of scientific knowledge. Using topological methods we explore the conceptual landscape of mathematical research, focusing on homological holes, regions with low connectivity in the simplicial structure. We find that homological holes are ubiquitous, which suggests that they capture some essential feature of research practice in mathematics. Holes die when a subset of their concepts appear in the same article, hence their death may be a sign of the creation of new knowledge, as we show with some examples. We find a positive relation between the dimension of a hole and the time it takes to be closed: larger holes may represent potential for important advances in the field because they separate conceptually distant areas. We also show that authors' conceptual entropy is positively related with their contribution to homological holes, suggesting that polymaths tend to be on the frontier of research.

In this talk I will discuss how computational algebraic geometry and topology can be useful for studying questions arising in systems biology. In particular I will focus on the problem of comparing models and data through the lens of computational algebraic geometry and statistics. I will provide concrete examples of biological signalling systems that are better understood with the developed methods.

Please note that this will be held at Tsuzuki Lecture Theatre, St Annes College, Oxford.

Please note that you will need to register for this event via https://www.eventbrite.co.uk/e/qbiox-colloquium-trinity-term-2018-ticke…

Cancer causing mutations must become permanently fixed within tissues. 

Please note that this will be held at Tsuzuki Lecture Theatre, St Annes College, Oxford.

Please note that you will need to register for this event via https://www.eventbrite.co.uk/e/qbiox-colloquium-trinity-term-2018-ticke…

1600-1645 - Philip Maini
1645-1705 - Edward Morrissey
1705-1725 - Heather Harrington
1725-1800 - Drinks and networking

The talks will be followed by a drinks reception.

Tickets can be obtained from https://www.eventbrite.co.uk/e/qbiox-colloquium-trinity-term-2018-ticke….
(As ever, tickets are not necessary, but they do help in judging catering requirements.)

PHILIP MAINI

Does mathematics have anything to do with biology? In this talk, I will review a number of interdisciplinary collaborations in which I have been involved over the years that have coupled mathematical modelling with experimental studies to try to advance our understanding of processes in biology and medicine. Examples will include somatic evolution in tumours, collective cell movement in epithelial sheets, cell invasion in neural crest, and pattern formation in slime mold. These are examples where verbal reasoning models are misleading and insufficient, while mathematical models can enhance our intuition.

EDWARD MORRISEY

Fixation and spread of somatic mutations in adult human colonic epithelium Cancer causing mutations must become permanently fixed within tissues. I will describe how, by visualizing somatic clones, we investigated the means and timing with which this occurs in the human colonic epithelium. Modelling the effects of gene mutation, stem cell dynamics and subsequent lateral expansion revealed that fixation required two sequential steps. First, one of around seven active stem cells residing within each colonic gland has to be mutated. Second, the mutated stem cell has to replace neighbours to populate the entire gland. This process takes many years because stem cell replacement is infrequent (around once every 9 months). Subsequent clonal expansion due to gland fission is also rare for neutral mutations. Pro-oncogenic mutations can subvert both stem cell replacement to accelerate fixation and clonal expansion by gland fission to achieve high mutant allele frequencies with age. The benchmarking and quantification of these behaviours allows the advantage associated with different gene specific mutations to be compared and ranked irrespective of the cellular mechanisms by which they are conferred. The age related mutational burden of advantaged mutations can be predicted on a gene-by-gene basis to identify windows of opportunity to affect fixation and limit spread.

HEATHER HARRINGTON

Comparing models with data using computational algebra In this talk I will discuss how computational algebraic geometry and topology can be useful for studying questions arising in systems biology. In particular I will focus on the problem of comparing models and data through the lens of computational algebraic geometry and statistics. I will provide concrete examples of biological signalling systems that are better understood with the developed methods.

It is at first sight surprising that a minimizer of an integral of the calculus of variations may make the integrand infinite somewhere.

This talk will discuss some examples of this phenomenon, how it can be related to material defects, and related open questions from nonlinear elasticity and the theory of liquid crystals.

In this talk, I will review a number of interdisciplinary collaborations in which I have been involved over the years that have coupled mathematical
modelling with experimental studies to try to advance our understanding of processes in biology and medicine. Examples will include somatic evolution in
tumours, collective cell movement in epithelial sheets, cell invasion in neural crest, and pattern formation in slime mold. These are examples where
verbal reasoning models are misleading and insufficient, while mathematical models can enhance our intuition.

Please note that this will be held at Tsuzuki Lecture Theatre, St Annes College, Oxford.

Please note that you will need to register for this event via https://www.eventbrite.co.uk/e/qbiox-colloquium-trinity-term-2018-ticke…

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In the late 1880's Goursat investigated what we now call rigid local systems, classically described as linear differential equations without accessory parameters. In this talk I will discuss some arithmetic and geometric aspects of certain particular cases of Goursat's in rank four. For example, I will discuss what are likely to be all cases where the monodromy group is finite. This is joint work with Danylo Radchenko.

Markoff triples are integer solutions  of the equation $x^2+y^2+z^2=3xyz$ which arose in Markoff's spectacular and fundamental work (1879) on diophantine approximation and has been henceforth ubiquitous in a tremendous variety of different fields in mathematics and beyond.  After reviewing some of these, we will discuss  joint work with Bourgain and Sarnak on the connectedness of the set of solutions of the Markoff equation modulo primes under the action of the group generated by Vieta involutions, showing, in particular,  that for almost all primes the induced graph is connected.  Similar results for composite moduli enable us to establish certain new arithmetical properties of Markoff numbers, for instance the fact that almost all of them are composite.
Time permitting, we will also discuss recent joint work with Magee and Ronan on the asymptotic formula for integer points on Markoff-Hurwitz surfaces  $x_1^2+x_2^2 + \dots + x_n^2 = x_1 x_2 \dots x_n$, giving an interpretation for the exponent of growth in terms of certain conformal measure on the projective space.
 

Equivariant cohomology is adapted from ordinary cohomology to better capture the action of a group on a topological space. In Floer theory, given an autonomous Hamiltonian, there is a natural action of the circle on 1-periodic flowlines given by time translation. Combining these two ideas leads to the definition of  $S^1$-equivariant symplectic cohomology. In this talk, I will introduce these ideas and explain how they are related. I will not assume prior knowledge of Floer theory.

We discuss two Freidlin-Wentzell large deviation principles for McKean-Vlasov equations (MV-SDEs) in certain path space topologies. The equations have a drift of polynomial growth and an existence/uniqueness result is provided. We apply the Monte-Carlo methods for evaluating expectations of functionals of solutions to MV-SDE with drifts of super-linear growth.  We assume that the MV-SDE is approximated in the standard manner by means of an interacting particle system and propose two importance sampling (IS) techniques to reduce the variance of the resulting Monte Carlo estimator. In the "complete measure change" approach, the IS measure change is applied simultaneously in the coefficients and in the expectation to be evaluated. In the "decoupling" approach we first estimate the law of the solution in a first set of simulations without measure change and then perform a second set of simulations under the importance sampling measure using the approximate solution law computed in the first step. 

In laboratories around the world, scientists use magnetic stirrers to mix solutions and dissolve powders. It is well known that at high drive rates the stir bar jumps around erratically with poor mixing, leading to its nick-name 'flea'. Investigating this behaviour, we discovered a state in which the flea levitates stably above the base of the vessel, supported by magnetic repulsion between flea and drive magnet. The vertical motion is oscillatory and the angular motion a superposition of rotation and oscillation. By solving the coupled vertical and angular equations of motion, we characterised the flea’s behaviour in terms of two dimensionless quantities: (i) the normalized drive speed and (ii) the ratio of magnetic to viscous forces. However, Earnshaw’s theorem states that levitation via any arrangement of static magnets is only possible with additional stabilising forces. In our system, we find that these forces arise from the flea’s oscillations which pump fluid radially outwards, and are only present for a narrow range of Reynold's numbers. At slower, creeping flow speeds, only viscous forces are present, whereas at higher speeds, the flow reverses direction and the flea is no longer stable. We also use both the levitating and non-levitating states to measure rheological properties of the system.

We first consider multilevel Monte Carlo and stochastic collocation methods for determining statistical information about an output of interest that depends on the solution of a PDE with inputs that depend on random parameters. In our context, these methods connect a hierarchy of spatial grids to the amount of sampling done for a given grid, resulting in dramatic acceleration in the convergence of approximations. We then consider multifidelity methods for the same purpose which feature a variety of models that have different fidelities. For example, we could have coarser grid discretizations, reduced-order models, simplified physics, surrogates such as interpolants, and, in principle, even experimental data. No assumptions are made about the fidelity of the models relative to the “truth” model of interest so that unlike multilevel methods, there is no a priori model hierarchy available. However, our approach can still greatly accelerate the convergence of approximations.

The Vlasov-Poisson system is a kinetic equation that models collisionless plasma. A plasma has a characteristic scale called the Debye length, which is typically much shorter than the scale of observation. In this case the plasma is called ‘quasineutral’. This motivates studying the limit in which the ratio between the Debye length and the observation scale tends to zero. Under this scaling, the formal limit of the Vlasov-Poisson system is the Kinetic Isothermal Euler system. The Vlasov-Poisson system itself can formally be derived as the limit of a system of ODEs describing the dynamics of a system of N interacting particles, as the number of particles approaches infinity. The rigorous justification of this mean field limit remains a fundamental open problem. In this talk we present the rigorous justification of the quasineutral limit for very small but rough perturbations of analytic initial data for the Vlasov-Poisson equation in dimensions 1, 2, and 3. Also, we discuss a recent result in which we derive the Kinetic Isothermal Euler system from a regularised particle model. Our approach uses a combined mean field and quasineutral limit.

One of the main themes in geometric group theory is Gromov's program to classify finitely generated groups up to quasi-isometry. We show that under certain situations, a quasi-isometry preserves commensurator subgroups. We will focus on the case where a finitely generated group G contains a coarse PD_n subgroup H such that G=Comm(H). Such groups can be thought of as coarse fibrations whose fibres are cosets of H; quasi-isometries of G coarsely preserve these fibres. This  generalises work of Whyte and Mosher--Sageev--Whyte.

Nash-Williams showed that the collection of locally finite trees under the topological minor relation results in a BQO. Naturally, two interesting questions arise:

1.      What is the number \lambda of topological types of locally finite trees?

2.       What are the possible sizes of an equivalence class of locally finite trees?

 For (1), clearly, \omega_0 \leq \lambda \leq c and Matthiesen refined it to \omega_1 \leq \lambda \leq c. Thus, this question becomes non-trivial in the absence of the Continuum Hypothesis. In this paper we address both questions by showing - entirely within ZFC - that for a large collection of locally finite trees that includes those with countably many rays:

- \lambda = \omega_1, and

- the size of an equivalence class can only be either 1 or c.

TBA

In joint work with Gareths Boxall and Jones we prove a poly-logarithmic bound for the number of rational points on the graph of functions on the disc that exhibit a certain decay. I will present an application of this counting theorem to the arithmetic of dynamical systems. It concerns fields generated by the solutions of equations of the form $P^{\circ n}(z) = P^{\circ n}(y)$ for a polynomial $P$ of degree $D \geq 2$ where $y$ is a fixed algebraic number. The general goal is to show that the degree of such fields grows like a power of $D^n$.    

We give a description of the category of ordinary K3 surfaces over a finite field in terms of linear algebra data over Z. This gives an analogue for K3 surfaces of Deligne's description of the category of ordinary abelian varieties over a finite field, and refines earlier work by N.O. Nygaard and J.-D. Yu. Two important ingredients in the proof are integral p-adic Hodge theory, and a description of CM points on Shimura stacks in terms of associated Galois representations. References: arXiv:1711.09225, arXiv:1707.01236.

It is difficult to determine when a graph G can be edge-covered by edge-disjoint copies of a fixed graph F. That is, when it has an F-decomposition. However, if G is large and has a high minimum degree then it has an F-decomposition, as long as some simple divisibility conditions hold. Recent research allows us to prove bounds on the necessary minimum degree by studying a relaxation of this problem, where a fractional decomposition is sought.

I will show how a relatively simple random process can give a good approximation to a fractional decomposition of a dense graph, and how it can then be made exact. This improves the best known bounds for this problem.
 

In this talk, first we  address the convergence issues of a standard finite volume element method (FVEM) applied to simple elliptic problems. Then, we discuss discontinuous finite volume element methods (DFVEM) for elliptic problems  with emphasis on  computational and theoretical  advantages over the standard FVEM. Further, we present a natural extension of DFVEM employed for the elliptic problem to the Stokes problems. We also discuss suitability of these methods for the approximation of incompressible miscible displacement problems.
 

Fake accounts detection and users’ polarization are two very well known topics concerning the social media sphere, that have been extensively discussed and analyzed, both in the academic literature and in everyday life. Social bots are autonomous accounts that are explicitly created to increase the number of followers of a target user, in order to inflate its visibility and consensus in a social media context. For this reason, a great variety of methods for their detection have been proposed and tested. Polarisation, also known as confirmation bias, is instead the common tendency to look for information that confirms one's preexisting beliefs, while ignoring opposite ones. Within this environment, groups of individuals characterized by the same system of beliefs are very likely to form. In the present talk we will first review part of the literature discussing both these topics. Then we will focus on a new dataset collecting tweets from the last Italian parliament elections in 2018 and some preliminary results will be discussed.

After outlining the principles of Algebraic Quantum Field Theory (AQFT) I will describe the generalization of Hochschild cohomology that is relevant to describing deformations in AQFT. An interaction is described by a cohomology class.

Answering a question of Milnor, Grigorchuk constructed in the early eighties the
first examples of groups of intermediate growth, that is, finitely generated
groups with growth strictly between polynomial and exponential.
In  joint work with Laurent Bartholdi, we show that under a mild regularity assumption, any function greater than exp(n^a), where `a' is a solution of the equation
  2^(3-3/x)+ 2^(2-2/x)+2^(1-1/x)=2,
is a growth function of some group. These are the first examples of groups
of intermediate growth where the asymptotic of  the growth function is known.
Among applications of our results is the fact that any group of locally subexponential growth
can be embedded as a subgroup of some group of intermediate growth (some of these latter groups cannot be  subgroups in Grigorchuk groups).

In a recent work with Tianyi Zheng, we  provide  near optimal lower bounds
for Grigorchuk torsion groups, including the first Grigorchuk group. Our argument is by a construction of random walks with non-trivial Poisson boundary, defined by 
a measure with power law decay.

We consider a collisionless kinetic equation describing the probability density of particles moving in a one-dimensional domain subject to partly diffusive reflection at the boundary. It was shown in 2017 by Mokhtar-Kharroubi and Rudnicki that for large times such systems either converge to an invariant density or, if no invariant density exists, exhibit a so-called “sweeping phenomenon” in which the mass concentrates near small velocities. This dichotomy is obtained by means of subtle arguments relying on the theory of positive operator semigroups. In this talk I shall review some of these results before discussing how, under suitable assumptions both on the boundary operators (which in particular ensure that an invariant density exists) and on the initial density, one may even obtain estimates on the rate at which the system converges to its equilibrium. This is joint work with Mustapha Mokhtar-Kharroubi (Besançon).

I will present a mathematical model for the genetic evolution of a population which is divided in discrete colonies along a linear habitat, and for which the population size of each colony is random and constant in time. I will show that, under reasonable assumptions on the distribution of the population sizes, over large spatial and temporal scales, this population can be described by the solution to a stochastic partial differential equation with constant coefficients. These coefficients describe the effective diffusion rate of genes within the population and its effective population density, which are both different from the mean population density and the mean diffusion rate of genes at the microscopic scale. To do this, I will present a duality technique and a new convergence result for coalescing random walks in a random environment.

Recent tools make it possible to partition the space of rational Dehn 
surgery slopes for a knot (or in some cases a link) in a 3-manifold into 
domains over which the Heegaard Floer homology of the surgered manifolds 
behaves continuously as a function of slope. I will describe some 
techniques for determining the walls of discontinuity separating these 
domains, along with efforts to interpret some aspects of this structure 
in terms of the behaviour of co-oriented taut foliations. This talk 
draws on a combination of independent work, previous joint work with 
Jake Rasmussen, and work in progress with Rachel Roberts.

In this paper, we investigate the limit properties of frequency of empirical averages when random variables are described by a set of probability measures and obtain a law of large numbers for upper-lower probabilities. Our result is an extension of the classical Kinchin's law of large numbers, but the proof is totally different.

keywords: Law of large numbers,capacity, non-additive probability, sub-linear expectation, indepence

paper by: Zengjing Chen School of Mathematics, Shandong University and Qingyang Liu Center for Economic Research, Shandong University

3d N=2 Chern-Simons-matter theories have a large variety of boundary conditions that preserve 2d N=(0,2) supersymmetry, and support chiral algebras. I'll discuss some examples of how the chiral algebras transform across dualities. I'll then explain how to construct duality interfaces in 3d N=2 theories, and relate dualities *of* duality interfaces to "Pachner moves" in triangulations of 4-manifolds. Based on recent and upcoming work with K. Costello, D. Gaiotto, and N. Paquette.

ATP-sensitive potassium (KATP) channels are critical for coupling changes in blood glucose to insulin secretion. Gain-of-function mutations in KATP channels cause a rare inherited form of diabetes that manifest soon after birth (neonatal diabetes). This talk shows how understanding KATP channel function has enabled many neonatal diabetes patients to switch from insulin injections to sulphonylurea drugs that block KATP channel activity, with considerable improvement in their clinical condition and quality of life.   Using a mouse model of neonatal diabetes, we also found that as little as 2 weeks of diabetes led to dramatic changes in gene expression, protein levels and metabolite concentrations. This reduced glucose-stimulated ATP production and insulin release. It also caused substantial glycogen storage and β-cell apoptosis. This may help explain why older neonatal diabetes patients with find it more difficult to transfer to drug therapy, and why the drug dose decreases with time in many patients. It also suggests that altered metabolism may underlie both the progressive impairment of insulin secretion and reduced β-cell mass in type 2 diabetes.

Joint work with Abdul-Lateef Haji-Ali

This talk will discuss efficient numerical methods for estimating the probability of a large portfolio loss, and associated risk measures such as VaR and CVaR. These involve nested expectations, and following Bujok, Hambly & Reisinger (2015) we use the number of samples for the inner conditional expectation as the key approximation parameter in the Multilevel Monte Carlo formulation. The main difference in this case is the indicator function in the definition of the probability. Here we build on previous work by Gordy & Juneja (2010) who analyse the use of a fixed number of inner samples, and Broadie, Du & Moallemi (2011) who develop and analyse an adaptive algorithm. I will present the algorithm, outline the main theoretical results and give the numerical results for a representative model problem. I will also discuss the extension to real portfolios with a large number of options based on multiple underlying assets.

We will discuss the algebraicity of two quantities central to the computation of persistent homology. We will also connect persistent homology and algebraic optimization. Namely, we will express the degree corresponding to the distance variable of the offset hypersurface in terms of the Euclidean distance degree of the starting variety, obtaining a new way to compute these degrees. Finally, we will describe the non-properness locus of the offset construction and use this to describe the set of points that are topologically interesting (the medial axis and center points of the bounded components of the complement of the variety) and relevant to the computation of persistent homology.

We will discuss a basic framework to deal with coherent sheaves on schemes over $\mathbb{Z}$, involving infinite-dimensional results on the geometry of numbers. As an application, we will discuss basic results, old and new, on arithmetic ampleness, such as Serre vanishing, Nakai-Moishezon, and Bertini. This is joint work with Jean-Benoît Bost.

There are a huge number of nonlinear partial differential equations that do not have analytic solutions.   Often one can find similarity solutions, which reduce the number of independent variables, but still leads, generally, to a nonlinear equation.  This can, only sometimes, be solved analytically.  But always the solution is independent of the initial conditions.   What role do they play?   It is generally stated that the similarity  solution agrees with the (not determined) exact solution when (for some variable say t) obeys t >> t_1.   But what is  t_1?   How does it depend on the initial conditions?  How large must  t be for the similarity solution to be within 15, 10, 5, 1, 0.1, ….. percent of the real solution?   And how does this depend on the parameters and initial conditions of the problem?   I will explain how two such typical, but somewhat different, fundamental problems can be solved, both analytically and numerically,  and compare some of the results with small scale laboratory experiments, performed during the talk.  It will be suggested that many members of the audience could take away the ideas and apply them in their own special areas.

This talk is concerned with quantitative periodic homogenization in domains with boundaries. The quantitative analysis near boundaries leads to the study of boundary layers correctors, which have in general a nonperiodic structure. The interaction between the boundary and the microstructure creates geometric resonances, making the study of the asymptotics or continuity properties particularly challenging. The talk is based on work with S. Armstrong, T. Kuusi and J.-C. Mourrat, as well as work by Z. Shen and J. Zhuge

In the 80s, Hatcher and Thurston introduced the pants graph as a tool to prove that the mapping class group of a closed, orientable surface is finitely presented. The pants graph remains relevant for the study of the mapping class group, sitting between the marking graph and the curve graph. More precisely, there is a sequence of natural coarse lipschitz maps taking the marking graph via the pants graph to the curve graph.

A second motivation for studying the pants graph comes from Teichmüller theory. Brock showed that the pants graph can be interpreted as a combinatorial model for Teichmüller space with the Weil-Petersson metric.

In this talk I will introduce the pants graph, discuss some of its properties and state a few open questions.

Post-quantum cryptography has been one of the most active subfields of
cryptography in the last few years. This is especially true today as
standardization efforts are currently underway, with no less than 69
candidate cryptographic schemes proposed.

In this talk, I will present one of these schemes: Falcon, a signature
scheme based on the NTRU class of structured lattices. I will focus on
mathematical aspects of Falcon: for example how we take advantage of the
algebraic structure to speed up some operations, or how relying on the
most adequate probability divergence can go a long way in getting more
efficient parameters "for free". The talk will be concluded with a few
open problems.

Varieties admitting Frobenius splittings exhibit very nice properties.
For example, many nice properties of toric varieties can be deduced from
the fact that they are Frobenius split. Varieties admitting a diagonal
splitting exhibit even nicer properties. In this talk I will give an
overview over the consequences of the existence of such splittings and
then discuss criteria for toric varieties to be diagonally split.