Past

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…

[[{"fid":"51386","view_mode":"default","fields":{"format":"default"},"type":"media","attributes":{"class":"file media-element file-default"},"link_text":"Barbara Mahler"}]]

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.