Past

An extension of the expected shortfall as well as the value at risk to
model uncertainty has been proposed by P. Shige.
In this talk we will present a systematic extension of the general
class of optimized certainty equivalent that includes the expected
shortfall.
We show that its representation can be simplified in many cases for
efficient computations.
In particular we present some result based on a probability model
uncertainty derived from some Wasserstein metric and provide explicit
solution for it.
We further study the duality and representation of them.

This talk is based on a joint work with Daniel Bartlxe and Ludovic
Tangpi

The Boltzmann equation is a well-studied PDE that describes the statistical evolution of a dilute gas of spherical particles. However, much less is known — both from the physical and mathematical viewpoints — about the Boltzmann equation for non-spherical particles. In this talk, we present some new results on the non-existence and non-uniqueness of weak solutions to the initial-boundary value problem for N non-spherical particles which have importance for the Boltzmann equation.

We present work which was done jointly with L. Saint-Raymond (ENS Lyon), and also with P. Palffy-Muhoray (Kent State), E. Virga (Pavia) and X. Zheng (Kent State).

In this talk Michael Bonsall will explore how we can use mathematics to link between scales of organisation in biology. He will delve in to developmental biology, ecology and neurosciences, all illustrated and explored with real life examples, simple games and, of course, some neat maths.

Michael Bonsall is Professor of Mathematical Biology in Oxford.

7 February 2018, 5pm-6pm, Mathematical Institute, Oxford

Please email @email to register or watch online: https://livestream.com/oxuni/bonsall

CAT(0) spaces are defined as having triangles that are no fatter than Euclidean triangles, so it is no surprise that under special conditions  you find pieces of the Euclidean plane appearing in CAT(0) spaces. What is surprising though is how weak these special conditions seem to be. I will present some well known results of this phenomenon, along with detailed sketch proofs.

Large parts of the cryptography in use today,

key-agreement protocols and digital signatures based on the

hardness of factoring large integers or solving the

discrete-logarithm problem, are not secure against attackers

equipped with a large universal quantum computer. It is not

clear when such a large quantum computer will be built, but

continuous progress by various labs around the world suggests

that it may well be less than two decades until today's

cryptography will become insecure.

To address this issue, NIST started a public competition to

identify suitable replacements for today's cryptosystems. In

my talk, I will describe two of these systems: the

key-encapsulation mechanism Kyber and the digital signature

scheme Dilithium. Both schemes are based on the hardness of

solving problems in module lattices and they together form the

"Cryptographic Suite for Algebraic Lattices -- CRYSTALS".

Let L be a lattice in R^n and let Z in R^(m+n) a parameterized family of subsets Z_T of R^n. Starting from an old result of Davenport and using O-minimal structures, together with Martin Widmer, we proved for fairly general families Z an estimate for the number of points of L in Z_T, which is essentially best possible.
After introducing the problem and stating the result, we will present applications to counting algebraic integers of bounded height and to Manin’s Conjecture.

Let L be a lattice in R^n and let Z in R^(m+n) a parameterized family of subsets Z_T of R^n. Starting from an old result of Davenport and using O-minimal structures, together with Martin Widmer, we proved for fairly general families Z an estimate for the number of points of L in Z_T, which is essentially best possible. 
After introducing the problem and stating the result, we will present applications to counting algebraic integers of bounded height and to Manin’s Conjecture.

The question of what happens to the eigenvalues of a matrix after an additive perturbation has a long history, with notable contributions from Wilkinson, Sorensen, Golub, H\"ormander, Ipsen and Mehrmann, among many others. If the perturbed matrix $C \in \mathbb{C}^{n \times n}$ is given by $C = A + B$, these theorems typically consider the case where $A$ and/or $B$ are symmetric and $B$ has rank one. In this talk we will prove a theorem that bounds the number of distinct eigenvalues of $C$ in terms of the number of distinct eigenvalues of $A$, the diagonalisability of $A$, and the rank of $B$. This new theorem is more general in that it applies to arbitrary matrices $A$ perturbed by matrices of arbitrary rank $B$. We will also discuss various refinements of my bound recently developed by other authors.
 

It is well-known that nilpotent orbits in $\mathfrak{sl}_n(\mathbb C)$ correspond bijectively with the set of partitions of $n$, such that the closure (partial) ordering on orbits is sent to the dominance order on partitions. Taking dual partitions simply turns this poset upside down, so in type $A$ there is an order-reversing involution on the poset of nilpotent orbits. More generally, if $\mathfrak g$ is any simple Lie algebra over $\mathbb C$ then Lusztig-Spaltenstein duality is an order-reversing bijection from the set of special nilpotent orbits in $\mathfrak g$ to the set of special nilpotent orbits in the Langlands dual Lie algebra $\mathfrak g^L$.
It was observed by Kraft and Procesi that the duality in type $A$ is manifested in the geometry of the nullcone. In particular, if two orbits $\mathcal O_1<\mathcal O_2$ are adjacent in the partial order then so are their duals $\mathcal O_1^t>\mathcal O_2^t$, and the isolated singularity attached to the pair $(\mathcal O_1,\mathcal O_2)$ is dual to the singularity attached to $(\mathcal O_2^t,\mathcal O_1^t)$: a Kleinian singularity of type $A_k$ is swapped with the minimal nilpotent orbit closure in $\mathfrak{sl}_{k+1}$ (and vice-versa). Subsequent work of Kraft-Procesi determined singularities associated to such pairs in the remaining classical Lie algebras, but did not specifically touch on duality for pairs of special orbits.
In this talk, I will explain some recent joint research with Fu, Juteau and Sommers on singularities associated to pairs $\mathcal O_1<\mathcal O_2$ of (special) orbits in exceptional Lie algebras. In particular, we (almost always) observe a generalized form of duality for such singularities in any simple Lie algebra.
 

We consider a system of nonlinear partial differential equations modelling the steady motion of an incompressible non-Newtonian fluid, which is chemically reacting. The governing system consists of a steady convection-diffusion equation for the concentration and the generalized steady Navier–Stokes equations, where the viscosity coefficient is a power-law type function of the shear-rate, and the coupling between the equations results from the concentration-dependence of the power-law index. This system of nonlinear partial differential equations arises in mathematical models of the synovial fluid found in the cavities of moving joints. We construct a finite element approximation of the model and perform the mathematical analysis of the numerical method. Key technical tools include discrete counterparts of the Bogovski operator, De Giorgi’s regularity theorem and the Acerbi–Fusco Lipschitz truncation of Sobolev functions, in function spaces with variable integrability exponents.

Assortative mixing in networks is the tendency for nodes with the same attributes, or metadata, to link to each other. It is a property often found in social networks manifesting as a higher tendency of links occurring between people with the same age, race, or political belief. Quantifying the level of assortativity or disassortativity (the preference of linking to nodes with different attributes) can shed light on the factors involved in the formation of links and contagion processes in complex networks. It is common practice to measure the level of assortativity according to the assortativity coefficient, or modularity in the case of discrete-valued metadata. This global value is the average level of assortativity across the network and may not be a representative statistic when mixing patterns are heterogeneous. For example, a social network spanning the globe may exhibit local differences in mixing patterns as a consequence of differences in cultural norms. Here, we introduce an approach to localise this global measure so that we can describe the assortativity, across multiple scales, at the node level. Consequently we are able to capture and qualitatively evaluate the distribution of mixing patterns in the network. We find that for many real-world networks the distribution of assortativity is skewed, overdispersed and multimodal. Our method provides a clearer lens through which we can more closely examine mixing patterns in networks.

Link to arxiv paper:  https://arxiv.org/abs/1708.01236

I will describe how to recast perturbative quantum gravity using non-perturbative techniques from conformal field theory, focussing on the case of N=4 super Yang-Mills theory. By resolving the degeneracy among double trace operators at large N we are able to bootstrap one-loop supergravity corrections from the OPE of the CFT.
 

Numerical methods for SDEs typically use only the discretized increments of the driving Brownian motion. As one would expect, this approach is sensible and very well studied.

In addition to generating increments, it is also straightforward to generate time integrals of Brownian motion. These quantities give extra information about the Brownian path and are known to improve the strong convergence of methods for one-dimensional SDEs. Despite this, numerical methods that use time integrals alongside increments have received less attention in the literature.

In this talk, we will develop some underlying theory for these time integrals and introduce a new numerical approach to SDEs that does not require evaluating vector field derivatives. We shall also discuss the possible implications of this work for multi-dimensional SDEs.

The Z/2-equivariant Heegaard Floer cohomlogy of the double cover of S^3 along a knot, defined by Lipshitz, Hendricks, and Sarkar, 
is an isomorphism class of F_2[\theta]-modules. In this talk, we show that this invariant is natural, and is functorial under based cobordisms. 
Given a transverse knot K in the standard contact 3-sphere, we define an element of the Z/2-equivariant Heegaard Floer cohomology 
that depends only on the tranverse isotopy class of K, and is functorial under certain symplectic cobordisms.

Based on the notion of paracontrolled distributions, existence and uniqueness results are presented for rough convolution equations. In particular, this wide class of equations includes rough differential equations with possible delay, stochastic Volterra equations, and moving average equations driven by Lévy processes. The talk is based on a joint work with Mathias Trabs.

In real dimension two, the symplectic mapping class group of a surface agrees with its `classical' mapping class group, whose properties are well-understood. To what extend do these generalise to higher-dimensions? We consider specific pairs of symplectic manifolds (S, M), where S is a surface, together with collections of Lagrangian spheres in S and in M, say v_1, ...,v_k and V_1, ...,V_k, that have analogous intersection patterns, in a sense that we will make precise. Our main theorem is that any relation between the Dehn twists in the V_i must also hold between Dehn twists in the v_i. Time allowing, we will give some corollaries, such as embeddings of certain interesting groups into auto-equivalence groups of Fukaya categories.

I am interested in the moduli spaces of heterotic vacua. These are closely related to the moduli spaces of stable holomorphic bundles but in which the base and bundle vary simultaneously, together with additional constraints deriving from string theory. I will first summarise some pre-Brexit results we have derived. These include an explicit Kaehler metric and Kaehler potential for both the moduli space and its first cousin, the matter field space. I will secondly describe new, post-Brexit work in which these results are encased within an elegant geometry, which we call a universal heterotic geometry. Beyond compelling aesthetics, the framework is surprisingly useful giving both a concise derivation of our pre-Brexit results as well as some new results. 

Today, everyone knows everything: with only a quick trip through WebMD or Wikipedia, average citizens believe themselves to be on an equal intellectual footing with doctors and diplomats. All voices, even the most ridiculous, demand to be taken with equal seriousness, and any claim to the contrary is dismissed as undemocratic elitism. Tom Nichols argues that in this climate, democratic institutions themselves are in danger of falling either to populism or to technocracy- or in the worst case, a combination of both.

Tom Nichols is Professor of National Security Affairs at the US Naval War College, an adjunct professor at the Harvard Extension School, and a former aide in the U.S. Senate. His latest book is The Death of Expertise: The Campaign Against Established Knowledge and Why it Matters. This lecture is based on that book.

All welcome. No need to book.

In this talk I will discuss the upcoming REF2021 and its significance for early career researchers (research fellows and postdocs) including

• why it is so important to all UK maths departments
• why the timing of it could have important career consequences for ECRs
• publication issues such as quality versus quantity, and choice of journal
• the importance of Impact Case Studies
   

The ability of cells to sense and respond to the mechanical properties of their environments is fundamental to cellular behaviour, with stiffness found to be a key control parameter. The physical mechanisms underpinning mechanosensing are, however, not well understood. I here consider the key physical cellular behaviours of active contractility of the internal cytoskeleton and cell growth, coupling these into mechanical models. These models suggest new distinct mechanisms of mechanotransduction in cells and tissues.

This is the first part of a discussion of  Ezra Miller's paper https://arxiv.org/abs/1709.08155.

Mazur observed that in many cases where an elliptic curve E has a non-trivial element C in its Tate-Shafarevich group, one can find another elliptic curve E' such that ExE' admits an isogeny that kills C. For elements of order 2 and 3 one can prove that such an E' always exists. However, for order 4 this leads to a question about rational points on certain K3-surfaces. We show how to explicitly construct these surfaces and give some results on their rational points.

This is joint work with Tom Fisher.
 

We describe a locally finite graph naturally associated to each knot type K, called the Reidemeister graph. We determine several local and global properties of this graph and prove that the graph-isomorphism type is a complete knot invariant up to mirroring. Lastly (time permitting), we introduce another object, relating the Reidemeister and Gordian graphs, and briefly present an application to the study of DNA.

In this talk, I will present some recent results exploring the connections between dynamical systems and network science. I will particularly focus on large-scale structures and their dynamical interpretation. Those may correspond to communities/clusters or classes of dynamically equivalent nodes. If time allows, I will also present results where the underlying network structure is unknown and where communities are directly inferred from time series observed on the nodes.

The solution to the standard cost efficiency problem depends crucially on the fact that a single real-world measure P is available to the investor pursuing a cost-efficient approach. In most applications of interest however, a historical measure is neither given nor can it be estimated with accuracy from available data. To incorporate the uncertainty about the measure P in the cost efficient approach we assume that, instead of a single measure, a class of plausible prior models is available. We define the notion of robust cost-efficiency and highlight its link with the maxmin expected utility setting of Gilboa and Schmeidler (1989) and more generally with robust preferences in a possibly non expected utility setting.

This is joint work with Thibaut Lux and Steven Vanduffel (VUB)

Decision trees usefully represent the sparse, high dimensional and noisy nature of chemical data from experiments. Having learned a function from this data, we may want to thereafter optimise the function, e.g. for picking the best catalyst for a chemical process. This work studies a mixed-integer non-linear optimisation problem involving: (i) gradient boosted trees modelling catalyst behaviour, (ii) penalty functions mitigating risk, and (iii) penalties enforcing chemical composition constraints. We develop several heuristic methods to find feasible solutions, and an exact, branch and bound algorithm that leverages structural properties of the gradient boost trees and penalty functions. We computationally test our methods on an industrial instance from BASF.
This work was completed in collaboration with Mr Miten Mistry and Dr Dimitris Letsios at Imperial College London and Dr Robert Lee and Dr Gerhard Krennrich from BASF.
 

In this talk, we propose a model describing the growth of tree stems and vine, taking into account also the presence of external obstacles. The system evolution is described by an integral differential equation which becomes discontinuous when the stem hits the obstacle. The stem feels the obstacle reaction not just at the tip, but along the whole stem. This fact represents one of the main challenges to overcome, since it produces a cone of possible reactions which is not normal with respect to the obstacle. However, using the geometric structure of the problem and optimal control tools, we are able to prove existence and uniqueness of the solution for the integral differential equation under natural assumptions on the initial data.

I will talk about the properties of algebraic integers that can arise as stretch factors of pseudo-Anosoc maps. I will mention a conjecture of Fried on which numbers supposedly arise and Thurston’s theorem that proves a similar result in the context of automorphisms of free groups. Then I will talk about recent developments on the Fried conjecture namely, every Salem number has a power arising as a stretch factor. 

We characterise the embeddability of simply connected 2-dimensional simplicial complexes in 3-space in a way analogous to Kuratowski’s characterisation of graph planarity, by excluded minors. This answers questions of Lovász, Pardon and Wagner.

 

In this talk we discuss the convergence rate of the Newton method for finding the singularity point on vetor fields. It is well-known that the Newton Method has local quadratic convergence rate with nonsingularity and Lipschitz condition. Here we release Lipschitz condition. With only nonsingularity, the Newton Method has superlinear convergence. If we have enough time, we can quickly give the damped Newton method on finding singularity on vector fields with superlinear convergence under nonsingularity condition only.

In this talk we consider the issue of mass loss in fragmentation models due to 'shattering'. As a solution we propose a hybrid discrete/continuous model whereby the smaller particles are considered as having discrete mass, whilst above a certain cut-off, mass is taken to be a continuous variable. The talk covers the development of such a model, its initial analysis via the theory of operator semigroups and its numerical approximation using a finite volume discretisation.

Online discussions are the essence of many social platforms on the Internet. Discussion platforms are receiving increasing interest because of their potential to become deliberative spaces. Although previous studies have proposed approaches to measure online deliberation using the complexity of discussion networks as a proxy, little research has focused on how these networks are affected by changes of platform features.

In this talk, we will focus on how interfaces might influence the network structures of discussions using techniques like interrupted time series analysis and regression discontinuity design. Futhermore, we will review and extend state-of-the-art generative models of discussion threads to explain better the structure and growth of online discussions.
 

After a brief introduction to the synthetic notions of Ricci curvature lower bounds in terms of optimal transportation, due to Lott-Sturm-Villani, I will discuss some applications to smooth Riemannian manifolds. These include: rigidity and stability of Levy- Gromov inequality, an almost euclidean isoperimetric inequality motivated by the celebrated Perelman’s Pseudo-Locality Theorem for Ricci flow. Joint work with F. Cavalletti.

Let f be the planar Bargmann-Fock field, i.e. the analytic Gaussian field with covariance kernel exp(-|x-y|^2/2). We compute the critical point for the percolation model induced by the level sets of f. More precisely, we prove that there exists a.s. an unbounded component in {f>p} if and only if p<0. Such a percolation model has been studied recently by Beffara-Gayet and Beliaev-Muirhead. One important aspect of our work is a derivation of a (KKL-type) sharp threshold result for correlated Gaussian variables. The idea to use a KKL-type result to compute a critical point goes back to Bollobás-Riordan. This is joint work with Alejandro Rivera.

Rips filtrations over a finite metric space and their corresponding persistent homology are prominent methods in Topological Data Analysis to summarize the ``shape'' of data. For finite metric space $X$ and distance $r$  the traditional Rips complex with parameter $r$ is the flag complex whose vertices are the points in $X$ and whose edges are $\{[x,y]: d(x,y)\leq r\}$. From considering how the homology of these complexes evolves we can create persistence modules (and their associated barcodes and persistence diagrams). Crucial to their use is the stability result that says if $X$ and $Y$ are finite metric space then the bottleneck distance between persistence modules constructed by the Rips filtration is bounded by $2d_{GH}(X,Y)$ (where $d_{GH}$ is the Gromov-Hausdorff distance). Using the asymmetry of the distance function we construct four different constructions analogous to the persistent homology of the Rips filtration and show they also are stable with respect to the Gromov-Hausdorff distance. These different constructions involve ordered-tuple homology, symmetric functions of the distance function, strongly connected components and poset topology.

We shall explain, from variational point of view, why the  Laplaciian operator introduced by Ebin-Marsden using deformations is suitable to describe the fluid motion in a milieu with viscosity.

First, we will discuss sequences of closed minimal hypersurfaces (in closed Riemannian manifolds of dimension up to 7) that have uniformly bounded index and area. In particular, we explain a bubbling result which yields a bound on the total curvature along the sequence and, as a consequence, topological control in terms of index and area. We then specialise to minimal surfaces in ambient manifolds of dimension 3, where we use the bubbling analysis to obtain smooth multiplicity-one convergence under bounds on the index and genus. This is joint work with Lucas Ambrozio, Alessandro Carlotto, and Ben Sharp

M-theory on K3 surfaces and Heterotic Strings on T^3 give rise to dual theories in 7 dimensions. Applying this duality fibre-wise is expected to connect G2 manifolds with Calabi-Yau threefolds (together with vector bundles). We make these ideas explicit for a class of G2 manifolds realized as twisted connected sums and prove the equivalence of the spectra of the dual theories. This naturally gives us examples of singular TCS G2 manifolds realizing non-abelian gauge theories with non-chiral matter.

A panel discussion and Q&A, looking at some of the challenges and opportunities available for mathematicians outside universities. Featuring:

Madeleine Copin – North London Collegiate School
Josephine French – Health Data Insight, working in partnership with Public Health England
Martin Gould – Spotify
Dan Jones – Quadrature Capital
Adam Sardar – e-therapeutics

In contemporary ecology and mathematical biology undergraduate courses, textbooks focus on competition and predation models despite it being accepted that most species on Earth are involved in mutualist relationships. Mutualism is usually discussed more briefly in texts, often from an observational perspective, and obligate mutualism mostly not at all. Part of the reason for this is the lack of a simple math model to successfully explain the observations. Traditionally, particular nonlinearities  are used, which produce a variety of apparently disparate models.

The failure of the traditional linear model to describe coexisting mutualists has been documented from May (1973) through Murray (2001) to Bronstein (2015). Here we argue that this could be because of the use of carrying capacity, and propose the use of a nutrient pool instead, which implies the need for an autotroph (e.g. a plant) that converts nutrients into living resources for higher trophic levels. We show that such a linear model can successfully explain the major features of obligate mutualism when simple expressions for obligated growth are included.

I will discuss a new theoretical approach to information and decisions in signalling systems and relate this to new experimental results about the NF-kappaB signalling system. NF-kappaB is an exemplar system that controls inflammation and in different contexts has varying effects on cell death and cell division. It is commonly claimed that it is information processing hub, taking in signals about the infection and stress status of the tissue environment and as a consequence of the oscillations, transmitting higher amounts of information to the hundreds of genes it controls. My aim is to develop a conceptual and mathematical framework to enable a rigorous quantifiable discussion of information in this context in order to follow Francis Crick's counsel that it is better in biology to follow the flow of information than those of matter or energy. In my approach the value of the information in the signalling system is defined by how well it can be used to make the "correct decisions" when those "decisions" are made by molecular networks. As part of this I will introduce a new mathematical method for the analysis and simulation of large stochastic non-linear oscillating systems. This allows an analytic analysis of the stochastic relationship between input and response and shows that for tightly-coupled systems like those based on current models for signalling systems, clocks, and the cell cycle this relationship is highly constrained and non-generic.

In the 12 months from the middle of June 2016 to the middle of June 2017, a number of events occurred in a relatively short period of time, all of which either had, or had the potential to have,  a considerably volatile impact upon financial markets. The events referred to here are the Brexit  referendum (23 June 2016), the US election (8 November 2016), the 2017 French elections (23 April and 7 May 2017) and the surprise 2017 UK parliamentary election (8 June 2017). 
All of these events - the Brexit referendum and the Trump election in particular - were notable both for their impact upon financial markets after the event and the degree to which the markets failed to anticipate these events. A natural question to ask is whether these could have been predicted, given information freely available in the financial markets beforehand. In this talk, we focus on market expectations for price action around Brexit and the Trump election, based on information available in the traded foreign exchange options market. We also investigate the horizon date of 30 March 2019, when the two year time window that started with the Article 50 notification on 29 March 2017 will terminate.
Mathematically, we construct a mixture model corresponding to two scenarios for the GBPUSD exchange rate after the referendum vote, one scenario for “remain” and one for “leave”. Calibrating this model to four months of market data, from 24 February to 22 June 2016, we find that a “leave” vote was associated with a predicted devaluation of the British pound to approximately 1.37 USD per GBP, a 4.5% devaluation, and quite consistent with the observed post-referendum exchange rate move down from 1.4877 to 1.3622. We find similar predictive power for USDMXN in the case of the 2016 US presidential election. We argue that we can apply the same bimodal mixture model technique to construct two states of the world corresponding to soft Brexit (continued access to the single market) and hard Brexit (failure of negotiations in this regard).
 

James Clerk Maxwell (1831–1879) was, by any measure, a natural philosopher of the first rank who made wide-ranging contributions to science. He also, however, wrote poetry.

In this talk examples of Maxwell’s poetry will be discussed in the context of a biographical sketch. It will be  argued that not only was Maxwell a good poet, but that his poetry enriches our view of his life and its intellectual context.