Past

The classical separate treatments of competition and predation, and an inability to provide a sensible theoretical basis for mutualism, attests to the inability of traditional models to provide a synthesising framework to study trophic interactions, a fundamental component of ecology. Recent approaches to food web modelling have focused on consumer-resource interactions. We develop this approach to explicitly represent finite resources for each population and construct a rigorous unifying theoretical framework with Lotka-Volterra Conservative Normal (LVCN) systems. We show that mixotrophy, a ubiquitous trophic interaction in marine plankton, provides the key to developing a synthesis of the various ways of making a living. The LVCN framework also facilitates an explicit redefinition of facultative mutualism, illuminating the over-simplification of the traditional definition.

We demonstrate a continuum between trophic interactions and show that populations can continuously and smoothly evolve through most population interactions without losing stable coexistence. This provides a theoretical basis consistent with the evolution of trophic interactions from autotrophy through mixotrophy/mutualism to heterotrophy.

This seminar has been cancelled

TBA

NbTi-based superconducting wires have widespread use in engineering applications of superconductivity such as MRI and accelerator magnets. Tolerance to the effects of interactions with changing (external) magnetic fields is an important consideration in wire design, in order to make the most efficient use of the superconducting material. This project aims to develop robust analytical models of the power dissipation in real conductor geometries across a broad frequency range of external field changes, with a view to developing wire designs that minimise these effects.

I'll sketch a construction which associates a canonical p-adic L-function with a 'non-critically refined' cohomological cuspidal automorphic representation of GL(2) over an arbitrary number field F, generalizing and unifying previous results of many authors. These p-adic L-functions have good interpolation and growth properties, and they vary analytically over eigenvarieties. When F=Q this reduces to a construction of Pollack and Stevens. I'll also explain where this fits in the general picture of Iwasawa theory, and I'll point towards the iceberg of which this construction is the tip.

A sharp increase in the popularity of commodity investing in the past decade has triggered an unprecedented inflow of institutional funds into commodity futures markets. Such financialization of commodities coincided with significant booms and busts in commodity markets, raising concerns of policymakers. In this paper, we explore the effects of financialization in a model that features institutional investors alongside traditional futures markets participants. The institutional investors care about their performance relative to a commodity index. We find that if a commodity futures is included in the index, supply and demand shocks specific to that commodity spill over to all other commodity futures markets. In contrast, supply and demand shocks to a nonindex commodity affect just that commodity market alone. Moreover, prices and volatilities of all commodity futures go up, but more so for the index futures than for nonindex ones. Furthermore, financialization — the presence of institutional investors — leads to an increase in correlations amongst commodity futures as well as in equity-commodity correlations. Consistent with empirical evidence, the increases in the correlations between index commodities exceed those for nonindex ones. We model explicitly demand shocks which allows us to disentangle the effects of financialization from the effects of demand and supply (fundamentals). We perform a simple calibration and find that financialization accounts for 11% to 17% of commodity futures prices and the rest is attributable to fundamentals.

Several experimental studies have shown that the abundance distributions of proteins in a population of isogenic cells may display multiple distinct maxima. Each of these maxima may be associated with a subpopulation of a particular phenotype, the quantification of which is important for understanding cellular decision-making. I will present a novel methodology which allows us to quantify multi-modal gene expression distributions and single cell power spectra in gene regulatory networks. The method is based on an extension of the linear noise approximation; in particular we rigorously show that, in the limit of slow promoter dynamics, these distributions can be systematically approximated as a mixture of Gaussian components. The resulting closed-form approximation provides a practical tool for studying complex nonlinear gene regulatory networks that have thus far been amenable only to stochastic simulation. I will demonstrate the applicability of our approach to several examples and discuss some new dynamical characteristics e.g., how the interplay of transcriptional and translational regulation can be exploited to control the multimodality of gene expression distributions in two-promoter networks and how genetic oscillators can display concerted noise-induced bimodality and noise-induced oscillations.

This talk will be an introduction to the notion of D-modules on

prestacks. We will begin by discussing Grothendieck's definition of

crystals of quasi-coherent sheaves on a smooth scheme X, and briefly

indicate how the category of such objects is equivalent to that of

modules over the sheaf of differential operators on X. We will then

explain what we mean by a prestack and define the category of

quasi-coherent sheaves on them. Finally, we consider how the

crystalline approach may be used to give a suitable generalization

of D-modules to this derived setting.

Computer simulation of the propagation and interaction of linear waves
is a core task in computational science and engineering.
The finite element method represents one of the most common
discretisation techniques for Helmholtz and Maxwell's equations, which
model time-harmonic acoustic and electromagnetic wave scattering.
At medium and high frequencies, resolution requirements and the
so-called pollution effect entail an excessive computational effort
and prevent standard finite element schemes from an effective use.
The wave-based Trefftz methods offer a possible way to deal with this
problem: trial and test functions are special solutions of the
underlying PDE inside each element, thus the information about the
frequency is directly incorporated in the discrete spaces.

This talk is concerned with a family of those methods: the so-called
Trefftz-discontinuous Galerkin (TDG) methods, which include the
well-known ultraweak variational formulation (UWVF).
We derive a general formulation of the TDG method for Helmholtz
impedance boundary value problems and we discuss its well-posedness
and quasi-optimality.
A complete theory for the (a priori) h- and p-convergence for plane
and circular/spherical wave finite element spaces has been developed,
relying on new best approximation estimates for the considered
discrete spaces.
In the two-dimensional case, on meshes with very general element
shapes geometrically graded towards domain corners, we prove
exponential convergence of the discrete solution in terms of number of
unknowns.

This is a joint work with Ralf Hiptmair, Christoph Schwab (ETH Zurich,
Switzerland) and Ilaria Perugia (Vienna, Austria).

As the title says, in this talk I will be giving a casual introduction to higher categories. I will begin by introducing strict n-categories and look closely at the resulting structure for n=2. After discussing why this turns out to be an unsatisfying definition I will discuss in what ways it can be weakened. Broadly there are two main classes of models for weak n-categories: algebraic and geometric. The differences between these two classes will be demonstrated by looking at bicategories on the algebraic side and quasicategories on the geometric.

This is a report on joint work (still in progress) with Ellen Eischen, Jian-Shu Li,
and Chris Skinner.  I will describe the general structure of our construction of p-adic L-functions
attached to families of ordinary holomorphic modular forms on Shimura varieties attached to
unitary groups.  The complex L-function is studied by means of the doubling method;
its p-adic interpolation applies adelic representation theory to Ellen Eischen's Eisenstein 
measure.

In this talk we aim to introduce the key ideas of homotopy type theory and show how it draws on and has applications to the areas of logic, higher category theory, and homotopy theory. We will discuss how types can be viewed both as propositions (statements about mathematics) as well as spaces (mathematical objects themselves). In particular we will define identity types and explore their groupoid-like structure; we will also discuss the notion of equivalence of types, introduce the Univalence Axiom, and consider some of its implications. Time permitting, we will discuss inductive types and show how they can be used to define types corresponding to specific topological spaces (e.g. spheres or more generally CW complexes).\\

This talk will assume no prior knowledge of type theory; however, some very basic background in category theory (e.g. the definition of a category) and homotopy theory (e.g. the definition of a homotopy) will be assumed.

Let $A$ and $B$ be boundaries of CAT(0) spaces. A function $f:A \to B$ is called a {\em boundary isomorphism} if $f$ is a homeomorphism in the visual topology and

$f$ is an isometry in the Tits metric. A compact metrizable space $Y$ is said to be {\em Tits rigid}, if for any two CAT(0) group boundaries $Z_1$ and $Z_2$ homeomorphic to $Y$, $Z_1$ is boundary isomorphic to $Z_2$.

We prove that the join of two Cantor sets and its suspension are Tits rigid.

Mirror Symmetry predicts a surprising relationship between the virtual numbers of degree-d rational curves in a target space X and variations of Hodge structure on a different space X’, called the mirror to X.  Concretely, it predicts that one can compute genus-zero Gromov–Witten invariants (which are the virtual numbers of rational curves) in terms of hypergeometric functions (which are the solutions to a differential equation that controls the variation of Hodge structure).  Existing proofs of this rely on beautiful but fearsomely complicated localization calculations in equivariant cohomology.  I will describe a new proof of the Mirror Theorem, for a broad range of target spaces X, which is much simpler and more conceptual. This is joint work with Cristina Manolache.

Partial differential equations defined on surfaces appear in various applications, for example image processing and reconstruction of non-planar images. In this talk, I will present a penalty method for evolution equations, based on an implicit representation of the surface. I will derive a simple equation in the surrounding space, formulated with an extension operator, and then show some analysis and applications of the method.

Several problems lead to the question of how well can a fine grid function be approximated by a coarse grid function, such as preconditioning in finite element methods or data compression and image processing. Particular challenges in answering this question arise when the functions may be only piecewise-continuous, or when the coarse space is not nested in the fine space. In this talk, we solve the problem by using a stable approximation from a space of globally smooth functions as an intermediate step, thereby allowing the use of known approximation results to establish the approximability by a coarse space. We outline the proof, which relies on techniques from the theory of discontinuous Galerkin methods and on the theory of Helmholtz decompositions. Finally, we present an application of our to nonoverlapping domain decomposition preconditioners for hp-version DGFEM.

Cox processes arise as a natural extension of inhomogeneous Poisson Processes, when the intensity function itself is taken to be stochastic. In multiple applications one is often concerned with characterizing the posterior distribution over the intensity process (given some observed data). Markov Chain Monte Carlo methods have historically been successful at such tasks. However, direct methods are doubly intractable, especially when the intensity process takes values in a space of continuous functions.

In this talk I'll be presenting a method to overcome this intractability that is based on the idea of "thinning" and that does not resort to approximations.

I will show how families of concentrating stationary vortices for the shallow

water equations can be constructed and studied asymptotically. The main tool

is the study of asymptotics of solutions to a family of semilinear elliptic

problems. The same method also yields results for axisymmetric vortices for

the Euler equation of incompressible fluids.

Abstract: The signature of a path characterizes the non-commutative evolvements along the path trajectory. Nevertheless, one can extract local commutativities from the signature, thus leading to an inversion scheme.

It is well known that several solutions to the Skorokhod problem

optimize certain ``cost''- or ``payoff''-functionals. We use the

theory of Monge-Kantorovich transport to study the corresponding

optimization problem. We formulate a dual problem and establish

duality based on the duality theory of optimal transport. Notably

the primal as well as the dual problem have a natural interpretation

in terms of model-independent no arbitrage theory.

In optimal transport the notion of c-monotonicity is used to

characterize the geometry of optimal transport plans. We derive a

similar optimality principle that provides a geometric

characterization of optimal stopping times. We then use this

principle to derive several known solutions to the Skorokhod

embedding problem and also new ones.

This is joint work with Mathias Beiglböck and Alex Cox.

This talk is based on a joint work with B. Rémy (Lyon) in which we study some subgroups of topological Kac–Moody groups and the implications of this study on the subgroup structure of the ambient Kac–Moody group.

In this talk we will discuss when one right-angled Artin group is a subgroup of another one and explain how this basic algebraic problem may provide answers to questions in geometric group theory and model theory such as classification of right-angled Artin groups up to quasi-isometries and universal equivalence.

Numerical models provide valuable tools for integrating understanding of riverine processes and morphology. Moreover, they have considerable potential for use in investigating river responses to environmental change and catchment management, and for aiding the interpretation of alluvial deposits and landforms. For this potential to be realised fully, such models must be capable of representing diverse river styles, and the spatial and temporal transitions between styles that can be driven by environmental forcing. However, while numerical modelling of rivers has advanced significantly over the past few decades, this has been accomplished largely by developing separate approaches to modelling different styles of river (e.g., meanders and braided networks). In addition, there has been considerable debate about what should constitute the ‘basic ingredients’ of river models, and the degree to which the environmental processes governing river evolution can be simplified in such models. This seminar aims to examine these unresolved issues, with particular reference to the simulation of large rivers and their floodplains.

We know since almost a century that a ball can be decomposed into five pieces and these pieces rearranged so as to produce two balls of the same size as the original. This apparent paradox has led von Neumann to the notion of amenability which is now much studied in many areas of mathematics. However, the initial paradox has remained tied down to an elementary property of free groups of rotations for most of the 20th century. I will describe recent progress leading to new paradoxical groups.

Traditional diffusion models for random phenomena have paths with Holder

regularity just greater than 1/2 almost surely but there are situations

arising in finance and telecommunications where it is natural to look

for models in which the Holder regularity of the paths can vary.

Such processes are called multifractal and we will construct a class of

such processes on R using ideas from branching processes.

Using connections with multitype branching random walk we will be able

to compute the multifractal spectrum which captures the variability in

the Holder regularity. In addition, if we observe one of our processes

at a fixed resolution then we obtain a finite Markov representation,

which allows efficient simulation.

As an application, we fit the model to some AUD-USD exchange rate data.

Joint work with Geoffrey Decrouez and Ben Hambly

In this talk, I will present our new local existence result to the shallow water equations describing the motions of vertically averaged flows, which are closely related to the $2$-D isentropic Navier-Stokes equations for compressible fluids with density-dependent viscosity coefficients. Via introducing the notion of regular solutions, the local existence of classical solutions is established for the case that the viscosity coefficients are degenerate and the initial data are arbitrarily large with vacuum appearing in the far field.

We shall talk our recent works on the well-posedness of the Prandtl boundary layer equations both in two and three space variables. For the two-dimensional problem, we obtain the well-posedness in the Sobolev spaces by using an energy method under the monotonicity assumption of tangential velocity, and for the three-dimensional Prandtl equations, we construct a special solution by using the Corocco transformation, and obtain it is linearly stable with respect to any three-dimensional perturbation. These works are collaborated with R. Alexandre, C. J. Liu, C. Xu and T. Yang.

Seismic exploration in the oil industry is one example where wave equations are used as models. When the wave speed is spatially varying one is naturally concerned with questions of homogenisation or upscaling, where one would like to calculate an effective or average wave speed. As a canonical problem this short workshop will introduce the one-dimensional acoustic wave equation with a rapidly varying wave speed, perhaps even a periodic variation. Three questions will be asked: (i) how do you calculate a sensible average wave speed (ii) does the wave equation suffice or is there a change of form after averaging and (iii) if one can induce any particular excitation at one end of a finite one-dimensional medium, and make any observations that we like at that end, what - if anything - can be inferred about the spatial variability of the wave speed?

Since their introduction in the context of symplectic geometry, moment maps and symplectic quotients have been generalized in many different directions. In this talk I plan to give an introduction to the notions of hyperkähler moment map and hyperkähler quotient through two examples, apparently very different, but related by the so called ADHM construction of instantons; the moduli space of instantons and a space of complex matrices arising from monads.

We extend Kisin's construction of integral canonical models of Hodge-type Shimura

varieties to p=2, using Dieudonné display theory.

Adaptive network models, in which node states and network topology coevolve, arise naturally in models of social dynamics that incorporate homophily and social influence. Homophily relates the similarity between pairs of agents' states to their network coupling strength, whilst social influence causes the convergence of coupled agents' states. In this talk, I will describe a deterministic adaptive network model of attitude formation in social groups that incorporates these effects, and in which the attitudinal dynamics are represented by an activator-inhibitor process. I will show that consensus, corresponding to all nodes adopting the same attitudinal state and being fully connected, may destabilise via Turing instability, giving rise to chaotic dynamics. For the case where there are just two agents, I will illustrate, using numerical continuation, how such chaotic dynamics arise.

We consider evaluation methods for payoffs with an inherent

financial risk as encountered for instance for portfolios held

by pension funds and insurance companies. Pricing such payoffs

in a way consistent to market prices typically involves

combining actuarial techniques with methods from mathematical

finance. We propose to extend standard actuarial principles by

a new market-consistent evaluation procedure which we call `two

step market evaluation.' This procedure preserves the structure

of standard evaluation techniques and has many other appealing

properties. We give a complete axiomatic characterization for

two step market evaluations. We show further that in a dynamic

setting with continuous stock prices every evaluation which is

time-consistent and market-consistent is a two step market

evaluation. We also give characterization results and examples

in terms of $g$-expectations in a Brownian-Poisson setting.

Infinity categories simultaneously generalize topological spaces and categories. As a result, their study benefits from a combination of techniques from homotopy theory and category theory. While the theory of ordinary categories provides a suitable context to analyze objects up to isomorphism (e.g. abelian groups), the theory of infinity categories provides a reasonable framework to study objects up to a weaker concept of identification (e.g. complexes of abelian groups). In the talk, we will introduce infinity categories from scratch, mention some of the fundamental results, and try to illustrate some features in concrete examples.

PLEASE NOTE: THIS EVENT HAS BEEN CANCELLED

Brady showed that there are hyperbolic groups with non-hyperbolic finitely presented subgroups. I will present a new construction of this kind using Bestvina-Brady Morse theory.