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.
An important moral truth about the mapping class group of a closed orientable surface is the following: a generic mapping class has no power fixing a finite family of simple closed curves on the surface. Such "generic" elements are called pseudo-Anosov. In this talk I will discuss one instantiation of this principle, namely that the probability of a simple random walk on the mapping class group returning a non-pseudo Anosov element decays exponentially quickly.
A finitely generated group has the Haagerup property if it admits a proper isometric action on a Hilbert space. It was a long open question whether Haagerup property is a quasi-isometry invariant. The negative answer was recently given by Mathieu Carette, who constructed two quasi-isometric generalized Baumslag-Solitar groups, one with the Haagerup property, the other not. Elaborating on these examples, we proved (jointly with S. Arnt and T. Pillon) that the equivariant Hilbert compression is not a quasi-isometry invariant. The talk will be devoted to describing Carette's examples.
Given a model dynamical system, a model of any measuring apparatus relating states to observations, and a prior assessment of uncertainty, the probability density of subsequent system states, conditioned upon the history of the observations, is of some practical interest.
When observations are made at discrete times, it is known that the evolving probability density is a solution of the Bayesian filtering equations. This talk will describe the difficulties in approximating the evolving probability density using a Gaussian mixture (i.e. a sum of Gaussian densities). In general this leads to a sequence of optimisation problems and related high-dimensional integrals. There are other problems too, related to the necessity of using a small number of densities in the mixture, the requirement to maintain sparsity of any matrices and the need to compute first and, somewhat disturbingly, second derivatives of the misfit between predictions and observations. Adjoint methods, Taylor expansions, Gaussian random fields and Newton’s method can be combined to, possibly, provide a solution. The approach is essentially a combination of filtering methods and '4-D Var’ methods and some recent progress will be described.
Erdos, Faudree, Gould, Gyarfas, Rousseau and Schelp, conjectured that
whenever the edges of a complete graph are coloured using three colours
there always exists a set of at most three vertices which have at least
two-thirds of their neighbours in one of the colours. We will describe a
proof of this conjecture. This is joint work with Rahil Baber
Everybody has heard of the Faraday cage effect, in which a wire mesh does a good job of blocking electric fields and electromagnetic waves. For example, the screen on the front of your microwave oven keeps the microwaves from getting out, while light with its smaller wavelength escapes so you can see your burrito. Surely the mathematics of such a famous and useful phenomenon has been long ago worked out and written up in the physics books, right?
Well, maybe. Dave Hewett and I have communicated with dozens of mathematicians, physicists, and engineers on this subject so far, and we've turned up amazingly little. Everybody has a view of why the Faraday cage mathematics is obvious, and most of their views are different. Feynman discusses the matter in his Lectures on Physics, but so far as we can tell, he gets it wrong.
For the static case at least (the Laplace equation), Hewett and I have made good progress with numerical explorations based on Mikhlin's method backed up by a theorem. The effect seems to much weaker than we had imagined -- are we missing something? For time-harmonic waves (the Helmholtz equation), our simulations lead to further puzzles. We need advice! Where in the world is the literature on this problem?
Monotonicity formulas play a pervasive role in the study of variational inequalities and free boundary problems. In this talk we will describe a new approach to a classical problem, namely the thin obstacle (or Signorini) problem, based on monotonicity properties for a family of so-called frequency functions.
Let $F \in \mathbb{Z}[x_1,\ldots,x_n]$. Suppose $F(\mathbf{x})=0$ has infinitely many integer solutions $\mathbf{x} \in \mathbb{Z}^n$. Roughly how common should be expect the solutions to be? I will tell you what your naive first guess ought to be, give a one-line reason why, and discuss the reasons why this first guess might be wrong.
I then will apply these ideas to explain the intriguing parallels between the handling of the Brauer-Manin obstruction by Heath-Brown/Skorobogotov [doi:10.1007/BF02392841] on the one hand and Wei/Xu [arXiv:1211.2286] on the other, despite the very different methods involved in each case.
I will present a formula that relates a Higgs bundle on an algebraic curve and Gromov-Witten invariants. I will start with the simplest example, which is a rank 2 bundle over the projective line with a meromorphic Higgs field. The corresponding quantum curve is the Airy differential equation, and the Gromov-Witten invariants are the intersection numbers on the moduli space of pointed stable curves. The formula connecting them is exactly the path that Airy took, i.e., from wave mechanics to geometric optics, or what we call the WKB method, after the birth of quantum mechanics. In general, we start with a Higgs bundle. Then we apply a generalization of the topological recursion, originally found by physicists Eynard and Orantin in matrix models, to this context. In this way we construct a quantization of the spectral curve of the Higgs bundle.
Morrey's lower semicontinuity theorem for quasiconvex integrands is a
classical result that establishes the existence of minimisers to
variational problems by the Direct Method, provided the integrand
satisfies "standard" growth conditions (i.e. when the growth and
coercivity exponents match). This theorem has more recently been refined
to consider convergence in Sobolev Spaces below the growth exponent of
the integrand: such results can be used to show existence of solutions
to a "Relaxed minimisation problem" when we have "non-standard'" growth
conditions.
When the integrand satisfies linear coercivity
conditions, it is much more useful to consider the space of functions of
Bounded Variation, which has better compactness properties than
$W^{1,1}$. We review the key results in the standard growth case, before
giving an overview of recent results that we have obtained in the
non-standard case. We find that new techniques and ideas are required in
this setting, which in fact provide us with some interesting (and
perhaps unexpected) corollaries on the general nature of quasiconvex
functions.
In this talk I will describe a multiple frequency approach to the boundary control of Helmholtz and Maxwell equations. We give boundary conditions and a finite number of frequencies such that the corresponding solutions satisfy certain non-zero constraints inside the domain. The suitable boundary conditions and frequencies are explicitly constructed and do not depend on the coefficients, in contrast to the illuminations given as traces of complex geometric optics solutions. This theory finds applications in several hybrid imaging modalities. Some examples will be discussed.
The first half of this talk will be an introduction to the wonderful world of Higgs bundles. The last half concerns Fourier--Mukai transforms, and we will discuss how to merge the two concepts by constructing a Fourier--Mukai transform for Higgs bundles. Finally we will discuss some properties of this transform. We will along the way discuss why you would want to transform Higgs bundles.
One of the things sustainability applications have in common with industrial applications is their close connection with decision-making and policy. We will discuss how a decision-support viewpoint may inspire new mathematical questions. For example, the concept of resilience (of ecosystems, food systems, communities, economies, etc) is often described as the capacity of a system to withstand disturbance and retain its functional characteristics. This has several familiar mathematical interpretations, probing the interaction between transient dynamics and noise. How does a focus on resilience change the modeling, dynamical and policy questions we ask? I look forward to your ideas and discussion.
Let $G=SL(2,\R)\ltimes R^2$ and $\Gamma=SL(2,Z)\ltimes Z^2$. Building on recent work of Strombergsson we prove a rate of equidistribution for the orbits of a certain 1-dimensional unipotent flow of $\Gamma\G$, which projects to a closed horocycle in the unit tangent bundle to the modular surface. We use this to answer a question of Elkies and McMullen by making effective the convergence of the gap distribution of $\sqrt n$ mod 1.
We present a new model of financial markets that studies the evolution of wealth
among investment strategies. An investment strategy can be generated by maximizing utility
given some expectations or by behavioral rules. The only requirement is that any investment strategy
is adapted to the information filtration. The model has the mathematical structure of a random dynamical system.
We solve the model by characterizing evolutionary properties of investment strategies (survival, evolutionary stability, dominance).
It turns out that only a fundamental strategy investing according to expected relative dividends satisfies these evolutionary criteria.
Thermoacoustic oscillations occur in combustion chambers when heat release oscillations lock into pressure oscillations. They were first observed in lamps in the 18th century, in rockets in the 1930s, and are now one of the most serious problems facing gas turbine manufacturers.
This theoretical and numerical study concerns an infinite-rate chemistry diffusion flame in a tube, which is a simple model for a flame in a combustion chamber. The problem is linearized around the non-oscillating state in order to derive the direct and adjoint equations governing the evolution of infinitesimal oscillations.
The direct equations are used to predict the frequency, growth rate, and mode shape of the most unstable thermoacoustic oscillations. The adjoint equations are then used to calculate how the frequency and growth rate change in response to (i) changes to the base state such as the flame shape or the composition of the fuel (ii) generic passive feedback mechanisms that could be added to the device. This information can be used to stabilize the system, which is verified by subsequent experiments.
This analysis reveals that, as expected from a simple model, the phase delay between velocity and heat-release fluctuations is the key parameter in determining the sensitivities. It also reveals that this thermo-acoustic system is exceedingly sensitive to changes in the base state. This analysis can be extended to more accurate models and is a promising new tool for the analysis and control of thermo-acoustic oscillations.
For any infinite group with a distinguished family of normal subgroups of finite index -- congruence subgroups-- one can ask whether every finite index subgroup contains a congruence subgroup. A classical example of this is the positive solution for $SL(n,\mathbb{Z})$ where $n\geq 3$, by Mennicke and Bass, Lazard and Serre. \\
Groups acting on infinite rooted trees are a natural setting in which to ask this question. In particular, branch groups have a sufficiently nice subgroup structure to yield interesting results in this area. In the talk, I will introduce this family of groups and the congruence subgroup problem in this context and will present some recent results.
A virtual endomorphism of a group $G$ is a homomorphism $f : H \rightarrow G$ where $H$
is a subgroup of $G$ of fi nite index $m$: A recursive construction using $f$ produces a
so called state-closed (or, self-similar in dynamical terms) representation of $G$ on
a 1-rooted regular $m$-ary tree. The kernel of this representation is the $f$-core $(H)$;
i.e., the maximal subgroup $K$ of $H$ which is both normal in G and is f-invariant.
Examples of state-closed groups are the Grigorchuk 2-group and the Gupta-
Sidki $p$-groups in their natural representations on rooted trees. The affine group
$Z^n \rtimes GL(n;Z)$ as well as the free group $F_3$ in three generators admit state-closed
representations. Yet another example is the free nilpotent group $G = F (c; d)$ of
class c, freely generated by $x_i (1\leq i \leq d)$: let $H = \langle x_i^n | \
(1 \leq i \leq d) \rangle$ where $n$ is a
fi xed integer greater than 1 and $f$ the extension of the map $x^n_i
\rightarrow x_i$ $(1 \leq i \leq d)$.
We will discuss state-closed representations of general abelian groups and of
nitely generated torsion-free nilpotent groups.
Counting curves with given topological properties in a variety is a very old question. Example questions are: How many conics pass through five points in a plane, how many lines are there on a Calabi-Yau 3-fold? There are by now several ways to count curves and the numbers coming from different curve counting theories may be different. We would then like to have methods to compare these numbers. I will present such a general method and show how it works in the case of stable maps and stable quasi-maps.
Fix positive integers p and q with 2 \leq q \leq {p \choose 2}. An
edge-colouring of the complete graph K_n is said to be a (p,
q)-colouring if every K_p receives at least q different colours. The
function f(n, p, q) is the minimum number of colours that are needed for
K_n to have a (p,q)-colouring. This function was introduced by
Erdos and Shelah about 40 years ago, but Erdos and Gyarfas
were the first to study the function in a systematic way. They proved
that f(n, p, p) is polynomial in n and asked to determine the maximum
q, depending on p, for which f(n,p,q) is subpolynomial in n. We
prove that the answer is p-1.
We also discuss some related questions.
Joint work with Jacob Fox, Choongbum Lee and Benny Sudakov.
In 1890, G. H. Bryan demonstrated that when a ringing wine glass rotates, the shape of the vibration pattern precesses, and this effect is the basis for a family of high-precision gyroscopes. Mathematically, the precession can be described in terms of a symmetry-breaking perturbation due to gyroscopic effects of a geometrically degenerate pair of vibration modes. Unfortunately, current attempts to miniaturize these gyroscope designs are subject to fabrication imperfections that also break the device symmetry. In this talk, we describe how these devices work and our approach to accurate and efficient simulations of both ideal device designs and designs subject to fabrication imperfections.