I will discuss a recent joint work with A. Malchiodi (Pisa) and M. Micallef (Warwick) in which we show that not every harmonic map can be approximated by a sequence of $\alpha$-harmonic maps.
Abstract: Kolmogorov backward equations related to stochastic evolution equations (SEE) in Hilbert space, driven by trace class Gaussian noise have been intensively studied in the literature. In this talk I discuss the extension to non trace class Gaussian noise in the particular case when the leading linear operator generates an analytic semigroup. This natural generalization leads to several complications, requiring new existence and uniqueness results for SEE with initial singularities and a new notion of an extended transition semigroup. This is joint work with Arnulf Jentzen and Ryan Kurniawan (ETH).
We consider the non-linear equation $T^{-1} u+\partial_tu-\partial_x^2\pi(u)=\xi$
driven by space-time white noise $\xi$, which is uniformly parabolic because we assume that $\pi'$ is bounded away from zero and infinity. Under the further assumption of Lipschitz continuity of $\pi'$ we show that the stationary solution is - as for the linear case - almost surely Hölder continuous with exponent $\alpha$ for any $\alpha<\frac{1}{2}$ w. r. t. the parabolic metric. More precisely, we show that the corresponding local Hölder norm has stretched exponential moments.
On the stochastic side, we use a combination of martingale arguments to get second moment estimates with concentration of measure arguments to upgrade to Gaussian moments. On the deterministic side, we first perform a Campanato iteration based on the De Giorgi-Nash Theorem as well as finite and infinitesimal versions of the $H^{-1}$-contraction principle, which yields Gaussian moments for a weaker Hölder norm. In a second step this estimate is improved to the optimal
Hölder exponent at the expense of weakening the integrability to stretched exponential.
This is joint work with Felix Otto.
It is unknown whether a bound on axion field ranges exists within quantum gravity. We study axion field ranges using extended supersymmetry, in particular allowing an analysis within strongly coupled regions of moduli space. We apply this strategy to Calabi-Yau compactifications with one and two Kähler moduli. We relate the maximally allowable decay constant to geometric properties of the underlying Calabi-Yau geometry. In all examples we find a maximal field range close to the reduced Planck mass (with the largest field range being 3.25 $M_P$). On this perspective, field ranges relate to the intersection and instanton numbers of the underlying Calabi-Yau geometry.
In the same way that the classical Torelli theorem determines a curve from its polarized Jacobian we show that moduli spaces of parabolic bundles and parabolic Higgs bundles over a compact Riemann surface X also determine X. We make use of a theorem of Hurtubise on the geometry of algebraic completely integrable systems in the course of the proof. This is a joint work with I. Biswas and T. Gómez
Most networks and graphs encountered in empirical studies, including internet and web graphs, social networks, and biological and ecological networks, are very sparse. Standard spectral and linear algebra methods can fail badly when applied to such networks and a fundamentally different approach is needed. Message passing methods, such as belief propagation, offer a promising solution for these problems. In this talk I will introduce some simple models of sparse networks and illustrate how message passing can form the basis for a wide range of calculations of their structure. I will also show how message passing can be applied to real-world data to calculate fundamental properties such as percolation thresholds, graph spectra, and community structure, and how the fixed-point structure of the message passing equations has a deep connection with structural phase transitions in networks.
Humpback whales are iconic mammals at the top of the Antarctic food chain. Their large reserves of lipid-rich tissues such as blubber predispose them to accumulation of lipophilic contaminants throughout their lifetime. Changes in the volume and distribution of lipids in humpback whales, particularly during migration, could play an important role in the pharmacokinetics of lipophilic contaminants such as the organochlorine pesticide hexachlorobenzene (HCB). Previous models have examined constant feeding and nonmigratory scenarios. In the present study, the authors develop a novel heuristic model to investigate HCB dynamics in a humpback whale and its environment by coupling an ecosystem nutrient-phytoplankton-zooplankton-detritus (NPZD) model, a dynamic energy budget (DEB) model, and a physiologically based pharmacokinetic (PBPK) model. The model takes into account the seasonal feeding pattern of whales, their energy requirements, and fluctuating contaminant burdens in the supporting plankton food chain. It is applied to a male whale from weaning to maturity, spanning 20 migration and feeding cycles. The model is initialized with environmental HCB burdens similar to those measured in the Southern Ocean and predicts blubber HCB concentrations consistent with empirical concentrations observed in a southern hemisphere population of male, migrating humpback whales.
We provide a characterization in terms of Fatou property for weakly closed monotone sets in the space of P-quasisure bounded random variables, where P is a (eventually non-dominated) class of probability measures. Our results can be applied to obtain a topological deduction of the First Fundamental Theorem of Asset Pricing for discrete time processes, the dual representation of the superhedging price and more in general the robust dual representation for (quasi)convex increasing functionals.
This is a joint paper with T. Meyer-Brandis and G. Svindland.
“Market-Basket (MB) and Household (HH) data provide a fertile substrate for the inference of association between marketing activity (e.g.: prices, promotions, advertisement, etc.) and customer behaviour (e.g.: customers driven to a store, specific product purchases, joint product purchases, etc.). The main aspect of MB and HH data which makes them suitable for this type of inference is the large number of variables of interest they contain at a granularity that is fit for purpose (e.g.: which items are bought together, at what frequency are items bought by a specific household, etc.).
A large number of methods are available to researchers and practitioners to infer meaningful networks of associations between variables of interest (e.g.: Bayesian networks, association rules, etc.). Inferred associations arise from applying statistical inference to the data. In order to use statistical association (correlation) to support an inference of causal association (“which is driving which”), an explicit theory of causality is needed.
Such a theory of causality can be used to design experiments and analyse the resultant data; in such a context certain statistical associations can be interpreted as evidence of causal associations.
On observational data (as opposed to experimental), the link between statistical and causal associations is less straightforward and it requires a theory of causality which is formal enough to support an appropriate calculus (e.g.: do-calculus) of counterfactuals and networks of causation.
My talk will be focused on providing retail analytic problems which may motivate an interest in exploring causal calculi’s potential benefits and challenges.”
I will report on joint work with Arno Fehm in which we apply
our previous `existential transfer' results to the problem of
determining which fields admit diophantine nontrivial henselian
valuation rings and ideals. Using our characterization we are able to
re-derive all the results in the literature. Also, I will explain a
connection with Pop's large fields.
Iwasawa theory is a powerful technique for relating the behaviour of arithmetic objects to the special values of L-functions. Iwasawa originally developed this theory in order to study the class groups of number fields, but it has since been generalised to many other settings. In this talk, I will discuss some new results in the Iwasawa theory of the symmetric square of a modular form. This is a joint project with Sarah Zerbes, and the main tool in this work is the Euler system of Beilinson-Flach elements, constructed in our earlier works with Kings and Lei.
The large majority of risk-sharing transactions involve few agents, each of whom can heavily influence the structure and the prices of securities. This paper proposes a game where agents' strategic sets consist of all possible sharing securities and pricing kernels that are consistent with Arrow-Debreu sharing rules. First, it is shown that agents' best response problems have unique solutions, even when the underlying probability space is infinite. The risk-sharing Nash equilibrium admits a finite-dimensional characterisation and it is proved to exist for general number of agents and be unique in the two-agent game. In equilibrium, agents choose to declare beliefs on future random outcomes different from their actual probability assessments, and the risk-sharing securities are endogenously bounded, implying (amongst other things) loss of efficiency. In addition, an analysis regarding extremely risk tolerant agents indicates that they profit more from the Nash risk-sharing equilibrium as compared to the Arrow-Debreu one.
(Joint work with Michail Anthropelos)
Respiratory illnesses, such as asthma and chronic obstructive pulmonary disease, account for one in five deaths worldwide and cost the UK over £6 billion a year. The main form of treatment is via inhaled drug delivery. Typically, however, a low fraction of the inhaled dose reaches the target areas in the lung. Predictive numerical capabilities have the potential for significant impact in the optimisation of pulmonary drug delivery. However, accurate and efficient prediction is challenging due to the complexity of the airway geometries and of the flow in the airways. In addition, geometric variation of the airways across subjects has a pronounced effect on the aerosol deposition. Therefore, an accurate model of respiratory deposition remains a challenge.
High-fidelity simulations of the flow field and prediction of the deposition patterns motivate the use of direct numerical simulations (DNS) in order to resolve the flow. Due to the high grid resolution requirements, it is desirable to adopt an efficient computational strategy. We employ a robust immersed boundary method developed for curvilinear coordinates, which allows the use of structured grids to model the complex patient-specific airways, and can accommodate the inter-subject geometric variations on the same grid. The proposed approach reduces the errors at the boundary and retains the stability guarantees of the original flow solver.
A Lagrangian particle tracking scheme is adopted to model the transport of aerosol particles. In order to characterise deposition, we propose the use of an instantaneous Stokes number based on the local properties of the flow field. The effective Stokes number is then defined as the time-average of the instantaneous value. This effective Stokes number thus encapsulates the flow history and geometric variability. Our results demonstrate that the effective Stokes number can deviate significantly from the reference value based solely on a characteristic flow velocity and length scale. In addition, the effective Stokes number shows a clear correlation with deposition efficiency.
Functions are usually approximated numerically in a basis, a non-redundant and complete set of functions that span a certain space. In this talk we highlight a number of benefits of using overcomplete sets, in particular using the more general notion of a "frame". The main benefit is that frames are easily constructed even for functions of several variables on domains with irregular shapes. On the other hand, allowing for possible linear depencies naturally leads to ill-conditioning of approximation algorithms. The ill-conditioning is potentially severe. We give some useful examples of frames and we first address the numerical stability of best approximations in a frame. Next, we briefly describe special point sets in which interpolation turns out to be stable. Finally, we review so-called Fourier extensions and an efficient algorithm to approximate functions with spectral accuracy on domains without structure.
"We give a diophantine criterion for a polynomial with rational coefficients not to have any
rational zero, i.e. an existential formula in terms of the coefficients expressing this property. This can be seen as a kind of restricted
model-completeness for Q and answers a question of Koenigsmann."
One can ask whether the fundamental groups of 3-manifolds are distinguished by their sets of finite quotients. I will discuss the recent solution of this question for Seifert fibre spaces.
STAR-Vote is voting system that results from a collaboration between a number of
academics and the Travis County, Texas elections office, which currently uses a
DRE voting system and previously used an optical scan voting system. STAR-Vote
represents a rare opportunity for a variety of sophisticated technologies, such
as end-to-end cryptography and risk limiting audits, to be designed into a new
voting system, from scratch, with a variety of real world constraints, such as
election-day vote centers that must support thousands of ballot styles and run
all day in the event of a power failure.
We present and motivate the design of the STAR-Vote system, the benefits that we
expect from it, and its current status.
This is based on joint work with Josh Benaloh, Mike Byrne, Philip Kortum,
Neal McBurnett, Ron Rivest, Philip Stark, Dan Wallach
and the Office of the Travis County Clerk
Graded affine Hecke algebras were introduced by Lusztig for studying the representation theory of p-adic groups. In particular, some problems about extensions of representations of p-adic groups can be transferred to problems in the graded Hecke algebra setting. The study of extensions gives insight to the structure of various reducible modules. In this talk, I shall discuss some methods of computing Ext-groups for graded Hecke algebras.
The talk is based on arXiv:1410.1495, arXiv:1510.05410 and forthcoming work.
Recent results showed that the low energy expansion of closed superstring amplitudes can be expressed in terms of single-valued multiple elliptic polylogarithms. I will explain how these functions may be defined as iterated integrals on the torus and sketch how they arise from Feynman integrals.
A question by Poonen asks whether iterating the étale-Brauer set can give a finer obstruction set. We tackle the algebraic version of Poonen's question and give, in many cases, a negative answer.
I will explain some recent work using minimal surfaces to address problems in 3-manifold topology. Given a Heegaard splitting, one can sweep out a three-manifold by surfaces isotopic to the splitting, and run the min-max procedure of Almgren-Pitts and Simon-Smith to construct a smooth embedded minimal surface. If the original splitting were strongly irreducible (as introduced by Casson-Gordon), H. Rubinstein sketched an argument in the 80s showing that the limiting minimal surface should be isotopic to the original splitting. I will explain some results in this direction and how jointly with T. Colding and D. Gabai we can use such min-max minimal surfaces to complete the classification problem for Heegaard splittings of non-Haken hyperbolic 3-manifolds.
First order asymptotic of scalar renewal sequences with infinite mean characterized by regular variation has been classified in the 60's (Garsia and Lamperti). In the recent years, the question of higher order asymptotic for renewal sequences with infinite mean was motivated by obtaining 'mixing rates' for dynamical systems with infinite measure. In this talk I will present the recent results we have obtained on higher order expansion for renewal sequences with infinite mean (not necessarily generated by independent processes) in the regime of slow regular variation (with small exponents). I will also discuss some consequences of these results for error rates in certain limit theorems (such as arcsine law for null recurrent Markov processes).
The first reaction-diffusion equation developed and studied is the Fisher-KPP equation. Introduced in 1937, it accounts for the spatial spreading and growth of a species. Understanding this population-dynamics model is equivalent to understanding the distribution of the maximum particle in a branching Brownian motion. Various generalizations of this model have been studied in the eighty years since its introduction, including a model with non-local reaction for the cane toads of Australia introduced by Benichou et. al. I will begin the talk by giving an extended introduction on the Fisher-KPP equation and the typical behavior of its solutions. Afterwards, I will describe the model for the cane toads equations and give new results regarding this model. In particular, I will show how the model may be viewed as a perturbation of a local equation using a new Harnack-type inequality and I will discuss the super-linear in time propagation of the toads. The talk is based on a joint work with Bouin and Ryzhik.
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Writing is a part of any career in science or mathematics. I will make some remarks about the role writing has played in my life and the role it might play in yours.
The InFoMM CDT Group Meetings will follow the format of the OCIAM group meetings. We hope they will facilitate good communication between the Academic and Student community so that the research activities remain closely connected, opportunities for additional interaction are easily identified, and cross-fertilisation of ideas can be catalysed.
Tmoslav Plesa: Chemical Reaction Systems with a Homoclinic Bifurcation: An Inverse Problem, 25+5 min;
John Ockendon: Wave Homogenisation, 10 min + questions;
Hilary Ockendon: Sloshing, 10 min + questions
Given a family $F$ of elliptic curves defined over $Q$, we are interested in the set $H(Y)$ of curves $E$ in $F$, of positive rank, and for which the minimum of the canonical heights of non-torsion rational points on $E$ is bounded by some parameter $Y$. When one can show that this set is finite, it is natural to investigate statistical properties of arithmetic objects attached to elliptic curves in the set $H(Y)$. We will describe some problems related to this, and will state some results in the case of families of quadratic twists of a fixed elliptic curve.
Motivated by the European sovereign debt crisis, we propose a hybrid sovereign default model which combines an accessible part which takes into account the movement of the sovereign solvency and the impact of critical political events, and a totally inaccessible part for the idiosyncratic credit risk. We obtain closed-form formulas for the probability that the default occurs at political critical dates in a Markovian CEV process setting. Moreover, we introduce a generalized density framework for the hybrid default times and deduce the compensator process of default. Finally we apply the hybrid model and the generalized density to the valuation of sovereign bond and explain the significant jumps in the long-term government bond yield during the sovereign crisis.
When assigned with the task of extracting information from given image data the first challenge one faces is the derivation of a truthful model for both the information and the data. Such a model can be determined by the a-priori knowledge about the image (information), the data and their relation to each other. The source of this knowledge is either our understanding of the type of images we want to reconstruct and of the physics behind the acquisition of the data or we can thrive to learn parametric models from the data itself. The common question arises: how can we customise our model choice to a particular application? Or better how can we make our model adaptive to the given data?
Starting from the first modelling strategy this talk will lead us from nonlinear diffusion equations and subdifferential inclusions of total variation type functionals as the most successful image modeltoday to non-smooth second- and third-order variational models, with data models for Gaussian and Poisson distributed data as well as impulse noise. These models exhibit solution-dependent adaptivities in form of nonlinearities or non-smooth terms in the PDE or the variational problem, respectively. Applications for image denoising, inpainting and surface reconstruction are given. After a critical discussion of these different image and data models we will turn towards the second modelling strategy and propose to combine it with the first one using a PDE constrained optimisation method that customises a parametrised form of the model by learning from examples. In particular, we will consider optimal parameter derivation for total variation denoising with multiple noise distributions and optimising total generalised variation regularisation for its application in photography.
I will illustrate how to build families of expanders out of 'very mixing' actions on measure spaces. I will then define the warped cones and show how these metric spaces are strictly related with those expanders.
Point-free topology can often seem like an algebraic almost-topology,
> not quite the same but still interesting to those with an interest in
> it. There is also a tradition of it in computer science, traceable back
> to Scott's topological model of the untyped lambda-calculus, and
> developing through Abramsky's 1987 thesis. There the point-free approach
> can be seen as giving new insights (from a logic of observations),
> albeit in a context where it is equivalent to point-set topology. It was
> in that tradition that I wrote my own book "Topology via Logic".
>
> Absent from my book, however, was a rather deeper connection with logic
> that was already known from topos theory: if one respects certain
> logical constraints (of geometric logic), then the maps one constructs
> are automatically continuous. Given a generic point x of X, if one
> geometrically constructs a point of Y, then one has constructed a
> continuous map from X to Y. This is in fact a point-free result, even
> though it unashamedly uses points.
>
> This "continuity via logic", continuity as geometricity, takes one
> rather further than simple continuity of maps. Sheaves and bundles can
> be understood as continuous set-valued or space-valued maps, and topos
> theory makes this meaningful - with the proviso that, to make it run
> cleanly, all spaces have to be point-free. In the resulting fibrewise
> topology via logic, every geometric construction of spaces (example:
> point-free hyperspaces, or powerlocales) leads automatically to a
> fibrewise construction on bundles.
>
> I shall present an overview of this framework, as well as touching on
> recent work using Joyal's Arithmetic Universes. This bears on the logic
> itself, and aims to replace the geometric logic, with its infinitary
> disjunctions, by a finitary "arithmetic type theory" that still has the
> intrinsic continuity, and is strong enough to encompass significant
> amounts of real analysis.
In recent years there has been amazing progress in building
practical protocols for Multi-Party Computation (MPC).
So much progress in fact that there are now a number of
companies producing products utilizing this technology. A major issue with existing solutions is the high round
complexity of protocols involving more than two players. In this talk I will survey the main protocols for MPC
and recent ideas in how to obtain practical low round
complexity protocols.
I will describe a construction of the Weinstein symplectic category of Lagrangian correspondences in the context of shifted symplectic geometry. I will then explain how one can linearize this category starting from a "quantization" of (-1)-shifted symplectic derived stacks: we assign a perverse sheaf to each (-1)-shifted symplectic derived stack (already done by Joyce and his collaborators) and a map of perverse sheaves to each (-1)-shifted Lagrangian correspondence (still conjectural).
A "hole" in a graph is an induced subgraph which is a cycle of length > 3. The perfect graph theorem says that if a graph has no odd holes and no odd antiholes (the complement of a hole), then its chromatic number equals its clique number; but unrestricted graphs can have clique number two and arbitrarily large chromatic number. There is a nice question half-way between them - for which classes of graphs is it true that a bound on clique number implies some (larger) bound on chromatic number? Call this being "chi-bounded".
Gyarfas proposed several conjectures of this form in 1985, and recently there has been significant progress on them. For instance, he conjectured
We survey this and several related results. This is joint with Alex Scott and partly with Maria Chudnovsky.
We discuss moments of $L$-functions in function fields, in the hyperelliptic ensemble, focusing on the fourth moment of quadratic Dirichlet $L$-functions at the critical point. We explain how to obtain an asymptotic formula with some of the secondary main terms.