We will discuss various familiar properties of groups studied in geometric group theory, whether or not they are invariant under quasi-isometry, and why.
The concept of delegated quantum computing is a quantum extension of
the classical task of computing with encrypted data without decrypting
them first. Many quantum protocols address this challenge for a
futuristic quantum client-server setting achieving a wide range of
security properties. The central challenge of all these protocols to
be applicable for classical tasks (such as secure multi party
computation or fully homomorphic encryption) is the requirement of a
server with a universal quantum computer. By restricting the task to
classical computation only, we derive a protocol for unconditionally
secure delegation of classical computation to a remote server that has
access to basic quantum devices.
In the talk I will give an introduction to the Manin-Mumford conjecture and to the Pila-Zannier strategy for attacking it in the case of products of elliptic curves. if the permits it, I will also speak about how this same strategy has allowed to attack the analogous André-Oort conjecture for Shimura Varieties of abelian type.
Fix some positive integer r. A famous theorem of Schur states that if you partition Z/pZ into r colour classes then, provided p > p_0(r) is sufficiently large, there is a monochromatic triple {x, y, x + y}. By essentially the same argument there is also a monochromatic triple {x', y', x'y'}. Recently, Tom Sanders and I showed that in fact there is a
monochromatic quadruple {x, y, x+y, xy}. I will discuss some aspects of the proof.
Amplitudes in quantum field theory have discontinuities when regarded as
functions of
the scattering kinematics. Such discontinuities can be determined from
Cutkosky rules.
We present a structural analysis of such rules for massive quantum field
theory which combines
algebraic geometry with the combinatorics of Karen Vogtmann's Outer Space.
This is joint work with Spencer Bloch (arXiv:1512.01705).
Many theorems and conjectures in prime number theory are equivalent to finding solutions to certain linear equations in primes -- witness Goldbach's conjecture, the twin prime conjecture, Vinogradov's theorem, finding k-term arithmetic progressions, etcetera. Classically these problems were attacked using Fourier analysis -- the 'circle' method -- which yielded some success, provided that the number of variables was sufficiently large. More recently, a long research programme of Ben Green and Terence Tao introduced two deep and wide-ranging techniques -- so-called 'higher order Fourier analysis' and the 'transference principle' -- which reduces the number of required variables dramatically. In particular, these methods give an asymptotic formula for the number of k-term arithmetic progressions of primes up to X. In this talk we will give a brief survey of these techniques, and describe new work of the speaker, partially ongoing, which applies the Green-Tao machinery to count prime solutions to certain linear inequalities in primes -- a 'higher order Davenport-Heilbronn method'.
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.