Past

Higgs bundles have a rich structure and play a role in many different areas including gauge theory, hyperkähler geometry, surface group representations, integrable systems, nonabelian Hodge theory, mirror symmetry and Langlands duality. In this introductory talk I will explain some basic notions of G-Higgs – including the Hitchin fibration and spectral data - and illustrate how this relates to mirror symmetry.

We shall describe a new proof of the Mordell-Lang conjecture in positive characteristic, in the situation where the variety under scrutiny is a smooth subvariety of an abelian variety. Our proof is based on the theory of semistable sheaves in positive characteristic, in particular on Langer's theorem that the Harder-Narasimhan filtration of sheaves becomes strongly semistable after a finite number of iterations of Frobenius pull-backs. Our proof produces a numerical upper-bound for the degree of the finite morphism from an isotrivial variety appearing in the statement of the Mordell-Lang conjecture. This upper-bound is given in terms of the Frobenius-stabilised slopes of the cotangent bundle of the variety.

We shall describe a new proof of the Mordell-Lang conjecture in positive characteristic, in the situation where the variety under scrutiny is a smooth subvariety of an abelian variety. 
Our proof is based on the theory of semistable sheaves in positive characteristic, in particular on  Langer's theorem that the Harder-Narasimhan filtration of sheaves becomes strongly semistable after a finite number of iterations of Frobenius pull-backs. Our proof produces a numerical upper-bound for the degree of the finite morphism from an isotrivial variety appearing in the statement of the Mordell-Lang conjecture. This upper-bound is given in terms of the Frobenius-stabilised slopes of the cotangent bundle of the variety.

With few exceptions, optimal stopping assumes that the underlying system is stopped immediately after the decision is made. 
In fact, most stoppings take time. This has been variously referred to as "time-to-build", "investment lag" and "gestation period", 
which is often non negligible. 
In this talk, we consider a class of optimal stopping/switching problems with delivery lags, or equivalently, delayed information, 
by using reflected BSDE method. As an example, we study American put option with delayed exercise, and show that it can be decomposed 
as a European put option and a premium, the latter of which involves a new optimal stopping problem where the investor decides when to stop
to collect the Greek theta of such a European option. We also give a complete characterization of the optimal exercise boundary by resorting to free boundary analysis.  

Joint work with Zhou Yang and Mihail Zervos. 

Barbara Mahler: 15+5 min

Thomas Woolley: 15+5 min

Julian A. Garcia Grajales: 15+5 min
 

Abstract: n-homological algebra was initiated by Iyama
via his notion of n-cluster tilting subcategories.
It was turned into an abstract theory by the definition
of n-abelian categories (Jasso) and (n+2)-angulated categories
(Geiss-Keller-Oppermann).
The talk explains some elementary aspects of these notions.
We also consider the special case of an n-representation finite algebra.
Such an algebra gives rise to an n-abelian
category which can be "derived" to an (n+2)-angulated category.
This case is particularly nice because it is
analogous to the classic relationship between
the module category and the derived category of a
hereditary algebra of finite representation type.
 

To face the advent of multicore processors and the ever increasing complexity of hardware architectures, programming
models based on DAG parallelism regained popularity in the high performance, scientific computing community. Modern runtime systems offer a programming interface that complies with this paradigm and powerful engines for scheduling the tasks into which the application is decomposed. These tools have already proved their effectiveness on a number of dense linear algebra applications. 

In this talk we present the design of task-based sparse direct solvers on top of runtime systems. In the context of the
qr_mumps solver, we prove the usability and effectiveness of our approach with the implementation of a sparse matrix multifrontal factorization based on a Sequential Task flow parallel programming model. Using this programming model, we developed features such as the integration of dense 2D Communication Avoiding algorithms in the multifrontal method allowing for better scalability compared to the original approach used in qr_mumps.

Following this approach, we move to heterogeneous architectures where task granularity and scheduling strategies are critical to achieve performance. We present, for the multifrontal method, a hierarchical strategy for data partitioning and a scheduling algorithm capable of handling the heterogeneity of resources.   Finally we introduce a memory-aware algorithm to control the memory behavior of our solver and show, in the context of multicore architectures, an important reduction of the memory footprint for the multifrontal QR factorization with a small impact on performance.

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.

 "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.

TBA

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).