I will discuss a couple of techniques often useful to prove quasi-isometric rigidity results for isometry groups. I will then sketch how these were used by B. Kleiner and B. Leeb to obtain quasi-isometric rigidity for the class of fundamental groups of closed locally symmetric spaces of noncompact type.
In this talk, I'll discuss some personal experiences - good, bad, and
ugly - of disclosing vulnerabilities in a range of different cryptographic
standards and implementations. I'll try to draw some general lessons about
what works well and what does not.
Given an Artin stack $X$, there is growing evidence that there should be an associated `category of B-branes', which is some subcategory of the derived category of coherent sheaves on $X$. The simplest case is when $X$ is just a vector space modulo a linear action of a reductive group, or `gauged linear sigma model' in physicists' terminology. In this case we know some examples of what the category B-branes should be. Hori has conjectured a physical duality between certain families of GLSMs, which would imply that their B-brane categories are equivalent. We prove this equivalence of categories. As an application, we construct Homological Projective Duality for (non-commutative resolutions of) Pfaffian varieties.
The chromatic number of the Erdős–Rényi random graph G(n,p) has been an intensely studied subject since at least the 1970s. A celebrated breakthrough by Bollobás in 1987 first established the asymptotic value of the chromatic number of G(n,1/2), and a considerable amount of effort has since been spent on refining Bollobás' approach, resulting in increasingly accurate bounds. Despite this, up until now there has been a gap of size O(1) in the denominator between the best known upper and lower bounds for the chromatic number of dense random graphs G(n,p) where p is constant. In contrast, much more is known in the sparse case.
In this talk, new upper and lower bounds for the chromatic number of G(n,p) where p is constant will be presented which match each other up to a term of size o(1) in the denominator. In particular, they narrow down the optimal colouring rate, defined as the average colour class size in a colouring with the minimum number of colours, to an interval of length o(1). These bounds were obtained through a careful application of the second moment method rather than a variant of Bollobás' method. Somewhat surprisingly, the behaviour of the chromatic number changes around p=1-1/e^2, with a different limiting effect being dominant below and above this value.
I will describe 5 definitions of Euler characteristic for a space with perfect obstruction theory (i.e. a well-behaved moduli space), and their inter-relations. This is joint work with Yunfeng Jiang. Then I will describe work of Yuuji Tanaka on how to this can be used to give two possible definitions of Vafa-Witten invariants of projective surfaces in the stable=semistable case.
Adaptive cubic regularization methods have recently emerged as a credible alternative to line search and trust-region for smooth nonconvex optimization, with optimal complexity amongst second-order methods. Here we consider a general class of adaptive regularization methods, that use first- or higher-order local Taylor models of the objective regularized by a(ny) power of the step size. We investigate the worst-case complexity/global rate of convergence of these algorithms, in the presence of varying (unknown) smoothness of the objective. We find that some methods automatically adapt their complexity to the degree of smoothness of the objective; while others take advantage of the power of the regularization step to satisfy increasingly better bounds with the order of the models. This work is joint with Nick Gould (RAL) and Philippe Toint (Namur).
tba
Modular invariance is ubiquitous in string theory. This is the symmetry of genus-one amplitudes, as well as the non-perturbative duality symmetry of type IIb superstring in ten dimensions. The alpha’ expansion of string theory amplitudes leads to interesting new modular forms. In this talk we will describe the properties of the new modular forms. We will explain that the modular forms entering the alpha’ expansion of genus one type-II superstring amplitude are naturally expressed as particular values of single valued elliptic multiple polylogarithm. They are natural modular generalization of the single valued elliptic multiple-zeta introduced by Francis Brown.
Last year, G. Kings and D. Rossler related the degree zero part of the polylogarithm
on abelian schemes pol^0 with another object previously defined by V. Maillot and D.
Rossler. More precisely, they proved that the canonical class of currents constructed
by Maillot and Rossler provides us with the realization of pol^0 in analytic Deligne
cohomology.
I will show that, adding some properness conditions, it is possible to give a
refinement of Kings and Rossler’s result involving Deligne-Beilinson cohomology
instead of analytic Deligne cohomology.
I will consider higher derivative corrections to the graviton 3-point coupling within a weakly coupled theory of gravity. Lorentz invariance allows further structures beyond that of Einstein’s theory. I will argue that these structures are constrained by causality, and show that the problem cannot be fixed by adding conventional particles with spins J ≤ 2, but adding an infinite tower of massive particles with higher spins. Implications of this result in the context of AdS/CFT, quantum gravity in asymptotically flat space-times, and non-Gaussianity features of primordial gravitational waves are discussed.
If the abundances of the constituent molecules of a biochemical reaction system are sufficiently high then their concentrations are typically modelled by a coupled set of ordinary differential equations (ODEs). If, however, the abundances are low then the standard deterministic models do not provide a good representation of the behaviour of the system and stochastic models are used. In this talk, I will first introduce both the stochastic and deterministic models. I will then provide theorems that allow us to determine the qualitative behaviour of the underlying mathematical models from easily checked properties of the associated reaction network. I will present results pertaining to so-called ``complex-balanced'' models and those satisfying ``absolute concentration robustness'' (ACR). In particular, I will show how ACR models, which are stable when modelled deterministically, necessarily undergo an extinction event in the stochastic setting. I will then characterise the behaviour of these models prior to extinction.
We examine the Foreign Exchange (FX) spot price spreads with and without Last Look on the transaction. We assume that brokers are risk-neutral and they quote spreads so that losses to latency arbitrageurs (LAs) are recovered from other traders in the FX market. These losses are reduced if the broker can reject, ex-post, loss-making trades by enforcing the Last Look option which is a feature of some trading venues in FX markets. For a given rejection threshold the risk-neutral broker quotes a spread to the market so that her expected profits are zero. When there is only one venue, we find that the Last Look option reduces quoted spreads. If there are two venues we show that the market reaches an equilibrium where traders have no incentive to migrate. The equilibrium can be reached with both venues coexisting, or with only one venue surviving. Moreover, when one venue enforces Last Look and the other one does not, counterintuitively, it may be the case that the Last Look venue quotes larger spreads.
a working version of the paper may be found here
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2630662
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.