Past

The representation dimension of an algebra was introduced in the early 70's by M. Auslander, with the goal of measuring how far an algebra is from having finite number of finitely generated indecomposable modules (up to isomorphism). This invariant is not well understood. For instance, it was not until 2002 that O. Iyama proved that every algebra has finite representation dimension. This was done by constructing special quasihereditary algebras. In this talk I will give an introduction to this topic and I shall briefly explain Iyama's construction.

I will survey some applications of a special kind of stratification of an algebraic stack called a theta-stratification. The goal is to eventually be able to study semistability and wall-crossing 
in a large array of moduli problems beyond the well-known examples. The most general application is to studying the derived category of coherent sheaves on the stack, but one can use this to understand the topology (K-theory, Hodge-structures, etc.) of the semistable locus and how it changes as one varies the stability condition. I will also describe a ``virtual non-abelian localization theorem'' which computes the virtual index of certain classes in the K-theory of a stack with perfect obstruction theory. This generalizes the virtual localization theorem of Pandharipande-Graber and the K-theoretic localization formulas of Teleman and Woodward.

It is easy to see that if a tournament (a complete oriented graph) has a directed cycle then it has a directed 3-cycle. We investigate the analogous question for 3-tournaments, and more generally for oriented 3-graphs.

Abstract: This is a joint work with E. Breuillard.

A conjecture of Breuillard asserts that for every positive integer d, there is a positive constant c such that the following holds. Let S be a finite subset of GL(d,C) that generates a group, which is not virtually nilpotent. Then |S^n|>exp(cn) for all n.
Considering an algebraic number a that is not a root of unity and the semigroup generated by the affine transformations x-> ax+1, x-> ax+1, the above conjecture implies that the Mahler measure of a is at least 1+c' for some c'>0 depending on c. This property is known as Lehmer's conjecture.

I will talk about the converse of this implication, namely that Lehmer's conjecture implies the uniform growth conjecture of
Breuillard.

Hawking radiation and particle creation by an expanding Universe
are paradigmatic predictions of quantum field theory in curved spacetime.
Although the theory is a few decades old, it still awaits experimental
demonstration. At first sight, the effects predicted by the theory are too
small to be measured in the laboratory. Therefore, current experimental
efforts have been directed towards siumlating Hawking radiation and
studying quantum particle creation in analogue spacetimes.
In this talk, I will present a proposal to test directly effects of
quantum field theory in the Earth's spacetime using quantum technologies.
Under certain circumstances, real spacetime distortions (such as
gravitational waves) can produce observable effects in the state of
phonons of a Bose-Einstein condensate. The sensitivity of the phononic
field to the underlying spacetime can also be used to measure spacetime
parameters such as the Schwarzschild radius of the Earth.

In quite an elementary, hands-on talk, I will discuss some Ax-Lindemann type results in the setting of modular functions.  There are some very powerful results in this area due to Pila, but in nonclassical variants we have only quite weak results, for a rather silly reason to be discussed in the talk.

TBC

We will discuss the concept of $\ell^2$-stability of a group and show some of its rigidity consequences.  We provide moreover some very concrete examples of lattices in product of trees that have many interesting properties, $\ell^2$-stability being only one of them.

Abstract: 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 asymptotic for renewal sequences with infinite mean and their consequences for error rates in certain limit theorems (such as arcsine law for null recurrent Markov processes).

I will discuss theta-stability, a framework for analyzing moduli problems in algebraic geometry by finding a special kind of stratification called a theta-stratification, a notion which generalizes the Kempf-Ness stratification in geometric invariant theory and the Harder-Narasimhan-Shatz stratification of the moduli of vector bundles on a Riemann surface.

The talk will consider three well-defined problems which can be interpreted as mathematical tests of the physical reality of black holes: Rigidity, stability and formation of black holes.

We will discuss how the analysis of a stochastic mean-field model for
synaptic activity can be used to reconstruct some parameters about
neuronal networks.  The method is based on a non-standard analysis of the
Fokker-Planck equation and the asymptotic computation of the spectrum for
the nonself-adjoint operator. Applications concern Up- and Down- states
and bursting activity in neuronal networks.

Systemic risk in financial markets occurs when activities that are beneficial to an agent in isolation (e.g. reducing microprudential risk) cause unintended consequences due to collective interactions (usually called macroprudential risk).  I will discuss three different mechanisms through which this occurs in financial markets.   Contagion can propagate due to the market impact of trading among agents with strongly overlapping portfolios, or due to cascading failures from chains of default caused by networks of interlinked counterparty exposures.  A proper understanding of these phenomena must take both dynamics and network effects into account.  I will discuss four different examples that illustrate these points.  The first is a simple model of the market dynamics induced by Basel-style risk management, which from extremely simple assumptions shows that excessive leverage can give rise to a slowly rising price bubble followed by an abrupt crash with a time period of 10 - 15 years.  The model gives rise to a chaotic attractor whose time series closely resembles the Great Moderation and subsequent crisis.   We show that alternatives to Basel can provide a better compromise between micro and macro prudential risk.   The second example is a model of leveraged value investors that yields clustered volatility and fat-tailed returns similar to those in financial markets.  The third example is the DebtRank algorithm, which uses a similar method to PageRank to correctly quantify the way risk propagates through networks of counterparty exposures and can be used as the basis of a systemic risk tax.  The fourth example will  be work in progress to provide an early warning system for financial stress caused by overlapping portfolios.  Finally I will discuss an often neglected source of financial risk due to imbalances in market ecologies.

A basic result in Morse theory due to Reeb states that a compact manifold which admits a smooth function with only two, non-degenerate critical points is homeomorphic to the sphere. We shall apply this idea to distance function associated to a Riemannian metric to prove the diameter-sphere theorem of Grove-Shiohama: A complete Riemannian manifold with sectional curvature $\geq 1$ and diameter $> \pi / 2$ is homeomorphic to a sphere. I shall not assume any knowledge about curvature for the talk.

Following a quick introduction to derivatives markets and the classic theory of valuation, we describe the changes triggered by post 2007 events. We re-discuss the valuation theory assumptions and introduce valuation under counterparty credit risk, collateral posting, initial and variation margins, and funding costs. A number of these aspects had been investigated well before 2007. We explain model dependence induced by credit effects, hybrid features, contagion, payout uncertainty, and nonlinear effects due to replacement closeout at default and possibly asymmetric borrowing and lending rates in the margin interest and in the funding strategy for the hedge of the relevant portfolio. Nonlinearity manifests itself in the valuation equations taking the form of semi-linear PDEs or Backward SDEs. We discuss existence and uniqueness of solutions for these equations. We present an invariance theorem showing that the final valuation equations do not depend on unobservable risk free rates, that become purely instrumental variables. Valuation is thus based only on real market rates and processes. We also present a high level analysis of the consequences of nonlinearities, both from the point of view of methodology and from an operational angle, including deal/entity/aggregation dependent valuation probability measures and the role of banks treasuries. Finally, we hint at how one may connect these developments to interest rate theory under multiple discount curves, thus building a consistent valuation framework encompassing most post-2007 effects.

Damiano Brigo, Joint work with Andrea Pallavicini, Daniele Perini, Marco Francischello. 

The number of steps required by the Euclidean algorithm to find the greatest common divisor of a pair of integers $u,v$ with $1<u<v<n$ has been investigated since at least the 16th century, with an asymptotic for the mean number of steps being found independently by H. Heilbronn and J.D. Dixon in around 1970. It was subsequently shown by D. Hensley in 1994 that the number of steps asymptotically follows a normal distribution about this mean. Existing proofs of this fact rely on extensive effective estimates on the Gauss-Kuzman-Wirsing operator which run to many dozens of pages. I will describe how this central limit theorem can be obtained instead by a much shorter Tauberian argument. If time permits, I will discuss some related work on the number of steps for the binary Euclidean algorithm.

Much of the recent interest in complex networks has been driven by the prospect that network optimization will help us understand the workings of evolutionary pressure in natural systems and the design of efficient engineered systems.  In this talk, I will reflect on unanticipated attributes and artifacts in three classes of network optimization problems. First, I will discuss implications of optimization for the metabolic activity of living cells and its role in giving rise to the recently discovered phenomenon of synthetic rescues. Then I will comment on the problem of controlling network dynamics and show that theoretical results on optimizing the number of driver nodes/variables often only offer a conservative lower bound to the number actually needed in practice. Finally, I will discuss the sensitive dependence of network dynamics on network structure that emerges in the optimization of network topology for dynamical processes governed by eigenvalue spectra, such as synchronization and consensus processes.  Optimization is a double-edged sword for which desired and adverse effects can be exacerbated in complex network systems due to the high dimensionality of their dynamics.

The moduli space of tropical curves (and its variants) is one of the most-studied objects in tropical geometry. So far this moduli space has only been considered as an essentially set-theoretic coarse moduli space (sometimes with additional structure). As a consequence of this restriction, the tropical forgetful map does not define a universal curve
(at least in the positive genus case). The classical work of Knudsen has resolved a similar issue for the algebraic moduli space of curves by considering the fine moduli stacks instead of the coarse moduli spaces. In this talk I am going to give an introduction to these fascinating tropical moduli spaces and report on ongoing work with R. Cavalieri, M. Chan, and J. Wise, where we propose the notion of a moduli stack of tropical curves as a geometric stack over the category of rational polyhedral cones. Using this framework one can give a natural interpretation of the forgetful morphism as a universal curve. The coarse moduli space arises as the set of $\mathbb{R}_{\geq 0}$-valued points of the moduli stack. Given time, I will also explain how the process of tropicalization for these moduli stacks can be phrased in a more fundamental way using the language of logarithmic algebraic stacks.
 

This talk focuses on the direct search method, arguably one of the simplest optimization algorithms. The algorithm minimizes an objective function by iteratively evaluating it along a number of (polling) directions, which are typically taken from so-called positive spanning sets. It does not use derivatives.

We first introduce the worst case complexity theory of direct search, and discuss how to choose the positive spanning set to minimize the complexity bound. The discussion leads us to a long-standing open
problem in Discrete Geometry. A recent result on this problem enables us to establish the optimal order for the worst case complexity of direct search.

We then show how to achieve even lower complexity bound by using random polling directions. It turns out that polling along two random directions at each iteration is sufficient to guarantee the convergence
of direct search for any dimension, and the resultant algorithm enjoys lower complexity both in theory and in practice.

The last part of the talk is devoted to direct search based on inaccurate function values. We address three questions:
i) what kind of solution can we obtain by direct search if the function values are inaccurate? 
ii) what is the worst case complexity to attain such a solution? iii) given
the inaccuracy in the function values, when to stop the algorithm in order
to guarantee the quality of the solution and also avoid “over-optimization”?

This talk is based on joint works with F. Delbos, M. Dodangeh, S. Gratton, B. Pauwels, C. W. Royer, and L. N. Vicente.

It has been observed long time ago (by many people) that singular affine symplectic varieties come in pairs; that is often to an affine singular symplectic variety $X$ one can associate a dual variety $X^!$; the geometries of $X$ and $X^!$ (and their quantizations) are related in a non-trivial way. The purpose of the talk will be 3-fold:

1) Explain a set of conjectures of Braden, Licata, Proudfoot and Webster which provide an exact formulation of the relationship between $X$ and $X^!$

2) Present a list of examples of symplectically dual pairs (some of them are very recent); in particular, we shall explain how the symplectic duals to Nakajima quiver varieties look like.

3) Give a new approach to the construction of $X^!$ and a proof of the conjectures from part 1).

The talk is based on a work in progress with Finkelberg and Nakajima.

 If you fix a class of models and a construction method that allows you to construct a new model in that class from an old model in that class, you can consider the Kripke frame generated from any given model by iterating that construction method and define the modal logic of that Kripke frame.  We shall give a general definition of these modal logics in the fully abstract setting and then apply these ideas in a number of cases.  Of particular interest is the case where we consider the class of models of ZFC with the construction method of forcing:  in this case, we are looking at the so-called "generic multiverse".

A family of expanders is a sequence of finite graphs which are both sparse and highly connected. Firstly defined in the 80s, they had huge applications in applied maths and computer science. Moreover, it soon turned out that they also had deep implications in pure maths. In this talk I will introduce the expander graphs and I will illustrate a way to construct them by approximating actions of groups on probability spaces.

Let F be a global field and let A denote its adele ring. The usual Tamagawa number formula computes the (suitably normalized) volume of the quotient G(A)/G(F) in terms of values of the zeta-function of F at the exponents of G; here G is simply connected semi-simple group. When F is functional field, this computation is closely related to the Atiyah-Bott computation of the cohomology of the moduli space of G-bundles on a smooth projective curve.

I am going to present a (somewhat indirect) generalization of the Tamagawa formula to the case when G is an affine Kac-Moody group and F is a functional fiend. Surprisingly, the proof heavily uses the so called Macdonald constant term identity. We are going to discuss possible (conjectural) geometric interpretations of this formula (related to moduli spaces of bundles on surfaces).

This is joint work with D.Kazhdan.

Cycles are fundamental objects in graph theory. A spanning cycle in a graph is also called a Hamiltonian cycle. The celebrated Dirac's Theorem in 1952 shows that every graph on $n\ge 3$ vertices with minimum degree at least $n/2$ contains a Hamiltonian cycle. In recent years, there has been a strong focus on extending Dirac’s Theorem to hypergraphs. We survey the results along the line and mention some recent progress on this problem. Joint work with Yi Zhao.

 About 10 years ago Minahan and Zarembo made a remarkable discovery: the one-loop Dilatation Operator in the SO(6) sector of planar N=4 SYM can be identified with the Hamiltonian of an integrable spin chain. This one-loop Dilatation operator was obtained by computing a two-point correlation function at one loop, which is a completely off-shell quantity. Around the same time, Witten proposed a duality between N=4 SYM and twistor string theory, which initiated a revolution in the field of on-shell objects like scattering amplitudes. In this talk we illustrate that these techniques that have been sucessfully used for on-shell quantities can also be employed for the computation of off-shell quantities by computing the one-loop Dilatation Operator in the SO(6) sector. The first half of the talk will be dedicated to doing this calculation using MHV diagrams and the second half of the talk shows the computation in twistor space. 

These two short talks will be followed by an informal afternoon session for those interested in further details of these approaches, and in form factors in Class Room C2 from 2-4.30 pm then from 4.30pm in N3.12.  All are welcome.

In algebraic and arithmetic geometry, there is the ubiquitous notion of a moduli space, which informally is a variety (or scheme) parametrising a class of objects of interest. My aim in this talk is to explain concretely what we mean by a moduli space, going through the functor-of-points formalism of Grothendieck. Time permitting, I may also discuss (informally!) a natural obstruction to the existence of moduli schemes, and how one can get around this problem by taking a 2-categorical point of view.

 On November 25th 1915 Albert Einstein submitted his famous paper on the General Theory of Relativity. David Hilbert also derived the General Theory in November 1915 using quite different methods. In the same year Emmy Noether derived her remarkable ‘Noether’s Theorem’ which lies at the heart of much modern Physics. 1915 was a very good vintage indeed. We will take a brief walking tour of General Relativity using some of the ideas of Noether, Hilbert and Einstein to examine gravitational redshift, gravitational lensing, the impact of General Relativity on GPS systems and high precision atomic clocks, and Black holes all of which can be summarised by asking ‘how long is a piece of spacetime?’ 

Much of the arithmetic behaviour of an elliptic curve can be understood by examining its mod p reduction at some prime p. In this talk, we will aim to explain some of the ways we can define the mod p reduction, and the classifications of which reduction types occur.

Topics to be covered include the classical reduction types (good/multiplicative/additive), the Kodaira-Neron reduction types that refine them, and the Raynaud parametrisation of a semistable abelian variety. Time permitting, we may also discuss joint work with Vladimir Dokchitser classifying the semistable reduction types of 2-dimensional abelian varieties.

Abstract: We consider different notions of rough paths on manifolds and study some of the relations between these definitions. Furthermore, we explore extensions to manifolds modelled along infinite dimensional Banach spaces.

Abstract: We consider random walks and Lévy processes in the free nilpotent Lie group as rough paths. For any p > 1, we completely characterise (almost) all Lévy processes whose sample paths have finite p-variation, provide a Lévy-Khintchine formula for the characteristic function of the signature of a Lévy process treated as a rough path, and give sufficient conditions under which a sequence of random walks converges weakly to a Lévy process in rough path topologies. At the heart of our analysis is a criterion for tightness of p-variation for a collection of càdlàg strong Markov processes. We demonstrate applications of our results to weak convergence of stochastic flows.

It has been recently pointed out that maximal gauged supergravities in four dimensions often come in one-parameter families. The parameter measures the combination of electric and magnetic vectors that participate in the gauging. I will discuss the higher-dimensional origin of these dyonic gaugings, when the gauge group is chosen to be ISO(7). This gauged supergravity arises from consistent truncation of massive type IIA on the six-sphere, with its dyonically-gauging parameter identified with the Romans mass. The (AdS) vacua of the 4D supergravity give rise to new explicit AdS4 backgrounds of massive type IIA. I will also show that the 3D field theories dual to these AdS4 solutions are Chern-Simons-matter theories with a simple gauge group and level k also given by the Romans mass.

There is wide interest in the oceanographic and engineering communities as to whether linear models are satisfactory for describing the largest and steepest waves in open ocean. This talk will give some background on the topic before describing some recent modelling. This concludes that non-linear physics produces only small increases in amplitude over that expected in a linear model — however, there are significant changes to the shape and structure of extreme wave-group caused by the non-linear physics.

Simulating the long time and length scales associated with DNA self-assembly
and DNA nanotechnology is not currently feasible with models at an atomic level
of detail. We, therefore, developed oxDNA a coarse-grained representation of
DNA that aims to capture the fundamental structural, thermodynamic and
mechanical properties of double-stranded and single-stranded DNA, which we have
subsequently applied to study a wide variety of DNA biophysical properties and
DNA nanotechnological systems.

Lein’s confocal systems make accurate and precise measurements in many different applications. In applications where the object under test introduces variability and/or optical aberrations to the optical signal, the accuracy and precision may deteriorate. This technical challenge looks for mathematical solutions to improve the accuracy and precision of measurements made in such circumstances.

The presentation will outline the confocal principle, show “perfect” signals, give details of how we analyse such signals, then move on to less perfect signals and the effects on measurement accuracy and precision.

In the real, p-adic and motivic settings, we will present recent results on oscillatory integrals. In the reals, they are related to subanalytic functions and their Fourier transforms. In the p-adic and motivic case, there are furthermore transfer principles and applications in the Langlands program. This is joint work with Comte, Gordon, Halupczok, Loeser, Miller, Rolin, and Servi, in various combinations. 
 

I will discuss the theory of branched covers of cube complexes as a method of hyperbolisation. I will show recent results using this technique. Time permitting I will discuss a form of Morse theory on simplicial complexes and show how these methods combined with the earlier methods allow one to create groups with interesting finiteness properties.