Past

Given a Mori Dream Space X, we construct via tropicalisation a model dominating all the small Q-factorial modifications of X. Via this construction we recover a Minkowski basis for the Newton-Okounkov bodies of Cartier divisors on X and hence generators of the movable cone of X. 
This is joint work with Stefano Urbinati.
 

In the ongoing programme to classify noncommutative projective surfaces (connected graded noetherian domains of Gelfand-Kirillov dimension three) a natural question is:  what are the minimal models within a birational class?  It is not even clear a priori what the correct definition is of a minimal model in this context.

We show that a generic Sklyanin algebra (a noncommutative analogue of P^2) satisfies the surprising property that it has no birational connected graded noetherian overrings, and explain why this is a reasonable definition of 'minimal model.' We show also that the noncommutative versions of P^1xP^1 and of the Hirzebruch surface F_2 are minimal.
This is joint work in progress with Dan Rogalski and Toby Stafford.

 

I shall report on recent progress, in Australia and Germany, on the empirical evaluation of special values of L-functions by minors of period matrices whose elements include Feynman integrals from diagrams with up to 20 loops. Previously such relations were known only for diagrams with up to 6 loops.
 

For mapping class groups of surfaces it is well-understood that their homology stability is closely related to the fact that they give rise to an infinite loop space. Indeed, they define an operad whose algebras group complete to infinite loop spaces.

In recent work with Basterra, Bobkova, Ponto and Yaekel we define operads with homology stability (OHS) more generally and prove that they are infinite loop space operads in the above sense. The strong homology stability results of Galatius and Randal-Williams for moduli spaces of manifolds can be used to construct examples of OHSs. As a consequence the map to K-theory defined by the action of the diffeomorphisms on the middle dimensional homology can be shown to be a map of infinite loop spaces.

If a dynamical system has a conservation law, i.e. a constant along the trajectory of the motion, the study of its evolution along the trajectories of a perturbed system becomes interesting. Conservation laws can be seen everywhere, especially at the level of probability distributions of a reduced dynamic.  We explain this with a number of models, in which we see a singular perturbation problem and identify a conservation law, the latter is used to seek out the correct scale to work with and to reduce the complexity of the system. The reduced dynamic consists of a family of  ODEs with rapidly oscillating right hands side from which in the limit we obtain a Markov process. For stochastic completely integrable system, the limit describes the evolution of the level sets of the family of Hamiltonian functions over a very large time scale.

The Yang-Mills heat equation is the gradient flow corresponding to the Yang-Mills functional. It was initially introduced by S. K. Donaldson to study the existence of irreducible Yang-Mills connections on the projective plane. In this talk, we will consider this equation over compact three-manifolds with boundary. It is a nonlinear weakly parabolic equation, but we will see how one can prove long-time existence and uniqueness of solutions by gauge symmetry breaking. We will also demonstrate some strong regularization results for the solution and see how they lead to detailed short-time asymptotic estimates, as well as the long-time convergence of the Wilson loop functions. 

Mean Curvature Flow (MCF) is a canonical way to deform sub-manifolds to minimal sub-manifolds. It also improves the geometric properties of sub-manifolds along the flow. The condition of being Lagrangian is preserved for smooth solutions of MCF in a Kahler-Einstein manifold. We call it Lagrangian mean curvature flow (LMCF) when requires slices of the flow to be Lagrangian.

Unfortunately, singularities may occur and cause obstructions to continue MCF in general. It is thus very important to understand the singularities, particularly isolated singularities of the flow. Isolated singularity models on soliton solutions that include self-similar solutions and translating solutions. In this talk, I will report some of my work with my collaborators on studying singularities of LMCF. It includes soliton solutions with different important properties and an in-progress joint project with Dominic Joyce that aims to understand how singularities form and construct examples to demonstrate these behaviours.

We begin by reviewing the “Gravity = Gauge x Gauge” paradigm that has emerged over the last decade. In particular, we will consider the origin of gravitational scattering amplitudes, symmetries and classical solutions in terms of the product of two Yang-Mills theories. Motivated by these developments we begin to address the classification of gravitational theories admitting a “factorisation” into a product of gauge theories. Progress in this direction leads us to twin supergravity theories - pair of supergravities with distinct supersymmetries, but identical bosonic sectors - from the perspective of Yang-Mills squared. 

Do you find yourself agreeing to things when actually you want more – or less? In this session we will look at how to be clear about what you want, and how to use assertiveness and negotiation skills and strategies to achieve win-win outcomes when working with others. 

Many northern hemisphere climate records show a series of rapid climate changes - Dansgaard-Oesgher (D-O) cycles - that recurred on centennial to millennial timescales throughout most of the last glacial period.  They consist of sudden warming jumps of order 10°C, followed generally by a slow cooling lasting a few centuries, and then a rapid temperature drop into a cold period of similar length.  Most explanations for D-O events call on changes in the strength of the Atlantic meridional overturning circulation (AMOC), but the mechanism for triggering and pacing such changes is uncertain. Changes in freshwater delivery to the ocean are assumed to be important. 

Here, we investigate whether the proposed AMOC changes could have occurred as part of a natural relaxation oscillation, in which runoff from the northern hemisphere ice sheets varies in response to each warming and cooling event, and in turn provides the freshwater delivery that controls the ocean circulation.  In this mechanism the changes are buffered and paced by slow changes in salnity of the Arctic ocean.  We construct a simple model to investigate whether the timescales and magnitudes make this a viable mechanism.  

Oxford Nanopore Technologies aim to enable the analysis of any living thing, by any person, in any environment. The world's first and only nanopore DNA
sequencer, the MinION is a portable, real time, long-read, low cost device that has been designed to bring easy biological analyses to anyone, whether in
scientific research, education or a range of real world applications such as disease/pathogen surveillance, environmental monitoring, food chain
surveillance, self-quantification or even microgravity biology. Gordon will talk the about the technology, applications and future direction.
Stuart will talk about the nanopore signal, computational methods and informatics challenges associated with reading DNA directly.

Speaker: Yixuan Wang
Titile: Minimum resting time with market orders
Abstract:  Regulators have been discussing possible rules to control high frequency trading and decrease market speed, and minimum resting time is one of them. We develop a simple mathematical model, and derive an asymptotic expression of the expected PnL, which is also the performance criteria that a market maker would like to maximize by choosing the optimal depth at which she posts the limit order. We investigate the comparative statistics of the optimal depth with each parameters, an in particular the comparative statistics show that the minimum resting time will decrease the market liquidity, forcing the market makers to post limit orders of volume 1.


Speaker: Marco Pangallo
Title: Does learning converge in generic games?
Abstract: In game theory, learning has often been proposed as a convincing method to achieve coordination on an equilibrium. But does learning converge, and to what? We start investigating the drivers of instability in the simplest possible non-trivial setting, that is generic 2-person, 2-strategy normal form games. In payoff matrices with a unique mixed strategy equilibrium the players may follow the best-reply cycle and fail to converge to the Nash Equilibrium (NE): we rather observe limit cycles or low-dimensional chaos. We then characterize the cyclic structure of games with many moves as a combinatorial problem: we quantify exactly how many best-reply configurations give rise to cycles or to NE, and show that acyclic (e.g. coordination, potential, supermodular) games become more and more rare as the number of moves increases (a fortiori if the payoffs are negatively correlated and with more than two players).  In most games the learning dynamics ends up in limit cycles or high-dimensional chaotic attractors, preventing the players to coordinate. Strategic interactions would then be governed by learning in an ever-changing environment, rather than by rational and fully-informed equilibrium thinking.
Collaborators: J. D. Farmer, T. Galla, T. Heinrich, J. Sanders

to come

Using some new results in probabilistic recognition of black box groups
(Joint work with Sukru Yalcinkaya), I will discuss some emerging
structures that beg for a model-theoretic analysis.

In this talk on joint work with REBECCA BELLOVIN we discuss the “local ε-isomorphism” conjecture of Fukaya and Kato for (crystalline) families of G_{Q_p}-representations. This can be regarded as a local analogue of the global Iwasawa main conjecture for families, extending earlier work of Kato for rank one modules, of Benois and Berger for crystalline representations with respect to the cyclotomic extension as well as of Loeffler, Venjakob and Zerbes for crystalline representations with respect to abelian p-adic Lie extensions of Q_p. Nakamura has shown Kato’s - conjecture for (ϕ,\Gamma)-modules over the Robba ring, which means in particular only after inverting p, for rank one and trianguline families. The main ingredient of (the integrality part of) the proof consists of the construction of families of Wach modules generalizing work of Wach and Berger and following Kisin’s approach via a corresponding moduli space.
 

We consider impatient decision makers when their assets' prices are in undesirable low regions for a significant amount of time, and they are risk averse to negative price jumps. We wish to study the unusual reactions of investors under such adverse market conditions. In mathematical terms, we study the optimal exercising of an American call option in a random time-horizon under spectrally negative Lévy models. The random time-horizon is modeled by an alarm of the so-called Omega default clock in insurance, which goes off when the cumulative amount of time spent by the asset price in an undesirable low region exceeds an independent exponential random time. We show that the optimal exercise strategies vary both quantitatively and qualitatively with the levels of impatience and nervousness of the investors, and we give a complete characterization of all optimal exercising thresholds. 

In variational imaging and other inverse problem modeling, regularisation plays a major role. In recent years, high order regularizers such as the total generalised variation, the mean curvature and the Gaussian curvature are increasingly studied and applied, and many improved results over the widely-used total variation model are reported.
Here we first introduce the fractional order derivatives and the total fractional-order variation which provides an alternative  regularizer and is not yet formally analysed. We demonstrate that existence and uniqueness properties of the new model can be analysed in a fractional BV space, and, equally, the new model performs as well as the high order regularizers (which do not yet have much theory). 
In the usual framework, the algorithms of a fractional order model are not fast due to dense matrices involved. Moreover, written in a Bregman framework, the resulting Sylvester equation with Toeplitz coefficients can be solved efficiently by a preconditioned solver. Further ideas based on adaptive integration can also improve the computational efficiency in a dramatic way.
 Numerical experiments will be given to illustrate the advantages of the new regulariser for both restoration and registration problems.
 

Using results by Eisenbud, Schoutens and Zilber I will propose a model theoretic structure that aims to capture the algebra (or geometry) of a non reduced scheme over an algebraically closed field. 

In this seminar, we present a fast fully homomorphic encryption (FHE) based on GSW and its ring variants. The cryptosystem relies on the hardness of lattice problems in the unique domain (e.g. the LWE family). After a brief presentation of these lattice problems, with a few notes on their asymptotic and practical average case hardness, we will present our homomorphic cryptosystem TFHE, based on a ring variant of GSW. TFHE can operate in two modes: The first one is a leveled homomorphic mode, which has the ability to evaluate deterministic automata (or branching programs) at a rate of 1 transition every 50microseconds. For the second mode, we also show that this scheme can evaluate its own decryption in only 20milliseconds, improving on the the construction by Ducas-Micciancio, and of Brakerski-Perlman. This makes the scheme fully homomorphic by Gentry's bootstrapping principle, and for instance, suitable for representing fully dynamic encrypted databases in the cloud.

An expander is a family of finite graphs of uniformly bounded degree, increasing number of vertices and Cheeger constant bounded away from zero. They occur throughout mathematics and computer science; the most famous constructions of expanders rely on powerful results in geometric group theory and number theory, while expanders are used in everything from error-correcting codes, through disproving the strongest version of the Baum-Connes conjecture, to affine sieve theory and the twin prime, Mersenne prime and Hardy-Littlewood conjectures.

However, very little was known about how different the geometry of two expanders could be. This question was raised by Ostrovskii in 2013, and a year later Mendel and Naor gave the first example of two 'distinct' expanders.

In this talk I will construct a continuum of expanders which are, in a certain sense, geometrically incomparable. Once the existence of a single expander is accepted, the remainder of the proof is a heady mix of counting, addition, multiplication, and just for the experts, a little bit of division. Two very different - and very interesting - continuums of 'distinct' expanders have since been constructed by Khukhro-Valette and Das.