Past

Let X be a smooth projective curve (of genus greater than or equal to 2) over C and M the moduli space of vector bundles over X, of rank 2 and with fixed determinant of degree 1.Then the Fourier-Mukai functor from the bounded derived category of coherent sheaves on X to that of M, given by the normalised Poincare bundle, is fully faithful, except (possibly) for hyperelliptic curves of genus 3,4,and 5

 This result is proved by establishing precise vanishing theorems for a family of vector bundles on the moduli space M.

 Results on the deformation  and inversion of Picard bundles (already known) follow from the full faithfulness of the F-M functor

In this talk we will explore the convergence of Krylov methods when used to solve $Lu = f$ where $L$ is an unbounded linear operator.  We will show that for certain problems, methods like Conjugate Gradients and GMRES still converge even though the spectrum of $L$ is unbounded. A theoretical justification for this behavior is given in terms of polynomial approximation on unbounded domains.    

Let $G(n,d)$ be a random $d$-regular graph on $n$ vertices. In 2004 Kim and Vu showed that if $d$ grows faster than $\log n$ as $n$ tends to infinity, then one can define a joint distribution of $G(n,d)$ and two binomial random graphs $G(n,p_1)$ and $G(n,p_2)$ -- both of which have asymptotic expected degree $d$ -- such that with high probability $G(n,d)$ is a supergraph of $G(n,p_1)$ and a subgraph of $G(n,p_2)$. The motivation for such a coupling is to deduce monotone properties (like Hamiltonicity) of $G(n,d)$ from the simpler model $G(n,p)$. We present our work with A. Dudek, A. Frieze and A. Rucinski on the Kim-Vu conjecture and its hypergraph counterpart.

Sparse matrices occur in numerical simulations throughout science and engineering. In particular, it is often desirable to solve systems of the form Ax=b, where A is a sparse matrix with 100,000+ rows and columns. The order that the rows and columns occur in can have a dramatic effect on the viability of a direct solver e.g., the time taken to find x, the amount of memory needed, the quality of x,... We shall consider symmetric matrices and, with the help of playdough, explore how best to order the rows/columns using a nested dissection strategy. Starting with a straightforward strategy, we will discover the pitfalls and develop an adaptive strategy with the aim of coping with a large variety of sparse matrix structures.

Some of the talk will involve the audience playing with playdough, so bring your inner child along with you!

By classical results of Mumford and Donagi, Mori-Mukai, Verra, the moduli spaces A_g of principally polarized abelian varieties of dimension g are unirational for g≤5 and are of general type for g≥7. Answering a conjecture of Kanev, we provide a uniformization of A6 by a Hurwitz space parameterizing certain curve covers. Using this uniformization, we study the geometry of A6 and make advances towards determining its birational type. This is a joint work with Donagi-Farkas-Izadi-Ortega.

I prove that if markets are weak-form efficient, meaning current prices fully reflect all information available in past prices, then P = NP, meaning every computational problem whose solution can be verified in polynomial time can also be solved in polynomial time. I also prove the converse by showing how we can "program" the market to solveNP-complete problems. Since P probably does not equal NP, markets are probably not efficient. Specifically, markets become increasingly inefficient as the time series lengthens or becomes more frequent. An illustration by way of partitioning the excess returns to momentum strategies based on data availability confirms this prediction.

For more info please visit: http://philipmaymin.com/academic-papers#pnp

The general problem of shock formation in three space dimensions was solved by Christodoulou in 2007. In his work also a complete description of the maximal development of the initial data is provided. This description sets up the problem of continuing the solution beyond the point where the solution ceases to be regular. This problem is called the shock development problem. It belongs to the category of free boundary problems but in addition has singular initial data because of the behavior of the solution at the blowup surface. In my talk I will present the solution to this problem in the case of spherical symmetry. This is joint work with Demetrios Christodoulou.

Many hard problems in Diophantine geometry have analogues over function fields which are less hard. I will give some examples.

*/ /*-->*/ Let G be a locally compact Hausdorff topological group. Examples are Lie groups, p-adic groups, adelic groups, and discrete groups. The BC (Baum-Connes) conjecture proposes an answer to the problem of calculating the K-theory of the convolution C* algebra of G. Validity of the conjecture has implications in several different areas of mathematics --- e.g. Novikov conjecture, Gromov-Lawson-Rosenberg conjecture, Dirac exhaustion of the discrete series, Kadison-Kaplansky conjecture. An expander is a sequence  of finite graphs which is efficiently connected. Any discrete group which contains an expander as a sub-graph of its Cayley graph is a counter-example to  the BC conjecture with coefficients. Such discrete groups have been constructed by Gromov-Arjantseva-Delzant and by Damian Osajda. This talk will indicate how to make a correction in BC with coefficients. There are no known counter-examples to the corrected conjecture, and all previously known confirming examples remain confirming examples.

We show how to get coarse bounds on the number of (non-simple) closed geodesics on a surface, given upper bounds on both length and self-intersection number. Recent work by Mirzakhani and by Rivin has produced asymptotics for the growth of the number of simple closed curves and curves with one self-intersection (respectively) with respect to length. However, no asymptotics, or even bounds, were previously known for other bounds on self-intersection number. Time permitting, we will discuss some applications of this result

It has been conjectured that the fundamental theory of strings and branes has an $E_{11}$ symmetry. I will explain how this conjecture  leads to  a generalised space-time,  which is automatically equipped with its own geometry, as well as equations of motion for the fields that live on this generalised space-time.

 

Shear Thickening fluids such as cornstarch and water show remarkable response under impact, which allows, for example, a person to run on the surface of the suspension. We perform constant velocity impact experiments along with imaging and particle tracking in a shear thickening fluid at velocities lower than 500 mm/s and suspension heights of a few cm. In this regime, where inertial effects are insignificant, we find that a solid-like dynamically jammed region with a propagating front is generated under impact. The suspension is able to support large stresses like a solid only when the front reaches the opposite boundary. These impact-activated fronts are generated only above a critical velocity. We construct a model by taking into account that sufficiently large stresses are generated when this solid like region spans to the opposite boundary and the work necessary to deform this solid like material dissipates the kinetic energy of the impacting object. The model shows quantitative agreement of the measured penetration depth using high speed video of a person running on cornstarch and water suspensions.

Data in many areas of science and sociology is now routinely represented in the form of networks. A fundamental task often required is to compare two datasets (networks) to assess the level of similarity between them. In the context of biological sciences, networks often represent either direct or indirect molecular interactions and an active research area is to assess the level of conservation of interaction patterns across species.

Currently biological network comparison software largely relies on the concept of alignment where close matches between the nodes of two or more networks are sought. These node matches are based on sequence similarity and/or interaction patterns. However, because of the incomplete and error-prone datasets currently available, such methods have had limited success. Moreover, the results of network alignment are in general not amenable for distance-based evolutionary analysis of sets of networks. In this talk I will describe Netdis, a topology-based distance measure between networks, which offers the possibility of network phylogeny reconstruction.

In a series of recent papers David Masser and Umberto Zannier proved the relative Manin-Mumford conjecture for abelian surfaces, at least when everything is defined over the algebraic numbers. In a further paper with Daniel Bertrand and Anand Pillay they have explained what happens in the semiabelian situation, under the same restriction as above.

At present it is not clear that these results are effective. I'll discuss joint work with Philipp Habegger and Masser and with Harry Schimdt in which we show that certain very special cases can be made effective. For instance, we can effectively compute a bound on the order of a root of unity t such that the point with abscissa 2 is torsion on the Legendre curve with parameter t.

**Note change of room**

We present a dynamic theory for time-inconsistent stopping problems. The theory is developed under the paradigm of expected discounted
payoff, where the process to stop is continuous and Markovian. We introduce equilibrium stopping policies, which are imple-mentable
stopping rules that take into account the change of preferences over time. When the discount function induces decreasing impatience, we
establish a constructive method to find equilibrium policies. A new class of stopping problems, involving equilibrium policies, is
introduced, as opposed to classical optimal stopping. By studying the stopping of a one-dimensional Bessel process under hyperbolic discounting, we illustrate our theory in an explicit manner.

I describe joint work with Alastair Irving in which we improve a result of
D.H.J. Polymath on the length of intervals in $[N,2N]$ that can be shown to
contain $m$ primes. Here $m$ should be thought of as large but fixed, while $N$
tends to infinity.
The Harman sieve is the key to the improvement. The preprint will be
available on the Math ArXiv before the date of the talk.

Can we extend the FEM to general polytopic, i.e. polygonal and polyhedral, meshes while retaining 
the ease of implementation and computational cost comparable to that of standard FEM? Within this talk, I present two approaches that achieve just  that (and much more): the Virtual Element Method (VEM) and an hp-version discontinuous Galerkin (dG) method.

The Virtual Element spaces are like the usual (polynomial) finite element spaces with the addition of suitable non-polynomial functions. This is far from being a novel idea. The novelty of the VEM approach is that it avoids expensive evaluations of the non-polynomial "virtual" functions by basing all 
computations solely on the method's carefully chosen degrees of freedom. This way we can easily deal 
with complicated element geometries and/or higher continuity requirements (like C1, C2, etc.), while 
maintaining the computational complexity comparable to that of standard finite element computations.

As you might expect, the choice and number of the degrees of freedom depends on such continuity 
requirements. If mesh flexibility is the goal, while one is ready to  give up on global regularity, other approaches can be considered. For instance, dG approaches are naturally suited to deal with polytopic meshes. Here I present an extension of the classical Interior Penalty dG method which achieves optimal rates of convergence on polytopic meshes even under elemental edge/face degeneration. 

The next step is to exploit mesh flexibility in the efficient resolution of problems characterised by 
complicated geometries and solution features, for instance within the framework of automatic FEM 
adaptivity. I shall finally introduce ongoing work in this direction.

I will present some recent results concerning the higher gradient integrability of

σ-harmonic functions u with discontinuous coefficients σ, i.e. weak solutions of

div(σ∇u) = 0. When σ is assumed to be symmetric, then the optimal integrability

exponent of the gradient field is known thanks to the work of Astala and Leonetti

& Nesi. I will discuss the case when only the ellipticity is fixed and σ is otherwise

unconstrained and show that the optimal exponent is attained on the class of

two-phase conductivities σ: Ω⊂R27→ {σ1,σ2} ⊂M2×2. The optimal exponent

is established, in the strongest possible way of the existence of so-called

exact solutions, via the exhibition of optimal microgeometries.

(Joint work with V. Nesi and M. Ponsiglione.)

I will report on some joint work with Jacob Tsimerman
concerning multiplicative relations among singular moduli.
Our results rely on the ``Ax-Schanuel'' theorem for the j-function
recently proved by us, which I will describe.

I will discuss various types of filling functions on topological spaces, stating some results in the area. I will then go onto prove that a finitely presented subgroup of a hyperbolic group of cohomological dimension 2 is hyperbolic. On the way I will prove a stronger result about filling functions of subgroups of hyperbolic groups of cohomological dimension $n$.

We review a class of systems of non-linear PDEs, derived from the Cahn--Hilliard and Ohta--Kawasaki functionals, that describe the energy evolution of diblock copolymers. These are long chain molecules that can self assemble into repeating patterns as they cool. We are particularly interested in finite element numerical methods that approximate these PDEs in the two-phase (in which we model the polymer only) and three-phase (in which we imagine the polymer surrounded by, and interacting with, a void) cases.

We present a brief derivation of the underlying models, review a class of numerical methods to approximate them, and showcase some early results from our codes.

The talk will present ongoing work on medical image reconstruction from x-ray scanners. A suitable method for reconstruction of these undersampled systems is compressed sensing. The presentation will show respective reconstruction methods and their analysis. Furthermore, work in progress about extensions of the standard approach will be shown.

[based on joint work with Li Guo and  Bin Zhang]

 We apply to  the study of exponential sums on lattice points in
convex rational polyhedral cones, the generalised algebraic approach of
Connes and Kreimer to  perturbative quantum field theory.  For this purpose
we equip the space of    cones   with a connected coalgebra structure.
The  algebraic Birkhoff factorisation of Connes and Kreimer   adapted  and
generalised to this context then gives rise to a convolution factorisation
of exponential sums on lattice points in cones. We show that this
factorisation coincides with the classical Euler-Maclaurin formula
generalised to convex rational polyhedral cones by Berline and Vergne by
means of  an interpolating holomorphic function.
We define  renormalised conical zeta values at non-positive integers as the
Taylor coefficients at zero of the interpolating holomorphic function.  When
restricted to Chen cones, this  yields yet another way to renormalise
multiple zeta values  at non-positive integers.

We consider the system of nonlinear differential equations
\begin{equation}
(1) \qquad
\begin{cases}
\dot u_n(t) + \lambda^{2n} u_n(t) 
- \lambda^{\beta n} u_{n-1}(t)^2 + \lambda^{\beta(n+1)} u_n(t) u_{n+1}(t) = 0,\\
u_n(0) = a_n, n \in \mathbb{N}, \quad \lambda > 1, \beta > 0.
\end{cases}
\end{equation}
In this talk we explain why this system is a model for the Navier-Stokes equations of hydrodynamics. The natural question is to find a such functional space, where one could prove the existence and the uniqueness of solution. In 2008, A. Cheskidov proved that the system (1) has a unique "strong" solution if $\beta \le 2$, whereas the "strong" solution does not exist if $\beta > 3$. (Note, that the 3D-Navier-Stokes equations correspond to the value $\beta = 5/2$.) We show that for sufficiently "good" initial data the system (1) has a unique Leray-Hopf solution for all $\beta > 0$.

We will give a sketch overview of Scholze's theory of perfectoid spaces and the tilting equivalence, starting from Huber's geometric approach to valuation theory. Applications to weight-monodromy and p-adic Hodge theory we will only hint at, preferring instead to focus on examples which illustrate the philosophy of tilting equivalence.
 

Representations of free loop groups possess an operation, akin to
tensor product, under which they form a braided tensor category. I
will discuss a similar operation, which is present on the category of
representations of the based loop groups, and which equips it with the
structure of a monoidal cateogory. Finally, I will present a recent
result, according to which the Drinfel'd centre of the category of
representations of a based loop group is equivalent to the category of
representations of the corresponding free loop group.

: Estimating volatility from recent high frequency data, we revisit the question of the smoothness of the volatility process. Our main result is that log-volatility behaves essentially as a fractional Brownian motion with Hurst exponent H of order 0.1, at any reasonable time scale.

This leads us to adopt the fractional stochastic volatility (FSV) model of Comte and Renault.

We call our model Rough FSV (RFSV) to underline that, in contrast to FSV, H<1/2.

We demonstrate that our RFSV model is remarkably consistent with financial time series data; one application is that it enables us to obtain improved forecasts of realized volatility.

Furthermore, we find that although volatility is not long memory in the RFSV model, classical statistical procedures aiming at detecting volatility persistence tend to conclude the presence of long memory in data generated from it.

This sheds light on why long memory of volatility has been widely accepted as a stylized fact.

Finally, we provide a quantitative market microstructure-based foundation for our findings, relating the roughness of volatility to high frequency trading and order splitting.

This is joint work with Jim Gatheral and Thibault Jaisson.

Form factors form a bridge between the purely on-shell amplitudes and the purely off-shell correlation functions. In this talk, we study the form factors of general gauge-invariant local composite operators in N=4 SYM theory via on-shell methods. At tree-level and for a minimalnumber of external fields, the form factor exactly realises the spin-chain picture of N=4 SYM theory in the language of scattering amplitudes. Via generalised unitarity, we obtain the cut-constructible part of the one-loop correction to the minimal form factor of a generic operator. Its UV divergence yields the complete one-loop dilatation operator of the theory. At two-loop order, we employ unitarity to calculate the minimal form factors and thereby the dilatation operator for the Konishi primary operator and all operators in the SU(2) sector. For the former operator as well as other non-protected operators, important subtleties arise which require an extension of the method of unitarity.

We will discuss 2d Euler and Boussinesq (incompressible) flows related to a possible boundary blow-up scenario for the 3d axi-symmetric case suggested by G. Luo and T. Hou, together with some easier model problems relevant for that situation.

Classically, basin modelling is undertaken with very little a priori knowledge. Alongside the challenge of improving the general fidelity and utility of the modelling systems, is the challenge of constraining these systems with unknowns and uncertainties in such a way that models (and derived simulation results) can be readily regenerated/reevaluated in the light of new empirical data obtained during the course of exploration, development and production activities.

A result of Jeffrey Brock states that, given a hyperbolic 3-manifold which is a mapping torus over a surface $S$, its volume can be expressed in terms of the distance induced by the monodromy map in the pants graph of $S$. This is an abstract graph whose vertices are pants decompositions of $S$, and edges correspond to some 'elementary alterations' of those.
I will show how this theorem gives an estimate for the volume of hyperbolic complements of closed braids in the solid torus, in terms of braid properties. The core piece of such estimate is a generalization of a result of Masur, Mosher and Schleimer that train track splitting sequences (which I will define in the talk) induce quasi-geodesics in the marking graph.

In this talk, we aim to provide a valuation framework for counterparty credit risk based on a structural default model which incorporates jumps and dependence between the assets of interest. In this framework default is caused by the firm value falling below a prespecified threshold following unforeseeable shocks, which deteriorate its liquidity and ability to meet its liabilities. The presence of dependence between names captures wrong-way risk and right-way risk effects. The structural model traces back to Merton (1974), who considered only the possibility of default occurring at the maturity of the contract; first passage time models starting from the seminal contribution of Black and Cox (1976) extend the original framework to incorporate default events at any time during the lifetime of the contract. However, as the driving risk process used is the Brownian motion, all these models suffers of vanishing credit spreads over the short period - a feature not observed in reality. As a consequence, the Credit Value Adjustment (CVA) would be underestimated for short term deals as well as the so-called gap risk, i.e. the unpredictable loss due to a jump event in the market. Improvements aimed at resolving this issue include for example random default barriers, time dependent volatilities, and jumps. In this contribution, we adopt Lévy processes and capture dependence via a linear combination of two independent Lévy processes representing respectively the systematic risk factor and the idiosyncratic shock. We then apply this framework to the valuation of CVA and DVA related to equity contracts such as forwards and swaps. The main focus is on the impact of correlation between entities on the value of CVA and DVA, with particular attention to wrong-way risk and right-way risk, the inclusion of mitigating clauses such as netting and collateral, and finally the impact of gap risk. Particular attention is also devoted to model calibration to market data, and development of adequate numerical methods for the complexity of the model considered.

 Let $X$ be a smooth cubic hypersurface of dimension $m$ defined over a global field $K$. A conjecture of Colliot-Thelene(02) states that $X$ satisfies the Hasse Principle and Weak approximation as long as $m\geq 3$. We use a global version of Hardy-Littlewood circle method along with the theory of global $L$-functions to establish this for $m\geq 6$, in the case $K=\mathbb{F}_q(t)$, where $\text{char}(\mathbb{F}_{q})> 3$.

In this seminar I will present a semi-langrangian discretisation of the Monge-Ampère operator, which is of interest in optimal transport 
and differential geometry as well as in related fields of application.

I will discuss the proof of convergence to viscosity solutions. To address the challenge of uniqueness and convexity we draw upon the classical relationship with Hamilton-Jacobi-Bellman equations, which we extend to the viscosity setting. I will explain that the monotonicity of semi-langrangian schemes implies that they possess large stencils, which in turn requires careful treatment of the boundary conditions.

The contents of the seminar is based on current work with X Feng from the University of Tennessee.

Joint work with Anders Hansen (Cambridge), Olavi Nevalinna (Aalto) and Markus Seidel (Zwickau).             

Expansion, rank gradient and virtual splitting are all concepts of great interest in asymptotic group theory. We discuss a result of Marc Lackenby which demonstrates a surprising relationship between then, and give examples exhibiting different combinations of asymptotic behaviour.

When the Computing Laboratory discarded its hardcopy journals around 2008, I kept the first ten years or so of each of six classic numerical analysis journals, starting from volume 1, number 1.  This will not be a seminar in the usual sense but a mutual exploration.  Come prepared to look through a few of these old volumes yourself and perhaps to tell the group of something interesting you find.  Bring a pen and paper.  All are welcome.

Mathematics of Computation, from 1943
SIAM Journal, from 1953
Numerische Mathematik, from 1959
BIT, from 1961
SIAM Journal on Numerical Analysis, from 1964
Journal of the IMA, from 1965

We look at the construction of radial metrics with an isolated singularity for the constant fractional curvature equation. This is a semilinear, non-local equation involving the fractional Laplacian, and appears naturally in conformal geometry. 

I will review Bott's classical periodicity result about topological K-theory (with period 2 in the case of complex K-theory, and period 8 in the case of real K-theory), and provide an easy sketch of proof, based on the algebraic periodicity of Clifford algebras. I will then introduce the `higher real K-theory' of Hopkins and Miller, also known as TMF. I'll discuss its periodicity (with period 576), and present a conjecture about a corresponding algebraic periodicity of `higher Clifford algebras'. Finally, applications to physics will be discussed.