Past

When firms want to buy back their own shares, they often use the services of investment banks through ASR contracts. ASR contracts are execution contracts including exotic option characteristics (an Asian-type payoff and Bermudian/American exercise dates). In this talk, I will present the different types of ASR contracts usually encountered, and I will present a model in order to (i) price ASR contracts and (ii) find the optimal execution strategy for each type of contract. This model is inspired from the classical (Almgren-Chriss) literature on optimal execution and uses classical ideas from option pricing. It can also be used to price options on illiquid assets. Original numerical methods will be presented.

1. Calibration and Pricing of Financial Derivatives using Forward PDEs (Mariapragassam)

Nowadays, various calibration techniques are in use in the financial industry and the exact re-pricing of call options is a must-have standard. However, practitioners are increasingly interested in taking into account the quotes of other derivatives as well.
We describe our approach to the calibration of a specific Local-Stochastic volatility model proposed by the FX group at BNP Paribas. We believe that forward PDEs are powerful tools as they allow to achieve stable and fast best-fit routines. We will expose our current results on this matter.

Joint work with Prof. Christoph Reisinger

2. Infinite discrete-time investment and consumption problem (Li)

We study the investment and consumption problem in infinite discrete-time framework. In our problem setting, we do not need the wealth process to be positive at any time point. We first analyze the time-consistent case and give the convergence of value function for infinite-horizon problem by value functions of finite-horizon problems.

Then we discuss the time-consistent case, and hope the value functions of finite-horizon problems will still converge to the infinite-horizon problem.

I'll discuss work (part with Savitt, part with Dembele and Roberts) on two related questions: describing local factors at primes over p in mod p automorphic representations, and describing reductions of local crystalline Galois representations with prescribed Hodge-Tate weights.

In this talk we want to discuss results by Dimca, Papadima, and Suciu about the finiteness properties of Kähler groups. Namely, we will sketch their proof that for every $2\leq n\leq \infty$ there is a Kähler group with finiteness property $\mathcal{F}_n$, but not $FP_{n+1}$. Their proof is by explicit construction of examples. These examples all arise as subgroups of finite products of surface groups and they are the first known examples of Kähler groups with arbitrary finiteness properties. The talk does not require any prior knowledge of finiteness properties or of Kähler groups.

LiFePO4 is a commercially available battery material with good theoretical discharge capacity, excellent cycle life and increased safety compared with competing Li-ion chemistries. During discharge, LiFePO4 material can undergo phase separation, between a highly and lowly lithiated form. Discharge of LiFePO4 crystals has traditionally been modelled by one-phase Stefan problems, which assume that phase separation occurs.

Recent work has been using phase-field models based on the Cahn-Hilliard equation, which only phase-separates when thermodynamically favourable. In the past year or two, this work has been having considerable impact in both theoretical and experimental electrochemistry.

Unfortunately, these models are very difficult to solve numerically and involve large, coupled systems of nonlinear PDEs across several different size scales that include a range of different physics and cannot be homogenised effectively.

This talk will give an overview of recent developments in modelling LiFePO4 and the sort of strategies used to solve these systems numerically.

I shall show that for every conjugacy class O in a connected semisimple algebraic group G over an algebraically closed field of characteristic good for G one can find a special transversal slice S to the set of conjugacy classes in G such that O intersects S and dim O=codim S. The construction of the slice utilizes some new combinatorics related to invariant planes for the action of Weyl group elements in the reflection representation. The condition dim O=codim S is checked using some new mysterious results by Lusztig on intersection of conjugacy classes in algebraic groups with Bruhat cells.

We outline a path from polynomial numerical hulls to multicentric calculus for evaluating f(A). Consider
$$Vp(A) = {z ∈ C : |p(z)| ≤ kp(A)k}$$
where p is a polynomial and A a bounded linear operator (or matrix). Intersecting these sets over polynomials of degree 1 gives the closure of the numerical range, while intersecting over all polynomials gives the spectrum of A, with possible holes filled in.
Outside any set Vp(A) one can write the resolvent down explicitly and this leads to multicentric holomorphic functional calculus.
The spectrum, pseudospectrum or the polynomial numerical hulls can move rapidly in low rank perturbations. However, this happens in a very controlled way and when measured correctly one gets an identity which shows e.g. the following: if you have a low-rank homotopy between self-adjoint and quasinilpotent, then the identity forces the nonnormality to increase in exact compensation with the spectrum shrinking.
In this talk we shall mention how the multicentric calculus leads to a nontrivial extension of von Neumann theorem
$$kf(A)k ≤ sup |z|≤1
kf(z)k$$
where A is a contraction in a Hilbert space, and conclude with some new results on (nonholomorphic) functional calculus for operators for which p(A) is normal at a nontrivial polynomial p. Notice that this is always true for matrices.

"We consider certain distinguished extensions of the field F_p((t)) of formal Laurent series over F_p, and look at questions about their model theory and Galois theory, with a particular focus on decidability."

Viktor Mayer-Schönberger is Professor of Internet Governance and Regulation at the University of Oxford's Internet Institute. He is also a faculty affiliate of Harvard's Belfer Center for Science and International Affairs. Together with Kenneth Cukier he is the co-author of the international bestseller Big Data.

Dinits, Karzanov and Lomonosov showed that the minimal edge cuts of a finite graph have the structure of a cactus, a tree-like graph constructed from cycles. Evangelidou and Papasoglu extended this to minimal cuts separating the ends of an infinite graph. In this talk we will discuss a similar structure theorem for minimal vertex cuts separating the ends of a graph; these can be encoded by a succulent, a mild generalization of a cactus that is still tree-like.

Counterexamples to Vaught's Conjecture regarding the number of countable
models of a theory in a logical language, may felicitously be studied by investigating a tree
of types of different arities and belonging to different languages. This
tree emerges from a category of topological spaces, and may be studied as such, without
reference to the original logic. The tree has an intuitive character of absoluteness
and of self-similarity. We present theorems expressing these ideas, some old and some new.

Stallings' folding technique lets us factor a map of graphs as a sequence of "folds" (edge identifications) followed by an immersion. We will show how this technique gives an algorithm to express a free-group automorphism as the product of Whitehead automorphisms (and hence Nielsen transformations), as well as proving finite generation for some subgroups of the automorphism group of a free group.

http://hottformath.wikispaces.com

Note: joint with Algebra seminar.

String algebras are tame - their finite-dimensional representations have been classified - and the Auslander-Reiten quiver of such an algebra shows some of the morphisms between them.  But not all.  To see the morphisms which pass between components of the Auslander-Reiten quiver, and so obtain a more complete picture of the category of representations, we should look at certain infinite-dimensional representations and use ideas and techniques from the model theory of modules.

This is joint work with Rosie Laking and Gena Puninski:
G. Puninski and M. Prest,  Ringel's conjecture for domestic string algebras, arXiv:1407.7470;
R. Laking, M. Prest and G. Puninski, Krull-Gabriel dimension of domestic string algebras, in preparation.

String algebras are tame - their finite-dimensional representations have been classified - and the Auslander-Reiten quiver of such an algebra shows some of the morphisms between them.  But not all.  To see the morphisms which pass between components of the Auslander-Reiten quiver, and so obtain a more complete picture of the category of representations, we should look at certain infinite-dimensional representations and use ideas and techniques from the model theory of modules.

This is joint work with Rosie Laking and Gena Puninski:
G. Puninski and M. Prest,  Ringel's conjecture for domestic string algebras, arXiv:1407.7470;
R. Laking, M. Prest and G. Puninski, Krull-Gabriel dimension of domestic string algebras, in preparation.

Traditionally threshold partial pivoting is used to ensure stability of sparse LDL^T factorizations of symmetric matrices. This involves comparing a candidate pivot with all entries in its row/column to ensure that growth in the size of the factors is limited by a threshold at each stage of the factorization. It is capabale of delivering a scaled backwards error on the level of machine precision for practically all real world matrices. However it has significant flaws when used in a massively parallel setting, such as on a GPU or modern supercomputer. It requires all entries of the column to be up-to-date and requires an all-to-all communication for every column. The latter requirement can be performance limiting as the factorization cannot proceed faster than k*(communication latency), where k is the length of the longest path in the sparse elimination tree.

We introduce a new family of communication-avoiding pivoting techniques that reduce the number of messages required by a constant factor allowing the communication cost to be more effectively hidden by computation. We exhibit two members of this family. The first deliver equivalent stability to threshold partial pivoting, but is more pessimistic, leading to additional fill in the factors. The second provides similar fill levels as traditional techniques and, whilst demonstrably unstable for pathological cases, is able to deliver machine accuracy on even the worst real world examples.

More than twenty years ago Erdős conjectured that a triangle-free graph $G$ of chromatic number $k$ contains cycles of at least $k^{2−o(1)}$ different lengths. In this talk we prove this conjecture in a stronger form, showing that every such $G$ contains cycles of $ck^2\log k$ consecutive lengths, which is tight. Our approach can be also used to give new bounds on the number of different cycle lengths for other monotone classes of $k$-chromatic graphs, i.e.,  clique-free graphs and graphs without odd cycles.

Joint work with A. Kostochka and J. Verstraete.

The relationship of diagonal dominance ideas to the convergence of stationary iterations is well known. There are a multitude of situations in which such considerations can be used to guarantee convergence when the splitting matrix (the preconditioner) is positive definite. In this talk we will describe and prove sufficient conditions for convergence of a stationary iteration based on a splitting with an indefinite preconditioner. Simple examples covered by this theory coming from Optimization and Economics will be described.

This is joint work with Michael Ferris and Tom Rutherford

We will discuss the relation between perturbative gauge theory and
perturbative gravity, and look at how this relation extends to some exact
classical solutions. First, we will review the double copy prescription that
takes gauge theory amplitudes into gravity amplitudes, which has been
crucial to progress in perturbative studies of supergravity. Then, we will
see how the relation between the two theories can be made manifest when we
restrict to the self-dual sector, in four dimensions. A key role is played
by a kinematic algebraic structure mirroring the colour structure, which can
be extended from the self-dual sector to the full theory, in any number of
dimensions. Finally, we will see how these ideas can be applied also to some
exact classical solutions, namely black holes and plane waves. This leads to
a relation of the type Schwarzschild as (Coulomb charge)^2.

We consider the two-phase Stefan problem with p-degenerate diffusion, p larger than two, and we prove continuity up to the boundary for weak solutions, providing a modulus of continuity which we conjecture to be optimal. Since our results are proven in the form of a priori estimates for appropriate regularized problems, as corollary we infer the existence of a globally continuous weak solution for continuous Cauchy-Dirichlet datum.

Consider a nonsingular projective variety $X$ defined by a system of $R$ forms of the same degree $d$. The circle method proves the Hasse principle and Manin's conjecture for $X$ when $\text{dim}X > C(d,R)$. I will describe how to improve the value of $C$ when $R$ is large. I use a technique for estimating mean values of exponential sums which I call a ``moat lemma". This leads to a novel and intriguing system of auxiliary inequalities.

In the Erdös-Rényi random graph process, starting from an empty graph, in each step a new random edge is added to the evolving graph. One of its most interesting features is the `percolation phase transition': as the ratio of the number of edges to vertices increases past a certain critical density, the global structure changes radically, from only small components to a single giant component plus small ones.

In this talk we consider Achlioptas processes, which have become a key example for random graph processes with dependencies between the edges.

Starting from an empty graph these proceed as follows: in each step two potential edges are chosen uniformly at random, and using some rule one of them is selected and added to the evolving graph. We discuss why, for a large class of rules, the percolation phase transition is qualitatively comparable to the classical Erdös-Rényi process.

Based on joint work with Oliver Riordan.

Given a 3-dimensional TQFT, the "conformal blocks" are the
values of that TQFT on closed Riemann surfaces.
The construction that we'll present (joint work with Douglas &
Bartels) takes as only input the value of the TQFT on discs. Towards
the end, I will explain to what extent the conformal blocks that we
construct agree with the conformal blocks constructed e.g. from the
theory of vertex operator algebras.

We give a necessary and sufficient condition for a map defined on a compact, quasiconvex and simply-connected space to factor through a tree. This condition can be checked using currents. In particular if the target is some Euclidean space and the map is H\"older continuous with exponent bigger than 1/2, such maps can be characterized by the vanishing of some integrals over the winding number. Moreover, this shows that if the target is the Heisenberg group equipped with the Carnot-Carath\'eodory metric and the H\"older exponent of the map is bigger than 2/3, the map factors through a tree.

I will discuss the implementation of explicit stabilisation of all closed string moduli in fluxed type IIB Calabi-Yau compactifications with chiral matter.  Using toric geometry we construct Calabi-Yau manifolds with del Pezzo singularities. D-branes located at such singularities can support the Standard Model gauge group and matter content. We consider Calabi-Yau manifolds with a discrete symmetry that reduces the effective number of complex structure moduli, which allows us to calculate the corresponding periods and find explicit flux vacua. We compute the values of the flux superpotential and the string coupling at these vacua. Starting from these explicit complex structure solutions, we obtain AdS and dS minima where the Kaehler moduli are stabilised by a mixture of D-terms, non-perturbative and perturbative alpha'-corrections as in the LARGE Volume Scenario.

Struggling for ideas at the weekends? Learn how to draw with colour and discover the creation of colour from our natural environment. Have a go at making your own natural paint colours. Create your own mini planet inspired by alchemy and the Radcliffe Observatory. Paint making demonstrations throughout the day with artist Nabil Al. All materials provided. Suitable for all ages from 6 to 60. Invite your friends.

We will present results from studies of the impact of the non-slow (typically fast) components of a rotating, stratified flow on its slow dynamics. We work in the framework of fast singular limits that derives from the work of Bogoliubov and Mitropolsky [1961], Klainerman and Majda [1981], Shochet [1994], Embid and Ma- jda [1996] and others.

In order to understand how the flow approaches and interacts with the slow dynamics we decompose the full solution, where u is a vector of all the unknowns, as

u = u α + u ′α where α represents the Ro → 0, F r → 0 or the simultaneous limit of both (QG for

quasi-geostrophy), with

P α u α = u α    P α u ′α = 0 ,

and where Pαu represents the projection of the full solution onto the null space of the fast operator. We use this decomposition to find evolution equations for the components of the flow (and the corresponding energy) on and off the slow manifold.

Numerical simulations indicate that for the geometry considered (triply periodic) and the type of forcing applied, the fast waves act as a conduit, moving energy onto the slow manifold. This decomposition clarifies how the energy is exchanged when either the stratification or the rotation is weak. In the quasi-geostrophic limit the energetics are less clear, however it is observed that the energy off the slow manifold equilibrates to a quasi-steady value.

We will also discuss generalizations of the method of cancellations of oscillations of Schochet for two distinct fast time scales, i.e. which fast time scale is fastest? We will give an example for the quasi-geostrophic limit of the Boussinesq equations.

At the end we will briefly discuss how understanding the role of oscillations has allowed us to develop convergent algorithms for parallel-in-time methods.

Beth A. Wingate - University of Exeter

Jared Whitehead - Brigham Young University

Terry Haut - Los Alamos National Laboratory

1. A Hybrid Monte-Carlo Partial Differential Solver for Stochastic  Volatility Models (Cozma)

In finance, Monte-Carlo and Finite Difference methods are the most popular approaches for pricing options. If the underlying asset is modeled by a multidimensional system of stochastic differential equations, an analytic solution is rarely available and working under a given computational budget comes at the cost of accuracy. The mixed Monte-Carlo partial differential solver introduced by Loeper and Pironneau (2009) is one way to overcome this issue and we investigate it thoroughly for a number of stochastic volatility models. Our main concern is to provide a rigorous mathematical proof of the convergence of the hybrid method under different frameworks, which in turn justifies the use of Monte-Carlo simulations to compute the expected discounted payoff of the financial derivative. Then, we carry out a quantitative assessment based on a European call option by comparison with alternative numerical methods.

2. tbc (Brackmann)

A G\"odel sentence is often described as a sentence saying about itself that it is not provable, and a Henkin sentence as a sentence stating its own provability. We discuss what it could mean for a sentence to ascribe to itself a property such as provability or unprovability. The starting point will be the answer Kreisel gave to Henkin's problem. We describe how the properties of the supposedly self-referential sentences depend on the chosen coding, the formulae expressing the properties and the way a fixed point for the formula is obtained. Some further examples of self-referential sentences are considered, such as sentences that \anf{say of themselves} that they are $\Sigma^0_n$-true (or $\Pi^0_n$-true), and their formal properties are investigated.

In this seminar I will discuss a function field analogue of classical problems in analytic number theory, concerning the auto-correlations of divisor functions, in the limit of a large finite field.

Historically, the study of positively curved manifolds has always been challenging. There are many reasons for this, but among them is the fact that the existence of a metric of positive curvature on a manifold imposes strong topological restrictions. In this talk we will discuss some of these topological implications and we will introduce the main results in this area. We will also present some recent results that relate positive curvature to the smooth structure of the manifold.

Let $G$ be a connected reductive group and $X$ a smooth complete curve, both defined over an algebraically closed field of characteristic zero. Let $Bun_G$ denote the stack of $G$-bundles on $X$. In analogy with the classical theory of Whittaker coefficients for automorphic functions, we construct a “Fourier transform” functor, called $coeff_{G}$, from the DG category of D-modules on $Bun_G$ to a certain DG category $Wh(G, ext)$, called the extended Whittaker category. Combined with work in progress by other mathematicians and the speaker, this construction allows to formulate the compatibility of the Langlands duality functor  $$\mathbb{L}_G : \operatorname{IndCoh}_{N}(LocSys_{\check{G}} ) \to D(Bun_G)$$ with the Whittaker model. For $G = GL_n$ and $G = PGL_n$, we prove that $coeff_G$ is fully faithful. This result guarantees that, for those groups, $\mathbb{L}_G$ is unique (if it exists) and necessarily fully faithful.

In numerical analysis the design and analysis of computational methods is often based on, and closely linked to, a well-posedness result for the underlying continuous problem. In particular the continuous dependence of the continuous model is inherited by the computational method when such an approach is used. In this talk our aim is to design a stabilised finite element method that can exploit continuous dependence of the underlying physical problem without making use of a standard well-posedness result such as Lax-Milgram's Lemma or The Babuska-Brezzi theorem. This is of particular interest for inverse problems or data assimilation problems which may not enter the framework of the above mentioned well-posedness results, but can nevertheless satisfy some weak continuous dependence properties. First we will discuss non-coercive elliptic and hyperbolic equations where the discrete problem can be ill-posed even for well posed continuous problems and then we will discuss the linear elliptic Cauchy problem as an example of an ill-posed problem where there are continuous dependence results available that are suitable for the framework that we propose.

After motivating why we would like to find examples of simple totally disconnected locally compact groups, I will describe a construction due to Banks, Elder and Willis which yields infinitely many such examples when given certain groups acting on a tree.

Given any family of varieties over a number field, if we have that the existence of local points everywhere is equivalent to the existence of a global point (for each member of the family), then we say that the family satisfies the Hasse principle. Of more interest, in this talk, is the case when the Hasse principle fails: we will give an overview of the "geography" of the currently known obstructions.