Past

One of the main obstacles to forecasting sea level rise over the coming centuries is the problem of predicting changes in the flow of ice sheets, and in particular their fast-flowing outlet glaciers. While numerical models of ice sheet flow exist, they are often hampered by a lack of input data, particularly concerning the bedrock topography beneath the ice. Measurements of this topography are relatively scarce, expensive to obtain, and often error-prone. In contrast, observations of surface elevations and velocities are widespread and accurate.

In an ideal world, we could combine surface observations with our understanding of ice flow to invert for the bed topography. However, this problem is ill-posed, and solutions are both unstable and non-unique. Conventionally, this problem is circumvented by the use of regularization terms in the inversion, but these are often arbitrary and the numerical methods are still somewhat unstable.

One philosophically appealing option is to apply a fully Bayesian framework to the problem. Although some success has been had in this area, the resulting distributions are extremely difficult to work with, both from an interpretive standpoint and a numerical one. In particular, certain forms of prior information, such as constraints on the bedrock slope and roughness, are extremely difficult to represent in this framework.

A more profitable avenue for exploration is a semi-Bayesian approach, whereby a classical inverse method is regularized using terms derived from a Bayesian model of the problem. This allows for the inclusion of quite sophisticated forms of prior information, while retaining the tractability of the classical inverse problem. In particular, we can account for the severely non-Gaussian error distribution of many of our measurements, which was previously impossible.

1. Minimising Regret in Portfolio Optimisation (Simões)

When looking for an "optimal" portfolio the traditional approach is to either try to minimise risk or maximise profit. While this approach is probably correct for someone investing their own wealth, usually traders and fund managers have other concerns. They are often assessed taking into account others' performance, and so their decisions are molded by that. We will present a model for this decision making process and try to find our own "optimal" portfolio.

2. Systemic risk in financial networks (Murevics)

Abstract: In this paper I present a framework for studying systemic risk and financial contagion in interbank networks. The current financial health of institutions is expressed through an abstract measure of robustness, and the evolution of robustness in time is described through a system of stochastic differential equations. Using this model I then study how the structure of the interbank lending network affects the spread of financial contagion through different contagion channels and compare the results for different network structures. Finally I outline the future directions for developing this model.

The aim of this talk is to provide a general setting in which a number of important dualities in mathematics can be framed uniformly.  The setting comes about as a natural generalisation of the Galois connection between ideals of polynomials with coefficients in a field K and affine varieties in K^n.  The general picture that comes into sight is that the topological representations of Stone, Priestley, Baker-Beynon, Gel’fand, or Pontryagin are to their respective classes of structures just as affine varieties are to K-algebras.

Let f be an elliptic modular newform of weight at least 2. The 
problem of the automorphy of the symmetric power L-functions of f is a 
key example of Langlands' functoriality conjectures. Recently, the 
potential automorphy of these L-functions has been established, using 
automorphy lifting techniques, and leading to a proof of the Sato-Tate 
conjecture. I will discuss a new approach to the automorphy of these 
L-functions that shows the existence of Sym^m f for m = 1,...,8.

Persistent homology is a tool for identifying topological features of (often high-dimensional) data. Typically, the data is indexed by a one-dimensional parameter space, and persistent homology is easily visualized via a persistence diagram or "barcode." Multi-dimensional persistent homology identifies topological features for data that is indexed by a multi-dimensional index space, and visualization is challenging for both practical and algebraic reasons. In this talk, I will give an introduction to persistent homology in both the single- and multi-dimensional settings. I will then describe an approach to visualizing multi-dimensional persistence, and the algebraic and computational challenges involved. Lastly, I will demonstrate an interactive visualization tool, the result of recent work to efficiently compute and visualize multi-dimensional persistent homology. This work is in collaboration with Michael Lesnick of the Institute for Mathematics and its Applications.

We investigate the effects of a finite set of agents interacting socially in an equilibrium pricing mechanism. A derivative written on non-tradable underlyings is introduced to the market and priced in an equilibrium framework by agents who assess risk using convex dynamic risk measures expressed by Backward Stochastic Differential Equations (BSDE). An agent is not only exposed to financial and non-financial risk factors, but he also faces performance concerns with respect to the other agents. The equilibrium analysis leads to systems of fully coupled multi-dimensional quadratic BSDEs.

Within our proposed models we prove the existence and uniqueness of an equilibrium. We show that aggregation of risk measures is possible and that a representative agent exists. We analyze the impact of the problem's parameters in the pricing mechanism, in particular how the agent's concern rates affect prices and risk perception.

For many tomographic imaging problems there are explicit inversion formulas, and depending on the completeness of the data these are unstable to differing degrees. Increasingly we are solving tomographic problems as though they were any other linear inverse problem using numerical linear algebra. I will illustrate the use of numerical singular value decomposition to explore the (in)stability for various problems. I will also show how standard techniques from numerical linear algebra, such as conjugate gradient least squares, can be employed with systematic regularization compared with the ad hoc use of slowly convergent iterative methods more traditionally used in computed tomography. I will mainly illustrate the talk with examples from three dimensional x-ray tomography but I will also touch on tensor tomography problems.
 

Dynamics of small particles in fluids have fascinated scientists for centuries. Phenomena such as Brownian motion, sedimentation, and electrophoresis continue to inspire cutting-edge research and innovations. The fluid in which the particles move is typically isotropic, such as water or a polymer solution. Recently, we started to explore what would happen if particles are placed in an anisotropic fluid: a liquid crystal. The study reveals that the liquid crystal changes dramatically both the statics and dynamics, leading to levitation of the particles, their anomalous Brownian motion and new mechanisms of electrokinetics. The new phenomena are rooted in anisotropy of the liquid crystal properties, such as different electric conductivity in the directions parallel and perpendicular to the average molecular orientation.

We will give an outline of the proof by Kahn and Markovic who showed that a closed hyperbolic 3-manifold $\textbf{M}$ contains a closed $\pi_1$-injective surface. This is done using exponential mixing to find many pairs of pants in $\textbf{M}$, which can then be glued together to form a suitable surface. This answers a long standing conjecture of Waldhausen and is a key ingredient in the proof of the Virtual Haken Theorem.

Cluster algebras are commutative algebras generated by a set S, obtained by an iterated mutation process of an initial seed. They were introduced by S. Fomin and A. Zelevinski in connection with canonical bases in Lie theory. Since then, many connections between cluster algebras and other areas have arisen.
This talk will focus on cluster algebras for which the set S is finite. These are called cluster algebras of finite type and are classified by Dynkin diagrams, in a similar way to many other objects.

http://hottformath.wikispaces.com

 In a previous work, we introduced a refinement of Juhasz’s sutured Floer homology, and constructed a minus theory for sutured manifolds, called sutured Floer chain complex. In this talk, we introduce a new description of sutured manifolds as “tangles” and describe a notion of cobordism between them. Using this construction, we define a cobordism map between the corresponding sutured Floer chain complexes. We also discuss some possible applications. This is a joint work with Eaman Eftekhary.

Rational Krylov subspaces have been proven to be useful for many applications, like the approximation of matrix functions or the solution of matrix equations. It will be shown that extended and rational Krylov subspaces —under some assumptions— can be retrieved without any explicit inversion or system solves involved. Instead we do the necessary computations of $A^{-1} v$ in an implicit way using the information from an enlarged standard Krylov subspace.

It is well-known that both for classical and extended Krylov spaces, direct unitary similarity transformations exist providing us the matrix of recurrences. In practice, however, for large dimensions computing time is saved by making use of iterative procedures to gradually gather the recurrences in a matrix. Unfortunately, for extended Krylov spaces one is required to frequently solve, in some way or another a system of equations. In this talk both techniques will be integrated. We start with an orthogonal basis of a standard Krylov subspace of dimension $m+m+p$. Then we will apply a unitary similarity built by rotations compressing thereby significantly the initial subspace and resulting in an orthogonal basis approximately spanning an extended or rational Krylov subspace.

Numerical experiments support our claims that this approximation is very good if the large Krylov subspace contains $A^{-(m+1)} v$, …, $A^{-1} v$ and thus can culminate in nonneglectable dimensionality reduction and as such also can lead to time savings when approximating, e.g., matrix functions.

Gyárfás conjectured in 1985 that if $G$ is a graph with no induced cycle of odd length at least 5, then the chromatic number of $G$ is bounded by a function of its clique number.  We prove this conjecture.  Joint work with Paul Seymour.

A stable algorithm to compute the roots of polynomials is presented. The roots are found by computing the eigenvalues of the associated companion matrix by Francis's implicitly-shifted $QR$ algorithm.  A companion matrix is an upper Hessenberg matrix that is unitary-plus-rank-one, that is, it is the sum of a unitary matrix and a rank-one matrix.  These properties are preserved by iterations of Francis's algorithm, and it is these properties that are exploited here. The matrix is represented as a product of $3n-1$ Givens rotators plus the rank-one part, so only $O(n)$ storage space is required.  In fact, the information about the rank-one part is also encoded in the rotators, so it is not necessary to store the rank-one part explicitly.  Francis's algorithm implemented on this representation requires only $O(n)$ flops per iteration and thus $O(n^{2})$ flops overall.  The algorithm is described, backward stability is proved under certain conditions on the polynomial coefficients, and an extensive set of numerical experiments is presented.  The algorithm is shown to be about as accurate as the (slow) Francis $QR$ algorithm applied to the companion matrix without exploiting the structure.  It is faster than other fast methods that have been proposed, and its accuracy is comparable or better.

In this talk I will present several results connected with the idea of magnetic relaxation for MHD, including some new commutator estimates (and a counterexample to the estimate in the critical case). (Joint work with various subsets of  D. McCormick, J. Robinson, C. Fefferman and J-Y. Chemin.)

Cramer conjectured a random model for the distribution of the primes, which would suggest that, on the scale of the average prime gap, the primes can be modelled by a Poisson process. In particular, the set of limit points of normalized prime gaps would be the whole interval $[0,\infty)$. I will describe joint work with Banks and Freiberg which shows that at least 1/8 of the positive reals are in the set of limit points. 

I will describe the Spatial Lambda-Fleming-Viot process, which is a model of evolution in a spatial continuum, and discuss the time and spatial scales on which selectively advantageous genes propagate through space. The appropriate scaling depends on the dimension of space, resulting in three distinct cases; d=1, d=2 and d>=3. In d=1 the limiting genealogy is the Brownian net whereas, by contrast, in d=2 local interactions give rise to a delicate damping mechanism and result in a finite limiting branching rate. This is joint work with Alison Etheridge and Daniel Straulino.

 Parametrized spectra are topological objects that represent
twisted forms of cohomology theories.  In this talk I will describe a theory
of parametrized spectra as highly structured bundle-like objects.  In
particular, we can make sense of the structure "group" of a bundle of
spectra.  This point of view leads to new examples and a good framework for
twisted equivariant cohomology theories.  

The theory of regularity structures allows one to give a meaning to several stochastic PDEs, including the Parabolic Anderson Model. So far, these equations have been considered on a torus. The goal of this talk is to explain how one can define the PAM on the whole space R^3. This is a joint work with Martin Hairer.

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.