Past

Convex regularization has become a popular approach to solve large scale inverse or data separation problems. A prominent example is the problem of identifying a sparse signal from linear samples my minimizing the l_1 norm under linear constraints. Recent empirical research indicates that many convex regularization problems on random data exhibit a phase transition phenomenon: the probability of successfully recovering a signal changes abruptly from zero to one as the number of constraints increases past a certain threshold. We present a rigorous analysis that explains why phase transitions are ubiquitous in convex optimization. It also describes tools for making reliable predictions about the quantitative aspects of the transition, including the location and the width of the transition region. These techniques apply to regularized linear inverse problems, to demixing problems, and to cone programs with random affine constraints. These applications depend on a new summary parameter, the statistical dimension of cones, that canonically extends the dimension of a linear subspace to the class of convex cones.

Joint work with Dennis Amelunxen, Mike McCoy and Joel Tropp.

Wei Wei

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Title: "Optimal Switching at Poisson Random Intervention Times"

(joint work with Dr Gechun Liang)

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Abstract: The paper introduces a new class of optimal switching problems, where

the player is allowed to switch at a sequence of exogenous Poisson

arrival times, and the underlying switching system is governed by an

infinite horizon backward stochastic differential equation system. The

value function and the optimal switching strategy are characterized by

the solution of the underlying switching system. In a Markovian setting,

the paper gives a complete description of the structure of switching

regions by means of the comparison principle.

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Julen Rotaetxe

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Title: Applicability of interpolation based finite difference method to problems in finance

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Abstract:

I will present the joint work with Christoph Reisinger on

the applicability of a numerical scheme relying on finite differences

and monotone interpolation to discretize linear and non-linear diffusion

equations. We propose suitable transformations to the process modeling

the underlying variable in order to overcome issues stemming from the

width of the stencil near the boundaries of the discrete spatial domain.

Numerical results would be given for typical diffusion models used in

finance in both the linear and non-linear setting.

We study the nonlinear wave equations on a class of asymptotically flat Lorentzian manifolds $(\mathbb{R}^{3+1}, g)$ with time dependent inhomogeneous metric g. Based on a new approach for proving the decay of solutions of linear wave equations, we give several applications to nonlinear problems. In particular, we show the small data global existence result for quasilinear wave equations satisfying the null condition on a class of time dependent inhomogeneous backgrounds which do not settle to any particular stationary metric.

Let $G$ be an addable minor-closed class of graphs. We prove that a zero-one law holds in monadic second-order logic (MSO) for connected graphs in G, and a convergence law in MSO for all graphs in $G$. For each surface $S$, we prove the existence of a zero-one law in first order logic (FO) for connected graphs embeddable in $S$, and a convergence law in FO for all graphs embeddable in $S$. Moreover, the limiting probability that a given FO sentence is satisfied is independent of the surface $S$. If $G$ is an addable minor-closed class, we prove that the closure of the set of limiting probabilities is a finite union of intervals, and it is the same for FO and MSO. For the class of planar graphs it consists of exactly 108 intervals. We give examples of non-addable classes where the results are quite different: for instance, the zero-one law does not hold for caterpillars, even in FO. This is joint work with Peter Heinig, Tobias Müller and Anusch Taraz.

A group is said to be quasirandom if all its unitary representations have “large” dimension. After introducing quasirandom groups and their basic properties, I shall turn to recent applications in two directions: constructions of expanders and non-existence of large product-free sets.

Let F be a field. A symplectic alternating algebra over F

consists of a symplectic vector space V over F with a non-degenerate

alternating form that is also equipped with a binary alternating

product · such that the law (u·v, w)=(v·w, u) holds. These algebraic

structures have arisen from the study of 2-Engel groups but seem also

to be of interest in their own right with many beautiful properties.

We will give an overview with a focus on some recent work on the

structure of nilpotent symplectic alternating algebras.

The concept of a matrix factorization was originally introduced by Eisenbud to study syzygies over local rings of singular hypersurfaces. More recently, interactions with mathematical physics, where matrix factorizations appear in quantum field theory, have provided various new insights. I will explain how matrix factorizations can be studied in the context of noncommutative algebraic geometry based on differential graded categories. We will see the relevance of the noncommutative analogue of de Rham cohomology in terms of classical singularity theory. Finally, I will outline how the Kapustin-Li formula for the noncommutative Serre duality pairing (originally computed via path integral methods) can be mathematically explained using a combination of homological perturbation theory and local duality.
Partly based on joint work with Daniel Murfet.

We propose a new algorithm for the approximate solution of large-scale high-dimensional tensor-structured linear systems. It can be applied to high-dimensional differential equations, which allow a low-parametric approximation of the multilevel matrix, right-hand side and solution in a tensor product format. We apply standard one-site tensor optimisation algorithm (ALS), but expand the tensor manifolds using the classical iterative schemes (e.g. steepest descent).  We obtain the rank--adaptive algorithm with the theoretical convergence estimate not worse than the one of the steepest descent, and fast practical convergence, comparable or even better than the convergence of more expensive two-site optimisation algorithm (DMRG).
The method is successfully applied for a high--dimensional problem of quantum chemistry, namely the NMR simulation of a large peptide.

This is a joint work with S.Dolgov (Max-Planck Institute, Leipzig, Germany), supported by RFBR and EPSRC grants.

Keywords: high--dimensional problems, tensor train format, ALS, DMRG, steepest descent, convergence rate, superfast algorithms, NMR.

The concept of a matrix factorization was originally introduced by Eisenbud to study syzygies over local rings of singular hypersurfaces. More recently, interactions with mathematical physics, where matrix factorizations appear in quantum field theory, have provided various new insights. I will explain how matrix factorizations can be studied in the context of noncommutative algebraic geometry based on differential graded categories. We will see the relevance of the noncommutative analogue of de Rham cohomology in terms of classical singularity theory. Finally, I will outline how the Kapustin-Li formula for the noncommutative Serre duality pairing (originally computed via path integral methods) can be mathematically explained using a combination of homological perturbation theory and local duality.
Partly based on joint work with Daniel Murfet.

We study a system of partial differential equations describing a steady flow of an incompressible generalized Newtonian fluid, wherein the Cauchy stress depends on concentration. Namely, we consider a coupled system of the generalized Navier-Stokes equations (viscosity of power-law type with concentration dependent power index) and convection-diffusion equation with non-linear diffusivity. We focus on the existence analysis of a weak solution for certain class of models by using a generalization of the monotone operator theory which fits into the framework of generalized Sobolev spaces with variable exponent (class of Sobolev-Orlicz spaces). Such results is then adapted for a suitable FEM approximation, for which the main tool of proof is a generalization of the Lipschitz approximation method.

In the first JAM seminar of 2013/2014, I will discuss the topic of singular perturbed hyperbolic systems of PDE arising in physical phenomena, particularly the St Venant equations of shallow water theory. Using a mixture of analytical and numerical techniques, I will demonstrate the dangers of approximating the dynamics of a system by the equations obtained upon taking a singular limit $\epsilon\rightarrow 0$ and furthermore how the dynamics of the system change when the parameter $\epsilon$ is taken to be small but finite. Problems of this type are ubiquitous in the physical sciences, and I intend to motivate another example arising in elastoplasticity, the subject of my DPhil study.

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Note: This seminar is not intended for faculty members, and is available only to current undergraduate and graduate students.

It is well known that one can attach Galois representations to certain modular forms, it is natural to ask how one might generalise this to produce more Galois representations. One such approach, due to Gross, defines objects called algebraic modular forms on certain types of reductive groups and then conjectures the existence of Galois representations attached to them. In this talk I will outline how for a particular choice of reductive group the conjectured Galois representations exist and are the classical modular Galois representations, thus providing some evidence that this is a good generalisation to consider.

The description of the behaviour of local minima or evolution problems for families of energies cannot in general be deduced from their Gamma-limit, which is a concept designed to treat static global minimum problems. Nevertheless this can be taken as a starting point. Various issues that have been addressed are:

Find criteria that ensure the convergence of local minimizers and critical points. In case this does not occur then modify the Gamma-limit in order to match this requirement. We note that in this way we `correct' some limit theories, finding (or `validating') other ones present in the literature;

Modify the concept of local minimizer, so that it may be more `compatible' with the process of Gamma-limit;

Treat evolution problems for energies with many local minima obtained by a time-discrete scheme introducing the notion of `minimizing movements along a sequence of functionals'. In this case the minimizing movement of the Gamma-limit can always be obtained by a choice of the space- and time-scale, but more interesting behaviors can be obtained at a critical ratio between them. In many cases a `critical scale' can be computed and an effective motion, from which all other minimizing movements are obtained by scaling.

Relate minimizing movements to general variational evolution results, in particular recent theories of quasistatic motion and gradient flow in metric spaces.

I will illustrate some of these points.

''Regression analysis aims to use observational data from multiple observations to develop a functional relationship relating explanatory variables to response variables, which is important for much of modern statistics, and econometrics, and also the field of machine learning. In this paper, we consider the special case where the explanatory variable is a stream of information, and the response is also potentially a stream. We provide an approach based on identifying carefully chosen features of the stream which allows linear regression to be used to characterise the functional relationship between explanatory variables and the conditional distribution of the response; the methods used to develop and justify this approach, such as the signature of a stream and the shue product of tensors, are standard tools in the theory of rough paths and seem appropriate in this context of regression as well and provide a surprisingly unified and non-parametric approach.''

The performance of the shares of a closed end bond fund is based on the returns of an underlying portfolio of bonds. The returns on closed end bond funds are typically higher than those of comparable open ended bond funds and this result is attributed to the use of leverage by closed end bond funds. This talk develops a simple model to assess the impact of leverage on the expected return and riskiness of a closed end bond fund. We illustrate the model with some examples

Let $A$ be a finite-dimensional $K$-algebra over an algebraically closed field $K$. Denote by $\Omega_A$ the syzygy operator on the category $\mod A$ of finite-dimensional right $A$-modules, which assigns to a module $M$ in $\mod A$ the kernel $\Omega_A(M)$ of a minimal projective cover $P_A(M) \to M$ of $M$ in $\mod A$. A module $M$ in $\mod A$ is said to be periodic if $\Omega_A^n(M) \cong M$ for some $n \geq 1$. Then $A$ is said to be a periodic algebra if $A$ is periodic in the module category $\mod A^e$ of the enveloping algebra $A^e = A^{\op} \otimes_K A$. The periodic algebras $A$ are self-injective and their module categories $\mod A$ are periodic (all modules in $\mod A$ without projective direct summands are periodic). The periodicity of an algebra $A$ is related with periodicity of its Hochschild cohomology algebra $HH^{*}(A)$ and is invariant under equivalences of the derived categories $D^b(\mod A)$ of bounded complexes over $\mod A$. One of the exciting open problems in the representation theory of self-injective algebras is to determine the Morita equivalence classes of periodic algebras.

We will present the current stage of the solution of this problem and exhibit prominent classes of periodic algebras.

We investigate symmetric quotient algebras of symmetric algebras,

with an emphasis on finite group algebras over a complete discrete

valuation ring R with residue field of positive characteristic p. Using elementary methods, we show that if an

ordinary irreducible character of a finite group gives

rise to a symmetric quotient over R which is not a matrix algebra,

then the decomposition numbers of the row labelled by the character are

all divisible by p. In a different direction, we show that if is P is a finite

p-group with a cyclic normal subgroup of index p, then every ordinary irreducible character of P gives rise to a

symmetric quotient of RP. This is joint work with Shigeo Koshitani and Markus Linckelmann.