The Fokas-Gel’fand theorem on the immersion formula of 2D-surfaces is related to the study of Lie symmetries of an integrable system. A rigorous proof of this theorem is presented which may help to better understand the immersion formula of 2D-surfaces in Lie algebras. It is shown, that even under weaker conditions, the main result of this theorem is still valid. A connection is established between three different analytic descriptions for immersion functions of 2D-surfaces, corresponding to the following three types of symmetries: gauge symmetries of the linear spectral problem, conformal transformations of the spectral parameter and generalized symmetries of the integrable system. The theoretical results are applied to the CP^{N-1} sigma model and several soliton-surfaces associated with these symmetries are constructed. It is shown that these surfaces are linked by the gauge transformations.

# On the Fokas-Gelfand theorem for integrable systems

# Elliptic PDEs in two dimensions

I will give a short survey of the several approaches to the regularity theory of elliptic equations in two dimensions. In particular I will focus on some old ideas of Bernstein and their application to the infinity Laplace equation and to the Bellman equation in two dimensions.

# Improved Algorithms for Computing Worst Value-at-Risk: Numerical Challenges and the Adaptive Rearrangement Algorithm // Thinking

Numerical challenges inherent in algorithms for computing worst Value-at-Risk in homogeneous portfolios are identified and solutions as well as words of warning concerning their implementation are provided. Furthermore, both conceptual and computational improvements to the Rearrangement Algorithm for approximating worst Value-at-Risk for portfolios with arbitrary marginal loss distributions are given. In particular, a novel Adaptive Rearrangement Algorithm is introduced and investigated. These algorithms are implemented using the R package qrmtools. Second speaker (Ricardas Zitikis): "The dose makes the poison," said Paracelsus, the father of toxicology. Thinking of risk, how much of it is good or bad? Despite all the illuminating results available in the literature on the topic, I will go back to the very basics: I will first briefly touch upon risk preferences, then look at some elusive behaviour of larger risks such as losses above deductibles, and finally venture into the tails. Those very few clarifications that will emerge from my talk will eventually be overshadowed by new questions and open problems, but I hope they will inspire some further work in the area.

# Integral hyperplane arrangements

Consider an arrangement of linear hyperplanes integral with respect to a given lattice. The lattice gives rise to a torus and the arrangement to a subdivision of the torus. We are interested in the combinatorics of this subdivision. We will describe questions and results for particular lattices associated to root systems and arrangements associated to graphs. This is joint work in progress with Swee Hong Chan.Consider an arrangement of linear hyperplanes integral with respect to a given lattice. The lattice gives rise to a torus and the arrangement to a subdivision of the torus. We are interested in the combinatorics of this subdivision. We will describe questions and results for particular lattices associated to root systems and arrangements associated to graphs. This is joint work in progress with Swee Hong Chan.

# Statistical Estimation Problems in Meta-Analysis

The principal statistical estimation problem in meta-analysis is to obtain a reliable confidence interval for the treatment effect. Several possible approaches and settings are described. In particular a Bayesian model with non-informative priors and the default data-dependent priors is discussed along with relevant optimization issues.

# System of points in saturated dg-categories

In this work we consider the question of realizing triangulated dg-categories by derived categories of algebraic varieties. For this, we introduce the notion of "system of points" in saturated dg-categories. We show that given such a system on a dg-category T, we can construct an algebraic space M, of finite type, smooth and separated, together with a dg-functor from T to a certain twisted dg-category of sheaves on M. We prove that this functor is furthermore an equivalence if and only if M is proper. All along this work we study t-strutcures on algebraic families of objects in T, which might be of independant interest.

Ref: http://arxiv.org/abs/1504.07748 (Joint work with Bertrand Toen)

# Weighted Hurwitz numbers, tau functions and topological recursion

Hurwitz numbers are classical combinatorial/geometric invariants that enumerate branched coverings of the Riemann sphere (or other Riemann surfaces). Combinatorially, they are understood equivalently as enumerating factorizations of elements of the symmetric group with given cycle structure. Extending results of Okounkov and Pandharipande on the use of KP tau-functions as generating functions for (single or double) Hurwitz numbers with only simple branching at all but one or two specified branch points to the general weighted case, with weights determined by a generating function with infinite parametric dependence, a KP tau-function of hypergeometric type is constructed which is shown to be the generating function for the latter.

This may be further generalized to include “quantum weighted” Hurwitz numbers, in terms of Macdonald polynomials, in which the weighted distribution is closely related to that for a quantum Bose gas with linear energy spectrum. An alternative type of generating function for weighted Hurwitz numbers may be constructed from the associated current correlation functions. These are shown to satisfy the equations of the Eynard-Orantin topological recursion scheme. The associated quantum spectral curve is derived, and the KP flows shown to be isomonodromic deformations of the corresponding differential operator in the spectral parameter.Hurwitz numbers are classical combinatorial/geometric invariants that enumerate branched coverings of the Riemann sphere (or other Riemann surfaces). Combinatorially, they are understood equivalently as enumerating factorizations of elements of the symmetric group with given cycle structure. Extending results of Okounkov and Pandharipande on the use of KP tau-functions as generating functions for (single or double) Hurwitz numbers with only simple branching at all but one or two specified branch points to the general weighted case, with weights determined by a generating function with infinite parametric dependence, a KP tau-function of hypergeometric type is constructed which is shown to be the generating function for the latter.

This may be further generalized to include “quantum weighted” Hurwitz numbers, in terms of Macdonald polynomials, in which the weighted distribution is closely related to that for a quantum Bose gas with linear energy spectrum. An alternative type of generating function for weighted Hurwitz numbers may be constructed from the associated current correlation functions. These are shown to satisfy the equations of the Eynard-Orantin topological recursion scheme. The associated quantum spectral curve is derived, and the KP flows shown to be isomonodromic deformations of the corresponding differential operator in the spectral parameter.

# The dimer model: universality and conformal invariance

The dimer model on a finite bipartite planar graph is a uniformly chosen set of edges which cover every vertex exactly once. It is a classical model of statistical mechanics, going back to work of Kasteleyn and Temperley/Fisher in the 1960s who computed its partition function.

After giving an overview, I will discuss some recent joint work with Benoit Laslier and Gourab Ray, where we prove in a variety of situations that when the mesh size tends to 0 the fluctuations are described by a universal and conformally invariant limit known as the Gaussian free field. A key novelty in our approach is that the exact solvability of the model plays only a minor role. Instead, we rely on a connection to imaginary geometry, where Schramm-Loewner Evolution curves are viewed as flow lines of an underlying Gaussian free field.

# Multivariate tests of associations based on univariate tests

For testing two random vectors for independence, we consider testing whether the distance of one vector from an arbitrary center point is independent from the distance of the other vector from an arbitrary center point by a univariate test. We provide conditions under which it is enough to have a consistent univariate test of independence on the distances to guarantee that the power to detect dependence between the random vectors increases to one, as the sample size increases. These conditions turn out to be minimal. If the univariate test is distribution-free, the multivariate test will also be distribution-free. If we consider multiple center points and aggregate the center-specific univariate tests, the power may be further improved. We suggest a specific aggregation method for which the resulting multivariate test will be distribution-free if the univariate test is distribution-free. We show that several multivariate tests recently proposed in the literature can be viewed as instances of this general approach.

# Nodal geometry of Steklov eigenfunctions

The eigenvalue and eigenfunction problem is fundamental and essential in mathematical analysis. The Steklov problem is an eigenvalue problem with spectrum at the boundary of a compact Riemannian manifold. Recently the study of Steklov eigenfunctions has been attracting much attention. We obtain the sharp doubling inequality for Steklov eigenfunctions on the boundary and interior of manifolds using delicate Carleman estimates. As an application, the optimal vanishing order is derived, which describes quantitative behavior of strong unique continuation property. We can ask Yau's type conjecture for the Hausdorff measure of nodal sets of Steklov eigenfunctions on the boundary and interior of the manifold. I will describe some recent progress about this challenging direction.

Part of work is joint with C. Sogge and X. Wang.