Related papers: Some problems and conjectures about $L^2$-invarian…
We discuss some of the issues that arise in attempts to classify automorphisms of compact abelian groups from a dynamical point of view. In the particular case of automorphisms of one-dimensional solenoids, a complete description is given…
Many 2D lattice models of physical phenomena are conjectured to have conformally invariant scaling limits: percolation, Ising model, self-avoiding polymers, ... This has led to numerous exact (but non-rigorous) predictions of their scaling…
The theory of Topological Modular Forms suggests the existence of deformation invariants for two-dimensional supersymmetric field theories that are more refined than the standard elliptic genus. In this note we give a physical definition of…
Galilei invariant equations for massive fields with various spins are found and classified. They have been obtained directly, i.e., by using requirement of Galilei invariance and the facts on representations of the Galilei group deduced in…
This essay summarizes the state of the art on some aspects of the dynamics of polynomial diffeomorphsms in complex dimension two, and it presents a number of open questions.
Automated program verification often proceeds by exhibiting inductive invariants entailing the desired properties.For numerical properties, a classical class of invariants is convex polyhedra: solution sets of system of linear…
Here we survey the compactness and geometric stability conjectures formulated by the participants at the 2018 IAS Emerging Topics Workshop on {\em Scalar Curvature and Convergence}. We have tried to survey all the progress towards these…
The main purpose of this note is to pose a couple of problems which are easily formulated thought some seem to be not yet solved. These problems are of general interest for discrete mathematics including a new twig of a bough of theory of…
The universal sl_2 invariant of string links has a universality property for the colored Jones polynomial of links, and takes values in the h-adic completed tensor powers of the quantized enveloping algebra of sl_2. In this paper, we…
In these talks, I discuss a few selected topics in integrable models that are of interest from various points of view. Some open questions are also described.
A previous work of A. Conway and the author introduced $L^2$-Burau maps of braids, which are generalizations of the Burau representation whose coefficients live in a more general group ring than the one of Laurent polynomials. This same…
We study Hadamard variation of eigenvalues of Laplacian with respect to general domain perturbations. We show their existence up to the second order rigorously and characterize the derivatives, using associated eigenvalue problems in finite…
Beyond normal surfaces there are several open questions concerning 2- dimensional spaces. We present some results and conjectures along this line.
We discuss two simple but useful observations that allow the construction of modular forms from given ones using invariant theory. The first one deals with elliptic modular forms and their derivatives, and generalizes the Rankin-Cohen…
We survey the field of nonparametric inference under shape constraints, providing a historical overview and a perspective on its current state. An outlook and some open problems offer thoughts on future directions.
In an attempt to progress towards proving the conjecture the numerical range W (A) is a 2--spectral set for the matrix A, we propose a study of various constants. We review some partial results, many problems are still open. We describe our…
We review computations of joint invariants on a linear symplectic space, discuss variations for an extension of group and space and relate this to other equivalence problems and approaches, most importantly to differential invariants.
Algebraic statistics is concerned with the study of probabilistic models and techniques for statistical inference using methods from algebra and geometry. This article presents a list of open mathematical problems in this emerging field,…
This paper surveys some results by the author and collaborators on the existence of invariant Lagrangian graphs for Tonelli Hamiltonian systems. The presentation is based on an invited talk by the author at XIX Congresso Unione Matematica…
A new relativistic invariant version of nonlinear Maxwell equations is offerred. Some properties of these equations are considered.