Related papers: Imperfect bifurcations via topological methods in …

We describe recent achievements in the theory of weight systems, which are functions on chord diagrams satisfying so-called $4$-term relations. Our main attention is devoted to constructions of weight systems. The two main sources of these…

We propose a framework to understand the unprecedented performance and robustness of deep neural networks using field theory. Correlations between the weights within the same layer can be described by symmetries in that layer, and networks…

We provide a combinatorial construction for linear codes attaining the maximum possible number of distinct weights. We then introduce the related problem of determining the existence of linear codes with an arbitrary number of distinct…

We use Polyak's skein relation to give a new proof that Milnor's string link homotopy invariants are finite type invariants, and to develop a recursive relation for their associated weight systems. We show that the obstruction to the…

Three aspects of supersymmetric theories are discussed: electroweak symmetry breaking, the issues of flavor, and gauge unification. The heavy top quark plays an important, sometimes dominant, role in each case. Additional symmetries lead to…

We study bifurcation for the constant scalar curvature equation along a one-parameter family of Riemannian metrics on the total space of a harmonic Riemannian submersion. We provide an existence theorem for bifurcation points and a…

We study complete non-compact manifolds of positive scalar curvature, with a focus on how curvature decay is constrained by topology at infinity. Our first main result shows that topological linking at infinity forces polynomial decay of…

Motivated by recent analytical and numerical work on two- and three-dimensional convection with imposed spatial periodicity, we analyse three examples of bifurcations from a continuous group orbit of spatio-temporally symmetric periodic…

We present a general approach to the bifurcation analysis of elastic frameworks with symmetry. While group-theoretic methods for bifurcation problems with symmetry are well known, their actual implementation in the context of elastic…

We define the parametric closure problem, in which the input is a partially ordered set whose elements have linearly varying weights and the goal is to compute the sequence of minimum-weight lower sets of the partial order as the weights…

A piecewise-linear model with a single degree of freedom is derived from first principles for a driven vertical cantilever beam with a localized mass and symmetric stops. The resulting piecewise-linear dynamical system is smoothed by a…

We propose a higher dimensional scenario to solve the gauge hierarchy problem. In our formulation, a crucial observation is that a supersymmetric structure is hidden in the 4d spectrum of any gauge invariant theories with compact extra…

We extend the well-known theoretical treatment of the spontaneous symmetry breaking (SSB) in two-component systems, combining linear coupling and self-attractive nonlinearity, to a system in which the linear coupling competes with repulsive…

We examine examples of weakly nonlinear systems whose steady states undergo a bifurcation with increasing forcing, such that a forced subsystem abruptly ceases to absorb additional energy, instead diverting it into an initially quiescent,…

Motivated by questions originating from the study of a class of shallow student-teacher neural networks, methods are developed for the analysis of spurious minima in classes of gradient equivariant dynamics related to neural nets. In the…

Packing problems in discrete geometry can be modeled as finding independent sets in infinite graphs where one is interested in independent sets which are as large as possible. For finite graphs one popular way to compute upper bounds for…

This paper is inspired by an imaging problem encountered in the framework of Electrical Resistance Tomography involving two different materials, one or both of which are nonlinear. Tomography with nonlinear materials is in the early stages…

Oscillatory dynamics are ubiquitous in biological networks. Possible sources of oscillations are well understood in low-dimensional systems, but have not been fully explored in high-dimensional networks. Here we study large networks…

The graph bisection problem is the problem of partitioning the vertex set of a graph into two sets of given sizes such that the sum of weights of edges joining these two sets is optimized. We present a semidefinite programming relaxation…

In this paper, we use the equivariant degree theory to establish a global bifurcation result for the existence of non-stationary branches of solutions to a nonlinear, two-parameter family of hyperbolic wave equations with local delay and…