Beyond wild walls there is algebraicity and exponential growth (of BPS indices)

textThe BPS spectrum of pure SU(3) four-dimensional super Yang-Mills with N=2 supersymmetry (a theory of class S(A)) exhibits a surprising phenomenon: there are regions of the Coulomb branch where the growth of BPS-indices with the charge is exponential. We show this using spectral networks and, independently, using wall-crossing formulae and quiver methods. The technique using spectral networks hints at a general property dubbed "algebraicity": generating series for BPS-indices in theories of class S(A) (a class of N=2 four-dimensional field theories) are secretly algebraic functions over the rational numbers. Kontsevich and Soibelman have an independent understanding of algebraicity using indirect techniques, however, spectral networks give a distinct reason for algebraicity with the advantage of providing explicit algebraic equations obeyed by generating series; along these lines, we provide a novel example of such an algebraic equation, and explore some relationships to Euler characteristics of Kronecker quiver stable moduli. We conclude by proving that exponential asymptotic growth is a corollary of algebraicity, leading to the slogan "there are either finitely many BPS indices or exponentially many" (in theories of class S(A)).Physic

Algorithms Seminar, 2002-2004

These seminar notes constitute the proceedings of a seminar devoted to the analysis of algorithms and related topics. The subjects covered include combinatorics, symbolic computation, and the asymptotic analysis of algorithms, data structures, and network protocols

Microstructure of Systems with Competition

The micro-structure of systems with competition often exhibits many universal features. In this thesis, we study certain aspects of these structural features as well as the microscopic interactions using disparate exact and approximate techniques. This thesis can be broadly divided into two parts. In the first part, we use statistical mechanics arguments to make general statements about length and timescales in systems with two-point interactions. We demonstrate that at high temperatures, the correlation function of general O(n) systems exhibits a universal form. This form enables the extraction of microscopic interaction potentials from the high temperature correlation functions. In systems with long range interactions, we find that the largest correlation length diverges in the limit of high temperatures. We derive an exact form for the correlation function in large-n systems with general two-point interactions at finite temperatures. From this, we obtain some features of the correlation and modulation lengths in general systems in the large-n limit. We derive a new exponent characterizing modulation lengths: or times) in systems in which the modulation length: or time) either diverges or becomes constant as a parameter, such as temperature exceeds a threshold value. In the second part of this thesis, we study the micro-structure of a metallic glass system using molecular dynamics simulations. We use both classical and first principles simulation to obtain atomic configurations in the liquid as well as the glassy phase. We analyze these using standard methods of local structure analysis - calculation of pair correlation function and structure factor, Voronoi construction, calculation of bond orientational order parameters and calculation of Honeycutt indices. We show the enhancement of icosahedral order in the glassy phase. Apart from this, we also use the techniques of community detection to obtain the inherent structures in the system using an algorithm which allows us to look at arbitrary length-scales

Computer Science for Continuous Data:Survey, Vision, Theory, and Practice of a Computer Analysis System

Building on George Boole's work, Logic provides a rigorous foundation for the powerful tools in Computer Science that underlie nowadays ubiquitous processing of discrete data, such as strings or graphs. Concerning continuous data, already Alan Turing had applied "his" machines to formalize and study the processing of real numbers: an aspect of his oeuvre that we transform from theory to practice.The present essay surveys the state of the art and envisions the future of Computer Science for continuous data: natively, beyond brute-force discretization, based on and guided by and extending classical discrete Computer Science, as bridge between Pure and Applied Mathematics

Hybrid convolution quadrature methods for modelling time-dependent waves with broadband frequency content

This work proposes two new hybrid convolution quadrature based discretisations of the wave equation for interior domains with broadband Neumann boundary data or source terms. The convolution quadrature method transforms the time domain wave problem into a series of Helmholtz problems with complex-valued wavenumbers, in which the boundary data and solutions are connected to those of the original problem through the Z-transform. The hybrid method terminology refers specifically to the use of different approximations of these Helmholtz problems, depending on the frequency. For lower frequencies we employ the boundary element method, while for more oscillatory problems we develop two alternative high frequency approximations based on plane wave decompositions of the acoustic field on the boundary. In the first approach we apply dynamical energy analysis to numerically approximate the plane wave amplitudes. The phases will then be reconstructed using a novel approach based on matching the boundary element solution to the plane wave ansatz in the frequency region where we switch between the low and high frequency methods. The second high frequency method is based on applying the Neumann-to Dirichlet map for plane waves to the given boundary data