CHAMP: A Cherednik Algebra Magma Package

We present a computer algebra package based on Magma for performing computations in rational Cherednik algebras at arbitrary parameters and in Verma modules for restricted rational Cherednik algebras. Part of this package is a new general Las Vegas algorithm for computing the head and the constituents of a module with simple head in characteristic zero which we develop here theoretically. This algorithm is very successful when applied to Verma modules for restricted rational Cherednik algebras and it allows us to answer several questions posed by Gordon in some specific cases. We could determine the decomposition matrices of the Verma modules, the graded G-module structure of the simple modules, and the Calogero-Moser families of the generic restricted rational Cherednik algebra for around half of the exceptional complex reflection groups. In this way we could also confirm Martino's conjecture for several exceptional complex reflection groups.Comment: Final version to appear in LMS J. Comput. Math. 41 pages, 3 ancillary files. CHAMP is available at http://thielul.github.io/CHAMP/. All results are listed explicitly in the ancillary PDF document (currently 935 pages). Please check the website for further update

Computing the homology of basic semialgebraic sets in weak exponential time

We describe and analyze an algorithm for computing the homology (Betti numbers and torsion coefficients) of basic semialgebraic sets which works in weak exponential time. That is, out of a set of exponentially small measure in the space of data the cost of the algorithm is exponential in the size of the data. All algorithms previously proposed for this problem have a complexity which is doubly exponential (and this is so for almost all data)

Faster real root decision algorithm for symmetric polynomials

In this paper, we consider the problem of deciding the existence of real solutions to a system of polynomial equations having real coefficients, and which are invariant under the action of the symmetric group. We construct and analyze a Monte Carlo probabilistic algorithm which solves this problem, under some regularity assumptions on the input, by taking advantage of the symmetry invariance property. The complexity of our algorithm is polynomial in $d^s, {{n+d} \choose d}$, and ${{n} \choose {s+1}}$, where $n$ is the number of variables and $d$ is the maximal degree of $s$ input polynomials defining the real algebraic set under study. In particular, this complexity is polynomial in $n$ when $d$ and $s$ are fixed and is equal to $n^{O(1)}2^n$ when $d=n$

S-functions, spectral functions of hyperbolic geometry, and vertex operators with applications to structure for Weyl and orthogonal group invariants

In this paper we analyze the quantum homological invariants (the Poincare polynomials of the sI(N) link homology). In the case when the dimensions of homologies of appropriate topological spaces are precisely known, the procedure of the calculation of the Kovanov-Rozansky type homology, based on the Euler-Poincare formula can be appreciably simplified. We express the formal character of the irreducible tensor representation of the classical groups in terms of the symmetric and spectral functions of hyperbolic geometry. On the basis of Labastida-Marino-Ooguri-Vafa conjecture, we derive a representation of the Chern-Simons partition function in the form of an infinite product in terms of the Ruelle spectral functions (the cases of a knot, unknot, and links have been considered). We also derive an infinite-product formula for the orthogonal Chem-Simons partition functions and analyze the singularities and the symmetry properties of the infinite-product structures. (C) 2016 The Authors. Published by Elsevier B.V.Peer reviewe

Critical Point Methods and Effective Real Algebraic Geometry: New Results and Trends

International audienceCritical point methods are at the core of the interplay between polynomial optimization and polynomial system solving over the reals. These methods are used in algorithms for solving various problems such as deciding the existence of real solutions of polynomial systems, performing one-block real quantifier elimination, computing the real dimension of the solution set, etc. The input consists of $s$ polynomials in $n$ variables of degree at most $D$. Usually, the complexity of the algorithms is $(s\, D)^{O(n^\alpha)}$ where $\alpha$ is a constant. In the past decade, tremendous efforts have been deployed to improve the exponents in the complexity bounds. This led to efficient implementations and new geometric procedures for solving polynomial systems over the reals that exploit properties of critical points. In this talk, we present an overview of these techniques and their impact on practical algorithms. Also, we show how we can tune them to exploit algebraic and geometric structures in two fundamental problems. The first one is real root finding of determinants of $n$-variate linear matrices of size $k\times k$. We introduce an algorithm whose complexity is polynomial in ${{n+k}\choose{k}}$ (joint work with S. Naldi and D. Henrion). This improves the previously known $k^{O(n)}$ bound. The second one is about computing the real dimension of a semi-algebraic set. We present a probabilistic algorithm with complexity $(s\, D)^{O(n)}$, that improves the long-standing $(s\, D)^{O(n^2)}$ bound obtained by Koi\-ran (joint work with E. Tsigaridas)

Computational aspects of Calogero-Moser spaces

We present a series of algorithms for computing geometric and representation-theoretic invariants of Calogero-Moser spaces and rational Cherednik algebras associated to complex reflection groups. Especially, we are concerned with Calogero-Moser families (which correspond to the $\mathbb{C}^\times$-fixed points of the Calogero-Moser space) and cellular characters (a proposed generalization by Rouquier and the first author of Lusztig's constructible characters based on a Galois covering of the Calogero-Moser space). To compute the former, we devised an algorithm for determining generators of the center of the rational Cherednik algebra (this algorithm has several further applications), and to compute the latter we developed an algorithmic approach to the construction of cellular characters via Gaudin operators. We have implemented all our algorithms in the Cherednik Algebra Magma Package (CHAMP) by the second author and used this to confirm open conjectures in several new cases. As an interesting application in birational geometry we are able to determine for many exceptional complex reflection groups the chamber decomposition of the movable cone of a $\mathbb{Q}$-factorial terminalization (and thus the number of non-isomorphic relative minimal models) of the associated symplectic singularity.Comment: 42 page

On tautological classes of fibre bundles and self-embedding calculus

In this thesis we study the ring of characteristic classes of smooth fibre bundles with a particular focus on tautological classes and the ring that they generate. In the first part we use tools from rational homotopy theory to compute analogues of tautological rings over fibrations, which provides upper bounds on the tautological rings of fibre bundles. In some cases we find an upper bound on the Krull dimension that is sharp. In the second part, we study the classifying space using the calculus of embeddings which provides a homotopy theoretic approximation. We construct cohomology classes on the self-embedding tower which extend certain characteristic classes that were introduced by Kontsevich. This construction is based on introducing configuration space integrals over the tower itself, which also has some consequences for tautological classes that we explore.Cambridge Trust and King's College Martin Thackeray Scholarshi

Multiscale Modeling and Simulation of Deformation Accumulation in Fault Networks

Strain accumulation and stress release along multiscale geological fault networks are fundamental mechanisms for earthquake and rupture processes in the lithosphere. Due to long periods of seismic quiescence, the scarcity of large earthquakes and incompleteness of paleoseismic, historical and instrumental record, there is a fundamental lack of insight into the multiscale, spatio-temporal nature of earthquake dynamics in fault networks. This thesis constitutes another step towards reliable earthquake prediction and quantitative hazard analysis. Its focus lies on developing a mathematical model for prototypical, layered fault networks on short time scales as well as their efficient numerical simulation. This exposition begins by establishing a fault system consisting of layered bodies with viscoelastic Kelvin-Voigt rheology and non-intersecting faults featuring rate-and-state friction as proposed by Dieterich and Ruina. The individual bodies are assumed to experience small viscoelastic deformations, but possibly large relative tangential displacements. Thereafter, semi-discretization in time with the classical Newmark scheme of the variational formulation yields a sequence of continuous, nonsmooth, coupled, spatial minimization problems for the velocities and states in each time step, that are decoupled by means of a fixed point iteration. Subsequently, spatial discretization is based on linear and piecewise constant finite elements for the rate and state problems, respectively. A dual mortar discretization of the non-penetration constraints entails a hierarchical decomposition of the discrete solution space, that enables the localization of the non-penetration condition. Exploiting the resulting structure, an algebraic representation of the parametrized rate problem can be solved efficiently using a variant of the Truncated Nonsmooth Newton Multigrid (TNNMG) method. It is globally convergent due to nonlinear, block Gauß–Seidel type smoothing and employs nonsmooth Newton and multigrid ideas to enhance robustness and efficiency of the overall method. A key step in the TNNMG algorithm is the efficient computation of a correction obtained from a linearized, inexact Newton step. The second part addresses the numerical homogenization of elliptic variational problems featuring fractal interface networks, that are structurally similar to the ones arising in the linearized correction step of the TNNMG method. Contrary to the previous setting, this model incorporates the full spatial complexity of geological fault networks in terms of truly multiscale fractal interface geometries. Here, the construction of projections from a fractal function space to finite element spaces with suitable approximation and stability properties constitutes the main contribution of this thesis. The existence of these projections enables the application of well-known approaches to numerical homogenization, such as localized orthogonal decomposition (LOD) for the construction of multiscale discretizations with optimal a priori error estimates or subspace correction methods, that lead to algebraic solvers with mesh- and scale-independent convergence rates. Finally, numerical experiments with a single fault and the layered multiscale fault system illustrate the properties of the mathematical model as well as the efficiency, reliability and scale-independence of the suggested algebraic solver

Spectral, Combinatorial, and Probabilistic Methods in Analyzing and Visualizing Vector Fields and Their Associated Flows

In this thesis, we introduce several tools, each coming from a different branch of mathematics, for analyzing real vector fields and their associated flows. Beginning with a discussion about generalized vector field decompositions, that mainly have been derived from the classical Helmholtz-Hodge-decomposition, we decompose a field into a kernel and a rest respectively to an arbitrary vector-valued linear differential operator that allows us to construct decompositions of either toroidal flows or flows obeying differential equations of second (or even fractional) order and a rest. The algorithm is based on the fast Fourier transform and guarantees a rapid processing and an implementation that can be directly derived from the spectral simplifications concerning differentiation used in mathematics. Moreover, we present two combinatorial methods to process 3D steady vector fields, which both use graph algorithms to extract features from the underlying vector field. Combinatorial approaches are known to be less sensitive to noise than extracting individual trajectories. Both of the methods are extensions of an existing 2D technique to 3D fields. We observed that the first technique can generate overly coarse results and therefore we present a second method that works using the same concepts but produces more detailed results. Finally, we discuss several possibilities for categorizing the invariant sets with respect to the flow. Existing methods for analyzing separation of streamlines are often restricted to a finite time or a local area. In the frame of this work, we introduce a new method that complements them by allowing an infinite-time-evaluation of steady planar vector fields. Our algorithm unifies combinatorial and probabilistic methods and introduces the concept of separation in time-discrete Markov chains. We compute particle distributions instead of the streamlines of single particles. We encode the flow into a map and then into a transition matrix for each time direction. Finally, we compare the results of our grid-independent algorithm to the popular Finite-Time-Lyapunov-Exponents and discuss the discrepancies. Gauss\'' theorem, which relates the flow through a surface to the vector field inside the surface, is an important tool in flow visualization. We are exploiting the fact that the theorem can be further refined on polygonal cells and construct a process that encodes the particle movement through the boundary facets of these cells using transition matrices. By pure power iteration of transition matrices, various topological features, such as separation and invariant sets, can be extracted without having to rely on the classical techniques, e.g., interpolation, differentiation and numerical streamline integration