The Sasaki Join, Hamiltonian 2-forms, and Sasaki-Einstein Metrics

By combining the join construction from Sasakian geometry with the Hamiltonian 2-form construction from K\"ahler geometry, we recover Sasaki-Einstein metrics discovered by physicists. Our geometrical approach allows us to give an algorithm for computing the topology of these Sasaki-Einstein manifolds. In particular, we explicitly compute the cohomology rings for several cases of interest and give a formula for homotopy equivalence in one particular 7-dimensional case. We also show that our construction gives at least a two dimensional cone of both Sasaki-Ricci solitons and extremal Sasaki metrics.Comment: 38 pages, paragraph added to introduction and Proposition 4.1 added, Proposition 4.15 corrected, Remark 5.5 added, and explanation for irregular Sasaki-Einstein structures expanded. Reference adde

Why Polyhedra Matter in Non-Linear Equation Solving

We give an elementary introduction to some recent polyhedral techniques for understanding and solving systems of multivariate polynomial equations. We provide numerous concrete examples and illustrations, and assume no background in algebraic geometry or convex geometry. Highlights include the following: (1) A completely self-contained proof of an extension of Bernstein's Theorem. Our extension relates volumes of polytopes with the number of connected components of the complex zero set of a polynomial system, and allows any number of polynomials and/or variables. (2) A near optimal complexity bound for computing mixed area -- a quantity intimately related to counting complex roots in the plane.Comment: 30 pages, 15 figures (26 ps or eps files), some in color. Paper corresponds to an invited tutorial talk delivered at a conference on Algebraic Geometry and Geometric Modelling (Vilnius, Lithuania, July 29-August 2, 2002), submitted for publicatio

Random matrices

We provide a self-contained introduction to random matrices. While some applications are mentioned, our main emphasis is on three different approaches to random matrix models: the Coulomb gas method and its interpretation in terms of algebraic geometry, loop equations and their solution using topological recursion, orthogonal polynomials and their relation with integrable systems. Each approach provides its own definition of the spectral curve, a geometric object which encodes all the properties of a model. We also introduce the two peripheral subjects of counting polygonal surfaces, and computing angular integrals.Comment: 196 pages, v2: major revision and expansion, 32 exercises adde

Charged particles constrained to a curved surface

We study the motion of charged particles constrained to arbitrary two-dimensional curved surfaces but interacting in three-dimensional space via the Coulomb potential. To speed-up the interaction calculations, we use the parallel compute capability of the Compute Unified Device Architecture (CUDA) of todays graphics boards. The particles and the curved surfaces are shown using the Open Graphics Library (OpenGL). The paper is intended to give graduate students, who have basic experiences with electrostatics and differential geometry, a deeper understanding in charged particle interactions and a short introduction how to handle a many particle system using parallel computing on a single home computerComment: 14 pages, 9 figure

Polyhedral Geometry in OSCAR

OSCAR is an innovative new computer algebra system which combines and extends the power of its four cornerstone systems - GAP (group theory), Singular (algebra and algebraic geometry), Polymake (polyhedral geometry), and Antic (number theory). Here, we give an introduction to polyhedral geometry computations in OSCAR, as a chapter of the upcoming OSCAR book. In particular, we define polytopes, polyhedra, and polyhedral fans, and we give a brief overview about computing convex hulls and solving linear programs. Three detailed case studies are concerned with face numbers of random polytopes, constructions and properties of Gelfand-Tsetlin polytopes, and secondary polytopes.Comment: 19 pages, 8 figure

ROOT Status and Future Developments

In this talk we will review the major additions and improvements made to the ROOT system in the last 18 months and present our plans for future developments. The additons and improvements range from modifications to the I/O sub-system to allow users to save and restore objects of classes that have not been instrumented by special ROOT macros, to the addition of a geometry package designed for building, browsing, tracking and visualizing detector geometries. Other improvements include enhancements to the quick analysis sub-system (TTree::Draw()), the addition of classes that allow inter-file object references (TRef, TRefArray), better support for templated and STL classes, amelioration of the Automatic Script Compiler and the incorporation of new fitting and mathematical tools. Efforts have also been made to increase the modularity of the ROOT system with the introduction of more abstract interfaces and the development of a plug-in manager. In the near future, we intend to continue the development of PROOF and its interfacing with GRID environments. We plan on providing an interface between Geant3, Geant4 and Fluka and the new geometry package. The ROOT GUI classes will finally be available on Windows and we plan to release a GUI inspector and builder. In the last year, ROOT has drawn the endorsement of additional experiments and institutions. It is now officially supported by CERN and used as key I/O component by the LCG project.Comment: Talk from the 2003 Computing in High Energy and Nuclear Physics (CHEP03), La Jolla, Ca, USA, March 2003, 5 pages, MSWord, pSN MOJT00

Critical exponents from parallel plate geometries subject to periodic and antiperiodic boundary conditions

We introduce a renormalized 1PI vertex part scalar field theory setting in momentum space to computing the critical exponents $\nu$ and $\eta$, at least at two-loop order, for a layered parallel plate geometry separated by a distance L, with periodic as well as antiperiodic boundary conditions on the plates. We utilize massive and massless fields in order to extract the exponents in independent ultraviolet and infrared scaling analysis, respectively, which are required in a complete description of the scaling regions for finite size systems. We prove that fixed points and other critical amounts either in the ultraviolet or in the infrared regime dependent on the plates boundary condition are a general feature of normalization conditions. We introduce a new description of typical crossover regimes occurring in finite size systems. Avoiding these crossovers, the three regions of finite size scaling present for each of these boundary conditions are shown to be indistinguishable in the results of the exponents in periodic and antiperiodic conditions, which coincide with those from the (bulk) infinite system.Comment: Modified introduction and some references; new crossover regimes discussion improved; Appendixes expanded. 48 pages, no figure

Random matrices

138 pages, based on lectures by Bertrand Eynard at IPhT, SaclayWe provide a self-contained introduction to random matrices. While some applications are mentioned, our main emphasis is on three different approaches to random matrix models: the Coulomb gas method and its interpretation in terms of algebraic geometry, loop equations and their solution using topological recursion, orthogonal polynomials and their relation with integrable systems. Each approach provides its own definition of the spectral curve, a geometric object which encodes all the properties of a model. We also introduce the two peripheral subjects of counting polygonal surfaces, and computing angular integrals

Gale duality, decoupling, parameter homotopies, and monodromy

2014 Spring.Numerical Algebraic Geometry (NAG) has recently seen significantly increased application among scientists and mathematicians as a tool that can be used to solve nonlinear systems of equations, particularly polynomial systems. With the many recent advances in the field, we can now routinely solve problems that could not have been solved even 10 years ago. We will give an introduction and overview of numerical algebraic geometry and homotopy continuation methods; discuss heuristics for preconditioning fewnomial systems, as well as provide a hybrid symbolic-numerical algorithm for computing the solutions of these types of polynomials and associated software called galeDuality; describe a software module of bertini named paramotopy that is scientific software specifically designed for large-scale parameter homotopy runs; give two examples that are parametric polynomial systems on which the aforementioned software is used; and finally describe two novel algorithms, decoupling and a heuristic that makes use of monodromy

On the symmetric squares of complex and quaternionic projective space

The problem of computing the integral cohomology ring of the symmetric square of a topological space has been of interest since the 1930s, but limited progress has been made on the general case until recently. In this work we offer a solution for the complex and quaternionic projective spaces $KP^n$, by taking advantage of their rich geometrical structure. Our description is in terms of generators and relations, and our methods entail ideas that have appeared in the literature of quantum chemistry, theoretical physics, and combinatorics. We deal first with the case $KP^\infty$, and proceed by identifying the truncation required for passage to finite n. The calculations rely upon a ladder of long exact cohomology sequences, which arises by comparing cofibrations associated to the diagonals of the symmetric square and the corresponding Borel construction. The cofibrations involve classic configuration spaces of unordered pairs of 1-dimensional subspaces of $K^{n+1}$, and their one-point compactifications; the latter are identified as Thom spaces by combining Lowdin's symmetric orthogonalisation process (and its quaternionic extension) with a dash of Pin geometry. The ensuing integral cohomology rings may be conveniently expressed using generalised Fibonacci polynomials. By way of validation, we note that our conclusions are compatible with mod 2 computations of Nakaoka and with homological results of Milgram.Comment: This version corrects many minor typos in v1, improves grammar, attempts to clarify a few mathematical obscurities, and includes an additional explanation in the Introduction (following Theorem 1.1) as to why the Theorem is generic for CW complexes whose integral cohomology is torsion free and concentrated in even dimension