Numerical Algebraic Geometry: A New Perspective on String and Gauge Theories

The interplay rich between algebraic geometry and string and gauge theories has recently been immensely aided by advances in computational algebra. However, these symbolic (Gr\"{o}bner) methods are severely limited by algorithmic issues such as exponential space complexity and being highly sequential. In this paper, we introduce a novel paradigm of numerical algebraic geometry which in a plethora of situations overcomes these short-comings. Its so-called 'embarrassing parallelizability' allows us to solve many problems and extract physical information which elude the symbolic methods. We describe the method and then use it to solve various problems arising from physics which could not be otherwise solved.Comment: 36 page

Serendipity Face and Edge VEM Spaces

We extend the basic idea of Serendipity Virtual Elements from the previous case (by the same authors) of nodal ($H^1$-conforming) elements, to a more general framework. Then we apply the general strategy to the case of $H(div)$ and $H(curl)$ conforming Virtual Element Methods, in two and three dimensions

Mini-Workshop: Projective Normality of Smooth Toric Varieties

The mini-workshop on ”Projective Normality of Smooth Toric Varieties” focused on the question of whether every projective embedding of a smooth toric variety is projectively normal. Equivalently, this question asks whether every lattice point in kP is the sum of k lattice points in P when P is a smooth (lattice) polytope. The workshop consisted of morning talks on different aspects of the problem, and afternoon discussion groups where participants from a variety of different backgrounds worked on specific examples and approaches

Two essays in computational optimization: computing the clar number in fullerene graphs and distributing the errors in iterative interior point methods

Fullerene are cage-like hollow carbon molecules graph of pseudospherical sym- metry consisting of only pentagons and hexagons faces. It has been the object of interest for chemists and mathematicians due to its widespread application in various fields, namely including electronic and optic engineering, medical sci- ence and biotechnology. A Fullerene molecular, Γ n of n atoms has a multiplicity of isomers which increases as N iso ∼ O(n 9 ). For instance, Γ 180 has 79,538,751 isomers. The Fries and Clar numbers are stability predictors of a Fullerene molecule. These number can be computed by solving a (possibly N P -hard) combinatorial optimization problem. We propose several ILP formulation of such a problem each yielding a solution algorithm that provides the exact value of the Fries and Clar numbers. We compare the performances of the algorithm derived from the proposed ILP formulations. One of this algorithm is used to find the Clar isomers, i.e., those for which the Clar number is maximum among all isomers having a given size. We repeated this computational experiment for all sizes up to 204 atoms. In the course of the study a total of 2 649 413 774 isomers were analyzed.The second essay concerns developing an iterative primal dual infeasible path following (PDIPF) interior point (IP) algorithm for separable convex quadratic minimum cost flow network problem. In each iteration of PDIPF algorithm, the main computational effort is solving the underlying Newton search direction system. We concentrated on finding the solution of the corresponding linear system iteratively and inexactly. We assumed that all the involved inequalities can be solved inexactly and to this purpose, we focused on different approaches for distributing the error generated by iterative linear solvers such that the convergences of the PDIPF algorithm are guaranteed. As a result, we achieved theoretical bases that open the path to further interesting practical investiga- tion

Invariants of plane curve singularities and Newton diagrams

We present an intersection-theoretical approach to the invariants of plane curve singularities $\mu$, $\delta$, $r$ related by the Milnor formula $2\delta=\mu+r-1$. Using Newton transformations we give formulae for $\mu$, $\delta$, $r$ which imply planar versions of well-known theorems on nondegenerate singularities

Decomposition in bunches of the critical locus of a quasi-ordinary map

A polar hypersurface P of a complex analytic hypersurface germ, f=0, can be investigated by analyzing the invariance of certain Newton polyhedra associated to the image of P, with respect to suitable coordinates, by certain morphisms appropriately associated to f. We develop this general principle of Teissier (see Varietes polaires. I. Invariants polaires des singularites d'hypersurfaces, Invent. Math. 40 (1977), 3, 267-292) when f=0 is a quasi-ordinary hypersurface germ and P is the polar hypersurface associated to any quasi-ordinary projection of f=0. We build a decomposition of P in bunches of branches which characterizes the embedded topological type of the irreducible components of f=0. This decomposition is characterized also by some properties of the strict transform of P by the toric embedded resolution of f=0 given by the second author in a paper which will appear in Annal. Inst. Fourier (Grenoble). In the plane curve case this result provides a simple algebraic proof of the main theorem of Le, Michel and Weber in "Sur le comportement des polaires associees aux germes de courbes planes", Compositio Math, 72, (1989), 1, 87-113