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Computational Number Theory, Geometry and Physics (Sage Days 53) September 23-29, 2013 Clay Mathematics Institute University of Oxford

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Computational Number Theory, Geometry and Physics (Sage Days 53) September 23-29, 2013 Clay Mathematics Institute University of Oxford Computational Number Theory, Geometry and Physics (Sage Days 53) September 23-29, 2013 Clay Mathematics Institute University of Oxford Abstracts Speaker: Jennifer Balakrishnan Introduction to number theory and Sage This talk will give some applications of Sage to number theory. No previous knowledge of Sage as a computer algebra system is assumed. Speaker: Volker Braun Introduction to mathematical physics and Sage This is the continuation of the first talk, giving some applications in geometry and physics. Speaker: tbc Scientific computing with Python and Cython We will give a quick review of Python and Cython, and how they are used in Sage to tie together scientific codes from various fields. Speaker: Simon King Implementing arithmetic operations in Sage (Coercions and Actions) Sage provides a framework for multiplication between objects of the same type (for example, ring operations) and between different types (for example, group actions). This talk will cover how to take advantage of it in your own code, especially if there is coercion involved. Speaker: Volker Braun Git and the new development workflow Git is a system to collaboratively edit a set of files, and is being used for anything from writing scientific articles to the Linux source code (~16 million lines of source code). I’ll start with a basic introduction to using git for distributed development that is of general interest. The second half will be about the new development workflow in Sage using Git. Speaker: Jean-Pierre Flori p-adics in Sage In this talk, I'll give a quick overview of the current and future implementations of p-adic numbers and related objects in Sage, with a special emphasis on their performance and the different ways offered to deal with inexact elements. Speaker: Ursula Whitcher & Adriana Salerno Zeta functions, point counting, and mirror symmetry Arithmetic mirror symmetry predicts that the zeta functions of mirror pairs of varieties should be closely related. We'll describe existing results involving highly symmetric hypersurfaces and suggest avenues for experimentation in Sage. Speaker: Kiran Kedlaya Some approaches to computing zeta functions of toric hypersurfaces A hypersurface in a complete toric variety is *nondegenerate* if it has transversal intersection with each component of the natural stratification of the toric variety; this class of varieties includes many examples of interest to both number theorists and physicists. We survey several approaches based on p- adic cohomology for computing the zeta functions of nondegenerate hypersurfaces over finite fields, including Lauder's deformation method, the Castryck-Denef-Vercauteren method, and the Abbott- Kedlaya-Roe method. We then propose a modification of the AKR method based on the idea of controlled reduction in de Rham cohomology, which we expect to work well in practice. Joint work in progress with David Harvey. Speaker: Amnon Besser and Francois Escriva Frobenius lifts and point counting for smooth curves We will sketch a new approach for point counting on smooth curves, which is a joint work with Rob de Jeu. A few new ideas are involved: computing the action of Frobenius on cohomology via residues, a general method for lifting Frobenius globally and a local version of this lift. Speaker: Andrey Novoseltsev Toric geometry in Sage We will give an overview of toric geometry capabilities of Sage, demonstrate its use on a few examples, and discuss the current code structure and potential future improvements and additions. Speaker: Jan Tuitman Counting points with the deformation method I will first give a short introduction to the deformation methods for computing zeta functions of varieties over finite fields. Then I will report on recent joint work with S. Pancratz that makes this method more practical for smooth projective hypersurfaces. Finally, I will try to show some examples computed with a very fast (FLINT) implementation of the algorithm. Speaker: Shaun Harker Computational homology via discrete Morse theory Homology of chain complexes can traditionally be computed via matrix algorithms, in particular by computing Smith Normal Form (SNF). However, SNF can be quite slow for large matrices. Additionally, many SNF algorithms provide only the invariant factors and not the transformation matrices. This is problematic for applications (such as computing the induced map on homology given a function between spaces) which require that representative cycles of homology generators be computed. Discrete Morse Theory can be used as a tool to accelerate homology computation. The essential idea is to quickly produce from a large complex a smaller one, known as the Morse Complex, which has isomorphic homology groups. Then standard techniques may be applied to the reduced complex at far less cost. Moreover, algorithms exist to lift representative cycles from the Morse Complex back to the original complex. We discuss the theory and the algorithms implemented in the CHomP software package. Speaker: John Voight Computing zeta functions of nondegenerate hypersurfaces with few monomials Using the cohomology theory of Dwork, as developed by Adolphson and Sperber, we exhibit a deterministic algorithm to compute the zeta function of a nondegenerate hypersurface defined over a finite field. This algorithm is particularly well suited to work with polynomials in small characteristic that have few monomials (relative to their dimension). Speaker: Jan Keitel Geometric engineering in toric F-theory An algorithm to systematically construct all Calabi-Yau elliptic fibrations realized as hypersurfaces in a toric ambient space for a given base and gauge group is described. This general method is applied to the particular question of constructing SU(5) GUTs with multiple U(1) gauge factors. The basic data consists of a top over each toric divisor in the base together with compactification data giving the embedding into a reflexive polytope. In order to ensure the existence of a low-energy gauge theory, the elliptic fibration must be flat, which is reformulated into conditions on the top and its embedding. Abelian gauge symmetries arising in toric F-theory compactifications are studied systematically. Using implementations in Sage, the toric Mordell-Weil groups determining the minimal number of U(1) factors are computed, an exhaustive list of SU(5) tops is compiled and several explicit fourfolds exemplifying the general algorithm are constructed..
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