Normal Form for Local Dynamical Systems

Normal Form is a theory that applies in the neighbourhood of an orbit of a vector field map. The theory provides an algorithmic way to generate a sequence of non-linear coordinate changes that eliminate as much non-linearity as possible at each order (where order refers to terms in Taylors series about an orbit).The normal form is intended to be the simplest form into which any system of the intended type can be transformed by changing the coordinates in a prescribed manner. Interestingly the form of non-linear that cannot be eliminated by such coordinate changes is determined by the structure of the linear part of the vector field map. This section consists of some background knowledge, theorems and definitions necessary for understanding the concept of normal form for local dynamical systems. We briefly discuss the concept of ring of invariants and module of equivariants, and use the Groebner basis methods to compute a Groebner basis for the ideal of relations among the basic invariants

Computational methods in algebra and analysis

This paper describes some applications of Computer Algebra to Algebraic Analysis also known as D-module theory, i.e. the algebraic study of the systems of linear partial differential equations. Gröbner bases for rings of linear differential operators are the main tools in the field. We start by giving a short review of the problem of solving systems of polynomial equations by symbolic methods. These problems motivate some of the later developed subjects.Ministerio de Ciencia y TecnologíaJunta de Andalucí

Landscaping with fluxes and the E8 Yukawa Point in F-theory

Integrality in the Hodge theory of Calabi-Yau fourfolds is essential to find the vacuum structure and the anomaly cancellation mechanism of four dimensional F-theory compactifications. We use the Griffiths-Frobenius geometry and homological mirror symmetry to fix the integral monodromy basis in the primitive horizontal subspace of Calabi-Yau fourfolds. The Gamma class and supersymmetric localization calculations in the 2d gauged linear sigma model on the hemisphere are used to check and extend this method. The result allows us to study the superpotential and the Weil-Petersson metric and an associated tt* structure over the full complex moduli space of compact fourfolds for the first time. We show that integral fluxes can drive the theory to N=1 supersymmetric vacua at orbifold points and argue that fluxes can be chosen that fix the complex moduli of F-theory compactifications at gauge enhancements including such with U(1) factors. Given the mechanism it is natural to start with the most generic complex structure families of elliptic Calabi-Yau 4-fold fibrations over a given base. We classify these families in toric ambient spaces and among them the ones with heterotic duals. The method also applies to the creating of matter and Yukawa structures in F-theory. We construct two SU(5) models in F-theory with a Yukawa point that have a point on the base with an $E_8$-type singularity on the fiber and explore their embeddings in the global models. The explicit resolution of the singularity introduce a higher dimensional fiber and leads to novel features.Comment: 150 page

Developments in multivariate post quantum cryptography.

Ever since Shor\u27s algorithm was introduced in 1994, cryptographers have been working to develop cryptosystems that can resist known quantum computer attacks. This push for quantum attack resistant schemes is known as post quantum cryptography. Specifically, my contributions to post quantum cryptography has been to the family of schemes known as Multivariate Public Key Cryptography (MPKC), which is a very attractive candidate for digital signature standardization in the post quantum collective for a wide variety of applications. In this document I will be providing all necessary background to fully understand MPKC and post quantum cryptography as a whole. Then, I will walk through the contributions I provided in my publications relating to differential security proofs for HFEv and HFEv−, key recovery attack for all parameters of HFEm, and my newly proposed multivariate encryption scheme, HFERP

Symmetric Ideals

Polynomials appear in many different fields such as statistics, physics and optimization. However, when the degrees or the number of variables are high, it generally becomes quite difficult to solve polynomials or to optimize polynomial functions. An approach that can often be helpful to reduce the complexity of such problems is to study symmetries in the problems. A relatively new field, that has gained a lot of traction in the last fifteen years, is the study of symmetry in polynomial rings in increasingly many variables. By considering the action of the symmetric groups on these polynomial rings, one can for instance show that certain sequences of symmetric ideals in increasingly larger polynomial rings are finitely generated up to orbits. In this thesis we will investigate some properties of such sequences. In particular the Hilbert Series and Gröbner bases of Specht ideals, a class of ideals arising from the representation theory of the symmetric group. We prove a conjectured Gröbner basis for Specht ideals of shape (n−k, 1^k) and give two different criteria for verifying the conjecture for other Specht ideals. We also build on a result from the representation theory of the symmetric group by showing that the leading monomials of the standard Specht polynomials span the vector space of leading monomials of Specht polynomials