Algebraic groups over the adeles

Die vorliegende Arbeit beschäftigt sich mit algebraischen Gruppen G definiert über einem algebraischen Zahlkörper k. Um Informationen über die k-Punkte von G zu erhalten können wir die adelischen Punkte G_A von G betrachten. Ein Ziel dieser Arbeit ist die Konstruktion von Fundamentalbereichen bzw. -mengen für G_k in G_A. Weiters sollen Kriterien für Kompaktheit sowie für endliches invarinates Volumen des Quotienten G_A/G_k gefunden werden. Ferner betrachten wir die Gruppe G_A^(1), definiert als der Schnitt über die Kerne aller k-Charaktere von G, und wollen auch hier Bedingungen finden, die garantieren, dass der Quotient G_A^(1)/G_k kompakt ist bzw. endliches invariantes Volumen hat. Am Ende dieser Arbeit betrachten wir Inklusionen i: H -> G von reduktiven algebraischen k-Gruppen und zeigen, dass der induzierte Morphismus i: H_A^(1)/H_k -> G_A^(1)/G_k eigentlich ist.This diploma thesis deals with algebraic groups G defined over algebraic number fields k. To gain information about the k-points of G we can consider the adelic points G_A of G. One aim of this thesis is to construct fundamental domains respectively sets for G_k in G_A. In addition, criteria for compactness and for the existence of a finite invariant volume for the quotient G_A/G_k shall be found. Furthermore, we consider the group G_A^(1) defined by the intersection of the kernels of all k-characters of G and again try to find conditions which guarantee that the quotient G_A^(1)/G_k is compact, has finite invariant volume respectively. At the end of this thesis we analyse inclusions i: H -> G of reductive algebraic k-groups and show that the induced morphism i: H_A^(1)/H_k -> G_A^(1)/G_k is proper

Quaternion algebras and quadratic forms towards Shimura curves

Postprint (published version

From hyperelliptic to superelliptic curves

In this long survey article we show that the theory of elliptic and hyperelliptic curves can be extended naturally to all superelliptic curves. We focus on automorphism groups, stratification of the moduli space $\mathcal{M}_g$, binary forms, invariants of curves, weighted projective spaces, minimal models for superelliptic curves, field of moduli versus field of definition, theta functions, Jacobian varieties, addition law in the Jacobian, isogenies among Jacobians, etc. Many recent developments on the theory of superelliptic curves are provided as well as many open problems.Comment: survey paper on hyperelliptic and superelliptic curves; 96 page

Ahlfors circle maps and total reality: from Riemann to Rohlin

This is a prejudiced survey on the Ahlfors (extremal) function and the weaker {\it circle maps} (Garabedian-Schiffer's translation of "Kreisabbildung"), i.e. those (branched) maps effecting the conformal representation upon the disc of a {\it compact bordered Riemann surface}. The theory in question has some well-known intersection with real algebraic geometry, especially Klein's ortho-symmetric curves via the paradigm of {\it total reality}. This leads to a gallery of pictures quite pleasant to visit of which we have attempted to trace the simplest representatives. This drifted us toward some electrodynamic motions along real circuits of dividing curves perhaps reminiscent of Kepler's planetary motions along ellipses. The ultimate origin of circle maps is of course to be traced back to Riemann's Thesis 1851 as well as his 1857 Nachlass. Apart from an abrupt claim by Teichm\"uller 1941 that everything is to be found in Klein (what we failed to assess on printed evidence), the pivotal contribution belongs to Ahlfors 1950 supplying an existence-proof of circle maps, as well as an analysis of an allied function-theoretic extremal problem. Works by Yamada 1978--2001, Gouma 1998 and Coppens 2011 suggest sharper degree controls than available in Ahlfors' era. Accordingly, our partisan belief is that much remains to be clarified regarding the foundation and optimal control of Ahlfors circle maps. The game of sharp estimation may look narrow-minded "Absch\"atzungsmathematik" alike, yet the philosophical outcome is as usual to contemplate how conformal and algebraic geometry are fighting together for the soul of Riemann surfaces. A second part explores the connection with Hilbert's 16th as envisioned by Rohlin 1978.Comment: 675 pages, 199 figures; extended version of the former text (v.1) by including now Rohlin's theory (v.2

The universal coefficient theorem and quantum field theory

During the end of the 1950's Alexander Grothendieck observed the importance of the coefficient groups in cohomology. Three decades later, he presented his ``Esquisse d'un Programme" to the main french funding body. This program also included the use of different coefficient groups in the definition of various (co)homologies. His proposal was rejected. Another three decades later, in the 21st century, his research proposal is considered one of the most inspiring and important collection of ideas in pure mathematics. His ideas brought together algebraic topology, geometry, Galois theory, etc. becoming the origin for several new branches of mathematics. Today, less than one year after his death, Grothendieck is considered one of the most influential mathematicians worldwide. His ideas were important for the proofs of some of the most remarkable mathematical problems like the Weil Conjectures, Mordell Conjectures and the solution of Fermat's last theorem. Grothendieck's dessins d'enfant have been used in mathematical physics in various domains. Seiberg-Witten curves, N=1 and N=2 gauge theories and matrix models are a few examples where his insights are relevant. In this thesis I try to connect the idea of cohomology with coefficients in various sheaves to some areas of modern research in physics. The applications are manifold: the universal coefficient theorem presents connections to the topological genus expansion invented by 't Hooft and applied to quantum chromodynamics (QCD) and string theory, but also to strongly coupled electronic systems or condensed matter physics. It also appears to give a more intuitive explanation for topological recursion formulas and the holomorphic anomaly equations. The counting of BPS states may also profit from this new perspective. Indeed, the merging of cohomology classes when a change in coefficient groups is implemented may be related to the wall-crossing formulas and the phenomenon of decay or coupling of BPS states while crossing stability walls. The $Ext$ groups appearing in universal coefficient theorems may be regarded as obstructions characterizing the phenomena occurring when BPS stability walls are being crossed. Another important aspect is the existence of dualities. These are the non-perturbative analogue of symmetry transformations. Until now, they were discovered more by accident or by educated guesswork. I show in this thesis that there exists an underlying structure to the dualities, a structure that connects them the number fields used as coefficients in (co)homologies. This observation makes a nontrivial connection between number theory and physics

UTRGV Graduate Catalog 2018-2019

https://scholarworks.utrgv.edu/utrgvcatalogs/1006/thumbnail.jp

UTRGV Graduate Catalog 2017-2018

https://scholarworks.utrgv.edu/utrgvcatalogs/1007/thumbnail.jp

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described