The objective of this work is the development of novel, efficient and reliable multidi- mensional adaptive quadrature routines defined over simplices (MAQS). MAQS pro- vides an approximation to the integral of a function defined over the unit hypercube and provides an error estimate that is used to drive a global subdivision strategy. The quadrature estimate is based on Lagrangian interpolation defined by using the vertices, edge nodes and interior points of a given simplex. The subdivision of a given simplex is chosen to allow for the reuse of points (thus function evaluations at those points) in successive refinements of the initial tessellation. While theory is developed for smooth functions, this algorithm is well suited for functions with discontinuities in dimensions three through six. Other advantages of this approach include straight-forward parallel implementation and application to integrals over polyhedral domains. / Ph. D.

Neste trabalho iremos apresentar uma releitura de um resultado clássico da topologia, na visão da topologia algébrica e em sua notação atual. A demonstração deste, apresentada por Cauchy (1813), é comentada de maneira crítica em Lima (1985a) e para esta apresentação destacaremos as definições, teoremas e entes básicos para o seu entendimento. / In this work we will present a rereading of a classic topology result, in the view of the algebraic topology in its current notation. The proof of this, presented by Cauchy (1813), is critically commented on Lima (1985a) for which we will present the definitions, theorems, basic entities for their understanding.

Simplices are the "simplest" examples of polytopes, and yet they exhibit much of the rich and subtle combinatorics and commutative algebra of their more general cousins. In this way they are sufficiently complicated --- insights gained from their study can inform broader research in Ehrhart theory and associated fields. In this dissertation we consider two previously unstudied properties of lattice simplices; one algebraic and one combinatorial. The first is the Poincar\'e series of the associated semigroup algebra, which is substantially more complicated than the Hilbert series of that same algebra. The second is the partial ordering of the elements of the fundamental parallelepiped associated to the simplex. We conclude with a proof-of-concept for using machine learning techniques in algebraic combinatorics. Specifically, we attempt to model the integer decomposition property of a family of lattice simplices using a neural network.

Neste trabalho iremos apresentar uma releitura de um resultado clássico da topologia, na visão da topologia algébrica e em sua notação atual. A demonstração deste, apresentada por Cauchy (1813), é comentada de maneira crítica em Lima (1985a) e para esta apresentação destacaremos as definições, teoremas e entes básicos para o seu entendimento. / In this work we will present a rereading of a classic topology result, in the view of the algebraic topology in its current notation. The proof of this, presented by Cauchy (1813), is critically commented on Lima (1985a) for which we will present the definitions, theorems, basic entities for their understanding.

Using algebra in geometry Pavel Paták Department: Department of Algebra Supervisor: Mgr. Pavel Růžička, Ph.D., Department of Algebra 1 Abstract In this thesis, we develop a technique that combines algebra, algebraic topology and combinatorial arguments and provides non-embeddability results. The novelty of our approach is to examine non- embeddability arguments from a homological point of view. We illustrate its strength by proving two interesting theorems. The first one states that k-dimensional skeleton of b 2k+2 k + k + 3 -dimensional simplex does not embed into any 2k-dimensional manifold M with Betti number βk(M; Z2) ≤ b. It is the first finite upper bound for Kühnel's conjecture of non-embeddability of simplices into manifolds. The second one is a very general topological Helly type theorem for sets in Rd : There exists a function h(b, d) such that the following holds. If F is a finite family of sets in Rd such that ˜βi ( G; Z2) ≤ b for any G F and every 0 ≤ i ≤ d/2 − 1, then F has Helly number at most h(b, d). If we are only interested whether the Helly numbers are bounded or not, the theorem subsumes a broad class of Helly types theorems for sets in Rd . Keywords: Homological Non-embeddability, Helly Type Theorem, Kühnel's conjecture of non-embeddability of ske- leta of simplices into manifolds

Let M be a finite set of random uniformly distributed points lying in a unit cube. Every four points from M make a tetrahedron and the tetrahedron can either contain some of the other points from M, or it can be empty. This diploma thesis brings an upper bound of the expected value of the number of empty tetrahedra with respect to size of M. We also show how precise is the upper bound in comparison to an approximation computed by a straightforward algorithm. In the last section we move from the three- dimensional case to a general dimension d. In the general d-dimensional case we have empty d-simplices in a d-hypercube instead of empty tetrahedra in a cube. Then we compare the upper bound for d-dimensional case to the results from another paper on this topic. 1