[pt] Para (Lambda) uma matriz diagonal real de espectro simples, consideramse O(Lambda), a variedade de matrizes reais, simétricas conjugadas a (Lambda), e Tau (Lambda), a variedade das matrizes tridiagonais em O(Lambda). Calcula-se as homologias das duas variedades, combinando técnicas de teoria de Morse e sistemas integráveis. Como conseqüência, mostra-se que a imersão de O(Lambda) no espaço vetorial de matrizes reais simétricas é tight e taut, o que tem implicações em teoria espectral numérica. / [en] For (Lambda) a real, diagonal matrix of simple spectrum, we consider O(lambda), the isospectral manifold of real, symmetric matrices conjugate to (Lambda), and (Tau)(Lambda), the isospectral manifold of tridiagonal matrices in O(Lambda).We compute the homologies of both manifolds, combining techniques of Morse theory and integrable systems. As a consequence, we show that the immersion of O(Lambda) in the vector space of real symmetric matrices is tight and taut, a fact with implications in numerical spectral theory.

We prove that the non-k-equal configuration space of a graph has a discretized model, analogous to the discretized model for configurations on graphs. We apply discrete Morse theory to the latter to give an explicit combinatorial formula for the ranks of homology and cohomology of configurations of two points on a tree. We give explicit presentations for homology and cohomology classes as well as pairings for ordered and unordered configurations of two and three points on a few simple trees, and show that the first homology group of ordered and unordered configurations of two points in any tree is generated by the first homology groups of configurations of two points in three particular graphs, K_{1,3}, K_{1,4}, and the trivalent tree with 6 vertices and 2 vertices of degree 3, via graph embeddings.

Neste trabalho, estudamos a existˆencia e multiplicidade de solu¸c˜oes de certos problemas p-sublineares envolvendo o operador p-laplaciano usando teoria de Morse. / The purpose of this text is to provide a didactic exposition of the paper “Solutions of p-sublinear p-Laplacian equation via Morse theory” by Yuxia Guo and Jiaquan Liu [8]. This paper addresses the existence and multiplicity of solutions for the problem where is a smooth, bounded domain of RN, p is the p-Laplacian operator and f satisfies certain conditions, in particular f is p-sublinear at 0. Morse theory is used to infer the existence of critical points of a functional associated to this problem. In Chapter 2, we introduce the necessary Morse theoretic concepts, assuming basic knowledge of singular homology theory. In Chapter 3, we introduce basic properties of the p-Laplacian operator, assuming knowledge of Sobolev spaces, including imbedding and compactness results. Finally, in Chapter 4, we follow Guo and Liu’s paper itself.

In this dissertation we study the excedance permutation statistic. We start by extending the classical excedance statistic of the symmetric group to the affine symmetric group eSn and determine the generating function of its distribution. The proof involves enumerating lattice points in a skew version of the root polytope of type A. Next we study the excedance set statistic on the symmetric group by defining a related algebra which we call the excedance algebra. A combinatorial interpretation of expansions from this algebra is provided. The second half of this dissertation deals with the topology of the Frobenius complex, that is the order complex of a poset whose definition was motivated by the classical Frobenius problem. We determine the homotopy type of the Frobenius complex in certain cases using discrete Morse theory. We end with an enumeration of Q-factorial posets. Open questions and directions for future research are located at the end of each chapter.

Neste trabalho, estudamos a existˆencia e multiplicidade de solu¸c˜oes de certos problemas p-sublineares envolvendo o operador p-laplaciano usando teoria de Morse. / The purpose of this text is to provide a didactic exposition of the paper “Solutions of p-sublinear p-Laplacian equation via Morse theory” by Yuxia Guo and Jiaquan Liu [8]. This paper addresses the existence and multiplicity of solutions for the problem where is a smooth, bounded domain of RN, p is the p-Laplacian operator and f satisfies certain conditions, in particular f is p-sublinear at 0. Morse theory is used to infer the existence of critical points of a functional associated to this problem. In Chapter 2, we introduce the necessary Morse theoretic concepts, assuming basic knowledge of singular homology theory. In Chapter 3, we introduce basic properties of the p-Laplacian operator, assuming knowledge of Sobolev spaces, including imbedding and compactness results. Finally, in Chapter 4, we follow Guo and Liu’s paper itself.

Orientadora: Profa. Dra. Ilma Aparecida Marques Silva / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Matemática , 2015.

Neste trabalho, estudamos a existˆencia e multiplicidade de solu¸c˜oes de certos problemas p-sublineares envolvendo o operador p-laplaciano usando teoria de Morse. / The purpose of this text is to provide a didactic exposition of the paper “Solutions of p-sublinear p-Laplacian equation via Morse theory” by Yuxia Guo and Jiaquan Liu [8]. This paper addresses the existence and multiplicity of solutions for the problem where is a smooth, bounded domain of RN, p is the p-Laplacian operator and f satisfies certain conditions, in particular f is p-sublinear at 0. Morse theory is used to infer the existence of critical points of a functional associated to this problem. In Chapter 2, we introduce the necessary Morse theoretic concepts, assuming basic knowledge of singular homology theory. In Chapter 3, we introduce basic properties of the p-Laplacian operator, assuming knowledge of Sobolev spaces, including imbedding and compactness results. Finally, in Chapter 4, we follow Guo and Liu’s paper itself.

We extend the concept of central configurations to the N-body problem in spaces of nonzero constant curvature. Based on the work of Florin Diacu on relative equilib- ria of the curved N-body problem and the work of Smale on general relative equilibria, we find a natural way to define the concept of central configurations with the effective potentials. We characterize the ordinary central configurations as constrained critical points of the cotangent potential, which helps us to establish the existence of ordi- nary central configurations for any given masses. After these fundamental results, we study central configurations on H2, ordinary central configurations in S3, and special central configurations in S3 in three separate chapters. For central configurations on H2, we generalize the theorem of Moulton on geodesic central configurations, the theorem of Shub on the compactness of central configurations, the theorem of Conley on the index of geodesic central configurations, and the theorem of Palmore on the lower bound for the number of central configurations. We show that all three-body central configurations that form equilateral triangles must have three equal masses. For ordinary central configurations in S3, we construct a class of S3 ordinary central configurations. We study the geodesic central configurations of two and three bodies. Three-body non-geodesic ordinary central configurations that form equilateral trian- gles must have three equal masses. We also put into the evidence some other classes of central configurations. For special central configurations, we show that for any N ≥ 3, there are masses that admit at least one special central configuration. We then consider the Dziobek special central configurations and obtain the central con- figuration equation in terms of mutual distances and volumes formed by the position vectors. We end the thesis with results concerning the stability of relative equilibria associated with 3-body special central configurations. We find that these relative equilibria are Lyapunov stable when confined to S1, and that they are linearly stable on S2 if and only if the angular momentum is bigger than a certain value determined by the configuration. / Graduate

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Determining the chromatic number of the partition graph P(33) poses a considerable challenge. We can bound it to 4 ≤ χ(P(33)) ≤ 6, with exhaustive search confirming χ(P(33)) = 6. A potential mathematical proof strategy for this equality involves identifying a Z2-invariant S4 with non-trivial homology in the box complex of the partition graph P(33), namely Bedge(︁P(33))︁, and applying the Borsuk-Ulam theorem to compute its Z2-index. This provides a robust topological lower bound for the chromatic number of P(33), termed the Lovász bound. We have verified the absence of such an S4 within certain sections of Bedge(︁P(33))︁. We also validated this approach through a case study on the Petersen graph. This thesis offers a thorough examination of various topological lower bounds for a graph’s chromatic number, complete with proofs and examples. We demonstrate instances where these lower bounds converge to a single value and others where they diverge significantly from a graph’s actual chromatic number. We also classify all vertex pairs, triples, and quadruples of P(33) into unique equivalence classes, facilitating the derivation of all maximal complete bipartite subgraphs. This classification informs the construction of all simplices of Bedge(︁P(33)). Following a detailed and technical exploration, we uncover both the maximal size of the pairwise intersections of its maximal simplices and their underlying structure. Our study proposes an algorithm for building the box complex of the partition graph P(33) using our method of identifying maximal complete bipartite subgraphs. This reduces time complexity to O(n3), marking a substantial enhancement over brute-force techniques. Lastly, we apply discrete Morse theory to construct a simplicial complex homotopy equivalent to the box complex of P(33), using two methods: elementary collapses and the determination of a discrete Morse function on the box complex. This process reduces the dimension of the box complex from 35 to 12, streamlining future calculations of the Z2-index and the Lovász bound.