First introduced by Forman in 1998, discrete Morse theory has become a standard tool in topological combinatorics. The main idea of discrete Morse theory is to pair cells in a cellular complex in a manner that permits cancellation via elementary collapses, reducing the complex under consideration to a homotopy equivalent complex with fewer cells. In chapter 1, we introduce the relevant background for discrete Morse theory. In chapter 2, we define a discrete Morse matching for a family of independence complexes that generalize the matching complexes of suitable "small" grid graphs. Using this matching, we determine the dimensions of the chain spaces for the resulting Morse complexes and derive bounds on the location of non-trivial homology groups. Furthermore, we determine the Euler characteristic for these complexes and prove that several of their homology groups are non-zero. In chapter 3, we introduce the notion of a homomorphism complex for partially ordered sets, placing particular emphasis on maps between chain posets and the Boolean algebras. We extend the notion of folding from general graph homomorphism complexes to the poset case, and we define an iterative discrete Morse matching for these Boolean complexes. We provide formulas for enumerating the number of critical cells arising from this matching as well as for the Euler characteristic. We end with a conjecture on the optimality of our matching derived from connections to 3-equal manifolds

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.

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.

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Interactions between combinatorics and statistical mechanics have provided many fruitful insights in both fields. A compelling example is Kuperberg’s solution to the alternating sign matrix conjecture, and its following generalisations. In this thesis we investigate two models from statistical mechanics which have received attention in recent years. The first is the fully packed loop model. A conjecture from 2001 by Razumov and Stroganov opened the field for a large ongoing investigation of the O(1) loop model and its connections to a refinement of the fully packed loop model. We apply a combinatorial bijection originally found by de Gier to an older conjecture made by Propp. The second model is the hard particle model. Recent discoveries by Fendley et al. and results by Jonsson suggests that the hard square model with cylindrical boundary conditions possess some beautiful combinatorial properties. We apply both topological and purely combinatorial methods to related independence complexes to try and gain a better understanding of this model.

<p>Interactions between combinatorics and statistical mechanics have provided many fruitful insights in both fields. A compelling example is Kuperberg’s solution to the alternating sign matrix conjecture, and its following generalisations. In this thesis we investigate two models from statistical mechanics which have received attention in recent years.</p><p>The first is the fully packed loop model. A conjecture from 2001 by Razumov and Stroganov opened the field for a large ongoing investigation of the O(1) loop model and its connections to a refinement of the fully packed loop model. We apply a combinatorial bijection originally found by de Gier to an older conjecture made by Propp.</p><p>The second model is the hard particle model. Recent discoveries by Fendley et al. and results by Jonsson suggests that the hard square model with cylindrical boundary conditions possess some beautiful combinatorial properties. We apply both topological and purely combinatorial methods to related independence complexes to try and gain a better understanding of this model.</p>

In this dissertation we first study partition posets and their topology. For each composition c we show that the order complex of the poset of pointed set partitions is a wedge of spheres of the same dimension with the multiplicity given by the number of permutations with descent composition c. Furthermore, the action of the symmetric group on the top homology is isomorphic to the Specht module of a border strip associated to the composition. We also study the filter of pointed set partitions generated by knapsack integer partitions. In the second half of this dissertation we study descent avoidance in permutations. We extend the notion of consecutive pattern avoidance to considering sums over all permutations where each term is a product of weights depending on each consecutive pattern of a fixed length. We study the problem of finding the asymptotics of these sums. Our technique is to extend the spectral method of Ehrenborg, Kitaev and Perry. When the weight depends on the descent pattern, we show how to find the equation determining the spectrum. We give two length 4 applications, and a weighted pattern of length 3 where the associated operator only has one non-zero eigenvalue. Using generating functions we show that the error term in the asymptotic expression is the smallest possible.

We generalize Forman’s discrete Morse theory, on one end by developing a discrete analogue of Morse-Bott theory for CW complexes, motivated by Morse-Bott theory in the smooth setting. On the other, motivated by J-N. Corvellec’s Morse theory for continuous functionals, we generalize Forman’s discrete Morse-floer theory by considering a vector field more general than the one extracted from a discrete Morse function, and defining a boundary operator from which the Betti numbers of the CW complex are obtained. We also do some Conley theory analysis.

Let G be a finite graph with vertex set V and edge set E. A graph complex on G is an abstract simplicial complex consisting of subsets of E. In particular, we may interpret such a complex as a family of subgraphs of G. The subject of this thesis is the topology of graph complexes, the emphasis being placed on homology, homotopy type, connectivity degree, Cohen-Macaulayness, and Euler characteristic. We are particularly interested in the case that G is the complete graph on V. Monotone graph properties are complexes on such a graph satisfying the additional condition that they are invariant under permutations of V. Some well-studied monotone graph properties that we discuss in this thesis are complexes of matchings, forests, bipartite graphs, disconnected graphs, and not 2-connected graphs. We present new results about several other monotone graph properties, including complexes of not 3-connected graphs and graphs not coverable by p vertices. Imagining the vertices as the corners of a regular polygon, we obtain another important class consisting of those graph complexes that are invariant under the natural action of the dihedral group on this polygon. The most famous example is the associahedron, whose faces are graphs without crossings inside the polygon. Restricting to matchings, forests, or bipartite graphs, we obtain other interesting complexes of noncrossing graphs. We also examine a certain "dihedral" variant of connectivity. The third class to be examined is the class of digraph complexes. Some well-studied examples are complexes of acyclic digraphs and not strongly connected digraphs. We present new results about a few other digraph complexes, including complexes of graded digraphs and non-spanning digraphs. Many of our proofs are based on Robin Forman's discrete version of Morse theory. As a byproduct, this thesis provides a loosely defined toolbox for attacking problems in topological combinatorics via discrete Morse theory. In terms of simplicity and power, arguably the most efficient tool is Forman's divide and conquer approach via decision trees, which we successfully apply to a large number of graph and digraph complexes. / QC 20100622

[pt] A teoria de Morse relaciona a topologia de um espaço aos elementos críticos de uma função escalar definida nele. Isso vale tanto para a teoria clássica quanto para a versão discreta proposta por Forman em 1995. Essas teorias de Morse permitem caracterizar a topologia do espaço a partir de funções definidas nele, mas também permite estudar funções a partir de construções tipológicas derivadas dela, como por exemplo o complexo de Morse-Smale. Apesar da teoria de Morse discreta se aplicar para complexos celulares gerais de forma inteiramente combinatória, o que torna a teoria particularmente bem adaptada para o computador, as funções usadas na teoria não são amostragens de funções contínuas, mas casamentos especiais no grafo que codifica as adjacências no complexo celular, chamadas de casamentos de Morse. Quando usar essa teoria para estudar um espaço topológico, procura- se casamentos de Morse ótimos, i.e. com o menor número possível de elementos críticos, para obter uma informação topológica do complexo sem redundância. Na primeira parte desta tese, investiga-se a complexidade parametrizada de encontrar esses casamentos de Morse ótimos. Por um lado, prova-se que o problema ERASABILITY, um problema fortemente relacionado à encontrar casamentos de Morse ótimos, é W [P ]-completo. Por outro lado, um algoritmo é proposto para calcular casamentos de Morse ótimos em triangulações de 3-variedades, que é FPT no parâmetro do tree- width de seu grafo dual. Quando usar a teoria de Morse discreta para estudar uma função escalar definida no espaço, procura-se casamentos de Morse que capturam a informação geométrica dessa função. Na segunda parte é proposto uma construção de casamentos de Morse baseada em casamentos estáveis. As garantias teóricas sobre a relação desses casamentos com a geometria são elaboradas a partir de provas surpreendentemente simples que aproveitam da caracterização local do casamento estável. A construção e as suas garantias funcionam em qualquer dimensão. Finalmente, resultados mais fortes são obtidos quando a função for suave discreta, uma noção definida nesta tese. / [en] Morse theory relates the topology of a space to the critical elements of a scalar function defined on it. This applies in both the classical theory and a discrete version of it defined by Forman in 1995. Those Morse theories permit to characterize a topological space from functions defined on it, but also to study functions based on topological constructions it implies, such as the Morse-Smale complex. While discrete Morse theory applies on general cell complexes in an entirely combinatorial manner, which makes it suitable for computation, the functions it considers are not sampling of continuous functions, but special matchings in the graph encoding the cell complex adjacencies, called Morse matchings. When using this theory to study a topological space, one looks for optimal Morse matchings, i.e. one with the smallest number of critical elements, to get highly succinct topological information about the complex. The first part of this thesis investigates the parameterized complexity of finding such optimal Morse matching. On the one hand the Erasability problem, a closely related problem to finding optimal Morse matchings, is proven to be W[P]-complete. On the other hand, an algorithm is proposed for computing optimal Morse matchings on triangulations of 3-manifolds which is fixed parameter tractable in the tree-width of its dual graph. When using discrete Morse theory to study a scalar function defined on the space, one looks for a Morse matching that captures the geometric information of that function. The second part of this thesis introduces a construction of Morse matchings based on stable matchings. The theoretical guarantees about the relation of such matchings to the geometry are established through surprisingly simple proofs that benefits from the local characterization of the stable matching. The construction and its guarantees work in any dimension. Finally stronger results are obtained if the function is discrete smooth on the complex, a notion defined in this thesis.