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A major impediment to scientific progress in many fields is the inability to make sense of the huge amounts of data that have been collected via experiment or computer simulation. This dissertation provides tools to visualize, represent, and analyze the collection of sensors and data all at once in a single combinatorial geometric object. Encoding and translating heterogeneous data into common language are modeled by supporting objects. In this methodology, the behavior of the system based on the detection of noise in the system, possible failure in data exchange and recognition of the redundant or complimentary sensors are studied via some related geometric objects. Applications of the constructed methodology are described by two case studies: one from wildfire threat monitoring and the other from air traffic monitoring. Both cases are distributed (spatial and temporal) information systems. The systems deal with temporal and spatial fusion of heterogeneous data obtained from multiple sources, where the schema, availability and quality vary. The behavior of both systems is explained thoroughly in terms of the detection of the failure in the systems and the recognition of the redundant and complimentary sensors. A comparison between the methodology in this dissertation and the alternative methods is described to further verify the validity of the sheaf theory method. It is seen that the method has less computational complexity in both space and time.

This dissertation examines the topological properties of simplicial complexes that arise from two distinct combinatorial objects. In 2003, A. Björner and M. de Longueville proved that the neighborhood complex of the stable Kneser graph SGn,k is homotopy equivalent to a k-sphere. Further, for n = 2 they showed that the neighborhood complex deformation retracts to a subcomplex isomorphic to the associahedron. They went on to ask whether or not, for all n and k, the neighborhood complex of SGn,k contains as a deformation retract the boundary complex of a simplicial polytope. Part one of this dissertation provides a positive answer to this question in the case k = 2. In this case it is also shown that, after partially subdividing the neighborhood complex, the resulting complex deformation retracts onto a subcomplex arising as a polyhedral boundary sphere that is invariant under the action induced by the automorphism group of SGn,2. Part two of this dissertation studies simplicial complexes that arise from non-attacking rook placements on a subclass of Ferrers boards that have ai rows of length i where ai > 0 and i ≤ n for some positive integer n. In particular, enumerative properties of their facets, homotopy type, and homology are investigated.

Making use of algebraic and combinatorial techniques, we study two topics: the arithmetic degree of squarefree strongly stable ideals and the h-vectors of matroid complexes. For a squarefree monomial ideal, I, the arithmetic degree of I is the number of facets of the simplicial complex which has I as its Stanley-Reisner ideal. We consider the case when I is squarefree strongly stable, in which case we give an exact formula for the arithmetic degree in terms of the minimal generators of I as well as a lower bound resembling that from the Multiplicity Conjecture. Using this, we can produce an upper bound on the number of minimal generators of any Cohen-Macaulay ideals with arbitrary codimension extending Dubreil’s theorem for codimension 2. A matroid complex is a pure complex such that every restriction is again pure. It is a long-standing open problem to classify all possible h-vectors of such complexes. In the case when the complex has dimension 1 we completely resolve this question and we give some partial results for higher dimensions. We also prove the 1-dimensional case of a conjecture of Stanley that all matroid h-vectors are pure O-sequences. Finally, we completely characterize the Stanley-Reisner ideals of matroid complexes.

Let V denote a vector space of finite positive dimension. An ordered triple of linear operators on V is said to be a Leonard triple whenever for each choice of element of the triple there exists a basis of V with respect to which the matrix representing the chosen element is diagonal and the matrices representing the other two elements are irreducible tridiagonal. A Leonard triple is said to be modular whenever for each choice of element there exists an antiautomorphism of End(V) which fixes the chosen element and swaps the other two elements. We study combinatorial structures associated with Leonard triples and modular Leonard triples. In the first part we construct a simplicial complex of Leonard triples. The simplicial complex of a Leonard triple is the smallest set of linear operators which contains the given Leonard triple with the property that if two elements of the set are part of a Leonard triple, then the third element of the triple is also in the set. In the second part we construct a Hamming association scheme from modular Leonard triples using a method used previously in the context of Grassmanian codes.

In 1990, Fröberg presented a combinatorial classification of the quadratic square-free monomial ideals with linear resolutions. He showed that the edge ideal of a graph has a linear resolution if and only if the complement of the graph is chordal. Since then, a generalization of Fröberg's theorem to higher dimensions has been sought in order to classify all square-free monomial ideals with linear resolutions. Such a characterization would also give a description of all square-free monomial ideals which are Cohen-Macaulay. In this thesis we explore one method of extending Fröberg's result. We generalize the idea of a chordal graph to simplicial complexes and use simplicial homology as a bridge between this combinatorial notion and the algebraic concept of a linear resolution. We are able to give a generalization of one direction of Fröberg's theorem and, in investigating the converse direction, find a necessary and sufficient combinatorial condition for a square-free monomial ideal to have a linear resolution over fields of characteristic 2.

Placement games are a subclass of combinatorial games which are played on graphs. In this thesis, we demonstrate that placement games could be considered as games played on simplicial complexes. These complexes are constructed using square-free monomials. We define new classes of placement games and the notion of Doppelgänger. To aid in exploring the simplicial complex of a game, we introduce the bipartite flip and develop tools to compare known bounds on simplicial complexes (such as the Kruskal-Katona bounds) with bounds on game complexes.

In the thesis we study two dimensional simplicial complexes and linear codes. We say that a linear code C over a field F is triangular representable if there exists a two dimensional simplicial complex ∆ such that C is a punctured code of the kernel ker ∆ of the incidence matrix of ∆ over F and dim C = dim ker ∆. We call this simplicial complex a geometric representation of C. We show that every linear code C over a primefield is triangular representable. In the case of finite primefields we construct a geometric representation such that the weight enumerator of C is obtained by a simple formula from the weight enumerator of the cycle space of ∆. Thus the geometric representation of C carries its weight enumerator. Our motivation comes from the theory of Pfaffian orientations of graphs which provides a polynomial algorithm for weight enumerator of the cut space of a graph of bounded genus. This algorithm uses geometric properties of an embedding of the graph into an orientable Riemann surface. Viewing the cut space of a graph as a linear code, the graph is thus a useful geometric representation of this linear code. We study embeddability of the geometric representations into Euclidean spaces. We show that every binary linear code has a geometric representation that can be embed- ded into R4 . We characterize...

<p>Applied topology is a rapidly growing discipline aiming at using ideas coming from algebraic topology to solve problems in the real world, including analyzing point cloud data, shape analysis, etc. Semi-algebraic geometry deals with studying properties of semi-algebraic sets that are subsets of $\mathbb{R}^n$ and defined in terms of polynomial inequalities. Semi-algebraic sets are ubiquitous in applications in areas such as modeling, motion planning, etc. Developing efficient algorithms for computing topological invariants of semi-algebraic sets is a rich and well-developed field.</p> <p>However, applied topology has thrown up new invariants---such as persistent homology and barcodes---which give us new ways of looking at the topology of semi-algebraic sets. In this thesis, we investigate the interplay between these two areas. We aim to develop new efficient algorithms for computing topological invariants of semi-algebraic sets, such as persistent homology, and to develop new mathematical tools to make such algorithms possible.</p>

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.