79 research outputs found
Quantum geometry of moduli spaces of local systems and representation theory
No full textLet G be a split semi-simple adjoint group, and S a colored decorated surface, given by an oriented surface with punctures, special boundary points, and a specified collection of boundary intervals.
We introduce a moduli space P(G,S) parametrizing G-local system on S with some boundary data, and prove that it carries a cluster Poisson structure, equivariant under the action of the cluster modular group M(G,S), containing the mapping class group of S, the group of outer automorphisms of G, and the product of Weyl / braid groups over punctures / boundary components. We prove that the dual moduli space A(G,S) carries a M(G,S)-equivariant cluster structure, and the pair (A(G,S), P(G,S)) is a cluster ensemble. These results generalize the works of V. Fock & the first author, and of I. Le.
We quantize cluster Poisson varieties X for any Planck constant h s.t. h>0 or |h|=1. First, we define a *-algebra structure on the Langlands modular double A(h; X) of the algebra of functions on X. We construct a principal series of representations of the *-algebra A(h; X), equivariant under a unitary projective representation of the cluster modular group M(X). This extends works of V. Fock and the first author when h>0.
Combining this, we get a M(G,S)-equivariant quantization of the moduli space P(G,S), given by the *-algebra A(h; P(G,S)) and its principal series representations. We construct realizations of the principal series *-representations. In particular, when S is punctured disc with two special points, we get a principal series *-representations of the Langlands modular double of the quantum group Uq(g).
We conjecture that there is a nondegenerate pairing between the local system of coinvariants of oscillatory representations of the W-algebra and the one provided by the projective representation of the mapping class group of S.234 pages. Updated following the referee report. Sections 2.4, 13.4, 17.8, 17.9 20.3, 20.4 ne
On some aspects of cluster algebras and combinatorial Hopf algebras
No full text"This dissertation deals with problems in cluster algebras and combinatorial Hopf algebras. Total positivity has been closely related to cluster algebras since their inception. Postnikov's totally nonnegative Grassmannian is a concrete example of total positivity with rich combinatorics. Our first problem is the computation of Pl\ufccker coordinates inside a generalization of the totally nonnegative Grassmannian. We provide a combinatorial formula in terms of edge weighted directed graphs embedded on a surface. The next problem we consider is the equality of a cluster algebra and its upper cluster algebra. Particular attention is paid to the coefficient ring of the cluster algebra. We give a sufficient condition for the cluster algebra and upper cluster algebra to coincide while allowing greater generality of coefficient ring than was previous known. The final problem we consider in cluster algebras is showing that log-canonical coordinates are as simple as possible (in a certain precise sense). Log-canonical coordinates are a fundamental part of the Poisson geometry approach to cluster algebras put forth by Gekhtman, Shapiro, and Vainshtein. In the theory of combinatorial Hopf algebras we compute a formula for the antipode in a Hopf algebra on simplicial complexes. This antipode formula generalizes Humpert and Martin's formula for graphs. We then use the character theory of Aguiar, Bergeron, and Sottile to realize a version of Stanley's chromatic symmetric function for simplicial complexes. We prove that the degree sequence of a uniform hypertree can be recovered from its chromatic symmetric function. We also show the chromatic symmetric function is not a complete invariant for uniform hypertrees."--Page ii.(Ph. D.)--Michigan State University. Mathematics, 2018Includes bibliographical references (pages 125-130
Cyclic shuffle-compatibility, cyclic permutation statistics, cyclic quasisymmetric functions and toric partitions
No full textThesis (Ph.D.)--Michigan State University. Mathematics - Doctor of Philosophy, 2024A permutation statistic \st is said to be shuffle-compatible if the distribution of \st over the set of shuffles of two disjoint permutations and depends only on \st\pi, \st\sigma, and the lengths of and . Shuffle-compatibility is implicit in Stanley's early work on -partitions, and was first explicitly studied by Gessel and Zhuang, who developed an algebraic framework for shuffle-compatibility centered around their notion of the shuffle algebra of a shuffle-compatible statistic. One of the places where shuffles are useful is in describing the product in the algebra of quasisymmetric functions. Recently Adin, Gessel, Reiner, and Roichman defined an algebra of cyclic quasisymmetric functions where a cyclic version of shuffling comes into play. This dissertation focuses on the study of cyclic shuffle-compatibility. We began by showing a result called the ``lifting lemma,'' which allows one (under certain nice conditions) to prove that a cyclic statistic is cyclic shuffle-compatible from the shuffle-compatibility of a related linear statistic. This lifting lemma can be used to prove the cyclic shuffle-compatibility of all four statistics \cDes, \cdes, \cPk, and \cpk. We then developed an algebraic framework for cyclic shuffle-compatibility centered around the notion of cyclic shuffle algebra of a cyclic shuffle-compatible statistic. Using this theory, we provide explicit descriptions for the cyclic shuffle algebras of various cyclic permutation statistics, which in turn gives algebraic proofs for their cyclic shuffle-compatibility. In particular, we developed the theory of enriched toric -partitions, which provides a characterization of the cyclic shuffle algebra of \cPk.Description based on online resource. Title from PDF t.p. (Michigan State University Fedora Repository, viewed ).Includes bibliographical references
Combinatorial properties of permutations and permutation statistics
No full textSome of the most interesting and fundamental objects in the study of combinatorics are permutations. Permutations are typically defined to be an arrangement of the numbers {1, 2, . . . \uD835\uDC5B}, and they appear in countless problems and applications throughout mathematics. Sometimes we are particularly interested in observing or counting some key property of a given permutation. A permutation statistic is a function that takes a particular permutation and returns specific information about it, such as how many numbers are only adjacent to smaller numbers. Each permutation statistic gives rise to many counting questions, such as finding all possible permutations which have that particular statistic, or perhaps finding a subset of those permutations where that statistic is maximized. These numbers can sometimes be described by generating functions or polynomials which give light to the beauty and structure of mathematics, sometimes even in fields outside combinatorics. Some examples of this will be given in Chapter 1. In the rest of this dissertation, we will explore several of these statistics including new ones and variations on some of the more well known statistics.In the Chapter 2, we focus on shuffle compatibility for cyclic permutations. The shuffle of two permutations is the set of all permutations that have both original permutations as sub-sequences. A statistic is said to be shuffle compatible if its values over all possible shuffles of two permutations is completely determined by the statistic on the two original permutations, together with their lengths. Shuffle compatibility is implicit in Stanley's work on \uD835\uDC43-partitions and was also studied by Gessel and Zhuang. Shuffle compatibility is also useful in studying mathematical objects outside of combinatorics, such as quasisymmetric functions. More recently, an analogous definition of shuffle compatibility has been defined for cyclic permutations, which are permutations arranged in a circle so that the last element is considered adjacent to the first. In Chapter 2, we study a Lifting Lemma that can prove shuffle compatibility for some statistics on circular permutations based on known results for those statistics on linear permutations.\u2117 One well-studied permutation statistic is the peak set, which is the set of all indices of a permutation where an element is adjacent to two smaller elements. Primarily spearheaded by Davis, Nelson, Petersen, and Tenner, there has been recent interest in studying an analogue of this statistic known as the pinnacle set, which are the values of the elements at the indices of the peak set. Davis et al. proposed a number of unanswered questions about this statistic, which later led to a series of papers on the topic. In Chapter 3 we will present multiple results that attempt to answer some of these questions, including some original formulas and also some alternate combinatorial proofs of known results. These will include a bijection for counting the number of sets that could be the pinnacle set of some permutation, formulas and recursions for counting the number of permutations with a given pinnacle set, along with a new proof for a weighted sum of those numbers, and a recursion for counting the number of distinct orderings in which the elements of a pinnacle set can appear within a permutation.Thesis (Ph. D.)--Michigan State University. Mathematics, 2023Includes bibliographical reference
ASPECTS OF COMBINATORIAL SPECTRAL THEORY AND COMMUTATIVE ALGEBRA
No full textThesis (Ph.D.)--Michigan State University. Mathematics - Doctor of Philosophy, 2025This dissertation advances algebraic and topological methods for data science through four lines of work.The first part introduces the path Dirac and hypergraph Dirac operators together with their persistent counterparts, and investigates their ability to capture harmonic and non-harmonic spec- tra while revealing informative subcomplex structure. Their sensitivity to filtration is analyzed, demonstrating how these operators adapt to topological changes, and their behavior is illustrated across diverse examples. A central application is to molecular science: strict preorders derived from molecular structure generate graphs and digraphs with rich path architecture, and the resulting path complexes encode information depth that varies with the underlying preorder classes. The second part develops Mayer Dirac operators on ?-chain complexes. These operators link an alternating sequence of Mayer Laplacians and generalize the classical identity ?2 = ?. An explicit Laplacian for ?-chain complexes induced by vertex sequences on finite sets is derived, and weighted Mayer Laplacian and Dirac operators are introduced to capture physical attributes more effectively. A generalized factorization of Laplacians as an operator product with its adjoint is also established. Persistent Mayer Dirac operators and extensions are applied to biological and chemical data, where they demonstrate practical utility. The third part establishes a persistent Stanley\u2013Reisner theory that connects commutative algebra with combinatorial algebraic topology, machine learning, and data science. The framework defines persistent h-vectors, persistent ? -vectors, persistent graded Betti numbers, persistent facet ideals, and facet persistence modules. Stability analysis confirms that these algebraic invariants are robust under geometric perturbations, and their predictive value is demonstrated on molecular datasets. The final part proposes Commutative Algebra k-mer Learning (CAKL), a nonlinear algebraic framework for comparative genomics that builds upon persistent Stanley\u2013Reisner theory. CAKL in- tegrates commutative algebra, algebraic topology, combinatorics, and machine learning to address genetic variant identification, phylogenetic tree inference, and viral genome classification. Across eleven datasets, CAKL outperforms five state-of-the-art sequence analysis methods\u2014particularly in viral classification\u2014and maintains stable predictive accuracy as dataset size increases, highlight- ing scalability and robustness. Collectively, these contributions provide new operators, invariants, and learning paradigms that unify algebraic, topological, and combinatorial perspectives on discrete structures and real-world data, yielding great performance in molecular science and genomics.Description based on online resource. Title from PDF t.p. (Michigan State University Fedora Repository, viewed ).Includes bibliographical references
Heritability Estimation of Reliable Connectome Features
No full textBrain imaging genetics is an emerging research field aimed at studying the underlying genetic architecture of brain structure and function by utilizing different imaging modalities. However, not all the changes in the brain are a direct result of the genetic effect. Furthermore, the imaging phenotypes are promising for genetic analyses are usually unknown. In this thesis, we focus on identifying highly heritable measures of structural brain networks derived from Diffusion Weighted Magnetic Resonance imaging data. Using data for twins that is made available by the Human Connectome Project (HCP), the reliability of edge-level measures, namely fractional anisotropy, fiber length, and fiber number in the structural connectome, as well as seven network-level measures, specifically assortativity coefficient, local efficiency, modularity, transitivity, cluster coefficient, global efficiency, and characteristic path length, were evaluated using intraclass correlation coefficients. In addition, estimates of the heritability of the reliable measures were also obtained. It was observed that across all 64,620 network edges between 360 brain regions in the Glasser parcellation, approximately 5% were significantly high heritability based on fractional anisotropy, fiber length, or fiber number. Moreover, all tested network level measures, that capture network integrity, segregation, or resilience, were found to be highly heritable, having a variance ranging from 59% to 77% that is attributable to an additive genetic effect
Rationality of Brauer-Severi surface bundles
No full textThesis (Ph.D.)--Michigan State University. Mathematics - Doctor of Philosophy, 2025Rationality problems of complex algebraic geometry has a long history. Recent developments, usually referred as specialization method in the literature, have given fruitful new examples of non-rational varieties.We give a sufficient condition for a Brauer-Severi surface bundle over a rational 3-fold to not be stably rational. Additionally, we present an example that satisfies this condition and demonstrate the existence of families of Brauer-Severi surface bundles whose general members are smooth and not stably rational.Description based on online resource. Title from PDF t.p. (Michigan State University Fedora Repository, viewed ).Includes bibliographical references
Cyclic Sieving and Cluster Duality of Grassmannian
No full textWe introduce a decorated configuration space Confˣₙ() with a potential function . We prove the cluster duality conjecture of Fock-Goncharov for Grassmannians, that is, the tropicalization of (Confˣₙ(), ) canonically parametrizes a linear basis of the homogeneous coordinate ring of the Grassmannian Grₐ(n) with respect to the Plücker embedding. We prove that (Confˣₙ(), ) is equivalent to the mirror Landau-Ginzburg model of the Grassmannian considered by Eguchi-Hori-Xiong, Marsh-Rietsch, and Rietsch-Williams. As an application, we show a cyclic sieving phenomenon involving plane partitions under a sequence of piecewise-linear toggles.We are grateful to Alexander Goncharov for the inspiration on the construction of the cluster dual space, and to Jiuzu Hong for many helpful discussions on the representation theoretical aspects of the cyclic sieving problem. We would also like to thank Michael Gekhtman, Li Li, Tim Magee, Gregg Musiker, Brendon Rhoades, Bruce Sagan, Lauren Williams, Eric Zaslow, and Peng Zhou for useful conversations in the process of drafting this paper. Finally, we thank the referees for their very careful reading of this paper and for many useful suggestions
Log-canonical poisson structures and non-commutative integrable systems
No full textLog-canonical Poisson structures are a particularly simple type of bracket which are given by quadratic expressions in local coordinates. They appear in many places, including the study of cluster algebras. A Poisson bracket is "compatible" with a cluster algebra structure if the bracket is log-canonical with respect to each cluster. In joint work with John Machacek, we prove a structural result about such Poisson structures, which justifies the use and significance of such brackets in cluster theory. The result says that no rational coordinate-changes can transform these brackets into a simpler form. The pentagram map is a discrete dynamical system on the space of plane polygons first intro- duced by Schwartz in 1992. It was proved to be Liouville integrable by Schwartz, Ovsienko, and Tabachnikov in 2010. Gekhtman, Shapiro, and Vainshtein studied Poisson geometry associated to certain networks embedded in a disc or annulus, and its relation to cluster algebras. These Poisson structures are log-canonical. Later, Gekhtman et al. and Tabachnikov reinterpreted the pentagram map in terms of these networks, and used the associated Poisson structures to give a new proof of integrability. In 2011, Mari Beffa and Felipe introduced a generalization of the pentagram map to certain Grassmannians, and proved it had a Lax representation. We reinterpret this Grassmann penta- gram map in terms of non-commutative algebra, in particular the double brackets of Van den Bergh, and generalize the approach of Gekhtman et al. to establish a non-commutative version of integrability. The integrability of the pentagram maps in both projective space and the Grass-mannian follow from this more general algebraic system by projecting to representation spaces.Thesis (Ph. D.)--Michigan State University. Mathematics, 2019Includes bibliographical references (pages 100-103
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