Browsing by Author "Huang, Dongzhou, committee member"

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Open Access
A local characterization of domino evacuation-shuffling
(Colorado State University. Libraries, 2024) McCann, Jacob, author; Gillespie, Maria, advisor; Peterson, Christopher, committee member; Huang, Dongzhou, committee member
We consider linear intersection problems in the Grassmanian (the space of k-dimensional subspaces of Cn), where the dimension of the intersection is 2. These spaces are called Schubert surfaces. We build of the previous work of Speyer [1] and Gillespie and Levinson [2]. Speyer showed there is a combinatorial interpretation for what happens to fibers of Schubert intersections above a "wall crossing", where marked points corresponding to the coordinates of partitions coincide. Building off Speyer's work, Levinson showed there is a combinatorial operation associated with the monodromy operator on Schubert curves, involving rectification, promotion, and shuffling of Littlewood-Richardson Young Tableaux, which overall is christened evacuation-shuffling. Gillespie and Levinson [2] further developed a localization of the evacuation-shuffling algorithm for Schubert curves. We fully develop a local description of the monodromy operator on certain classes of curves embedded inside Schubert surfaces [3].
Open Access
Connections between Hessenberg varieties, chromatic quasisymmetric functions, and q-series
(Colorado State University. Libraries, 2025) Salois, Kyle, author; Gillespie, Maria, advisor; Hulpke, Alexander, committee member; Cavalieri, Renzo, committee member; Huang, Dongzhou, committee member
In many ways, the combinatorics of symmetric functions can help us understand how other mathematical objects behave. For example, the Schur functions encode information about symmetric group representations as well as intersection theory in the Grassmannian. In this dissertation, we investigate connections between chromatic symmetric functions and Hessenberg varieties, and how each one can elevate the understanding of the other. Stanley and Stembridge conjectured in 1993 that the chromatic symmetric functions for unit interval graphs expanded with positive coefficients in the basis of elementary symmetric functions. This conjecture has been proved directly for several families of graphs, and a recent full proof was proposed by Hikita in 2024. Geometrically, this corresponds to showing that the cohomology rings of Hessenberg varieties, acted on by the symmetric group, decompose into permutation modules. Again, this result has been proven for several families of Hessenberg varieties, but in general remains open. For the Hessenberg function h=(h(1),n,...,n), the structure of the cohomology ring was determined by Abe, Horiguchi, and Masuda in 2017. In this dissertation, we define two new bases for this cohomology ring, one of which is a higher Specht basis, and the other of which is a permutation basis. We also examine the transpose Hessenberg variety, indexed by the Hessenberg function h' = ((n-1)n-m, nm), and show that analogous results hold. Further, we give combinatorial bijections between the monomials in the new basis and sets of P-tableaux, motivated by the work of Gasharov, and use P-tableaux to find a new formula for the Poincare polynomial of these Hessenberg varieties. Another open problem is to determine conditions for which the chromatic quasisymmetric function is symmetric. In 2024, Aliniaeifard et al. showed that if P is a path graph, then X_P(x;q) is symmetric if and only if the vertices of P are labeled in increasing or decreasing order, and if S is a star graph, then X_S(x;q) is not symmetric. In this dissertation, we extend this result, and show that if G is any tree, other than the path graph given above, then X_G(x;q) is not symmetric. We also construct a family of graphs called mixed mountain graphs, which are similar to unit interval graphs, and show that their chromatic quasisymmetric functions are symmetric.
Open Access
Tail dependence: application, exploration, and development of novel methods
(Colorado State University. Libraries, 2025) Wixson, Troy P., author; Cooley, Daniel, advisor; Shaby, Benjamin, advisor; Huang, Dongzhou, committee member; Wang, Tianying, committee member; Barnes, Elizabeth, committee member
The study of multivariate extreme events is largely concerned with modeling the dependence in the tail of the joint distribution. The understanding of extremal dependence and methodology for modeling that dependence has been an active research field over the past few decades and we contribute to that literature with three projects that are detailed in this dissertation. In the first project we consider the challenge of assessing the changing risk of wildfires. Wildfire risk is greatest during high winds after sustained periods of dry and hot conditions. This chapter is a statistical extreme event risk attribution study which aims to answer whether extreme wildfire seasons are more likely now than under past climate. This requires modeling temporal dependence at extreme levels. We propose the use of transformed-linear time series models which are constructed similarly to traditional ARMA models while having a dependence structure that is tied to a widely used framework for extremes (regular variation). We fit the models to the extreme values of the seasonally adjusted Fire Weather Index (FWI) time series to capture the dependence in the upper tail for past and present climate. Ten-thousand fire seasons are simulated from each fitted model and we compare the proportion of simulated high-risk fire seasons to quantify the increase in risk. Our method suggests that the risk of experiencing an extreme wildfire season in Grand Lake, Colorado under current climate has increased dramatically compared to the risk under the climate of the mid-20th century. Our method also finds some evidence of increased risk of extreme wildfire seasons in Quincy, California, but large uncertainties do not allow us to reject a null hypothesis of no change. In the second project we explore a fundamental characterization of tail dependence and develop a method to classify data into the two regimes. Classifying a data set as asymptotically dependent (AD) or asymptotically independent (AI) is a necessary early choice in the modeling of multivariate extremes. These two dependence regimes are defined asymptotically which complicates inference as practitioners have finite samples. We perform a series of experiments to determine whether a finite sample has enough information for a convolutional neural network to reliably distinguish between these regimes in the bivariate case. Along the way we develop a new classification tool for practitioners which we call nnadic as it is a Neural Network for Asymptotic Dependence/Independence Classification. This tool accurately classifies 95\% of test datasets and is robust to a wide range of sample sizes. The datasets which we are unable to correctly classify tend to either be nearly exactly independent or exhibit near perfect dependence, which are boundary cases for both the AD and AI models used for training. In the third project we consider the challenge of using likelihood methods for models developed for the tail of the distribution. Many multivariate extremes models have intractable likelihoods thus practitioners must use alternative fitting methods and likelihood-based methods for uncertainty quantification and model selection are unavailable. We develop a proxy-likelihood estimator for multivariate extremes models. Our method is based on the tail pairwise dependence (TPD) which is a summary measure of the dependence in the tail of any multivariate extremes model. The TPD parameter has a one-to-one relationship with the dependence parameter of the HR distribution. We use the HR distribution as a proxy for the likelihood in a composite likelihood approach. The method is demonstrated using the transformed linear extremes time series (TLETS) models of Mhatre & Cooley (2024).