Marisa Gaetz

Abstract. In Roger Howe's seminal 1989 paper "Remarks on classical invariant theory," he introduces the notion of Lie algebra dual pairs, and its natural analog in the groups context: a pair \((G_1,G_2)\) of reductive subgroups of an algebraic group \(G\) is a dual pair in \(G\) if \(G_1\) and \(G_2\) equal each other's centralizers in \(G\). While reductive dual pairs in the complex reductive Lie algebras have been classified, much less is known about algebraic group dual pairs, which were only fully classified in the context of certain classical matrix groups. In this paper, we classify the reductive dual pairs in \(PGL(n,\mathbb{C})\).

Abstract. The primary goal of this paper is to explicitly write down all semisimple dual pairs in the exceptional Lie algebras. (A dual pair in a reductive Lie algebra \(\mathfrak{g}\) is a pair of subalgebras such that each member equals the other's centralizer in \(\mathfrak{g}\).) In a 1994 paper, H. Rubenthaler outlined a process for generating a complete list of candidate dual pairs in each of the exceptional Lie algebras. However, the process of checking whether each of these candidate dual pairs is in fact a dual pair is not easy, and requires several distinct insights and methods. In this paper, we carry out this process and explain the relevant concepts as we go. We also give plenty of examples with the hopes of making Rubenthaler's 1994 result not only more complete but more usable and understandable.

Abstract. In this paper, we describe the possible disconnected complex reductive algebraic groups \(E\) with component group \(\Gamma=E/E_0\). We show that there is a natural bijection between such groups \(E\) and algebraic extensions of \(\Gamma\) by \(Z(E_0)\).

Abstract. In this paper, we present substantial progress towards formalizing and generalizing the biologically-motivated pathogenesis signal processing method introduced in 2022 by Thakkar and Famulare for COVID-19 modeling. In particular, we describe a generalization of this signal processing method, the accuracy of which relies on having the right notion of the effective rank of a matrix. Rather than adopting the Roy-Vetterli effective rank (which was used by Thakkar and Famulare for COVID-19), we introduce an alternative notion of effective rank and argue that it is the correct notion of effective rank for the purposes of pathogenesis signal processing. This generalized approach has the potential to be used in the analysis of other infectious diseases (besides COVID-19) as well as for multipathogen analysis.

Abstract. Self-efficacy and digital literacy are key predictors to incarcerated people's success in the modern workplace. While digitization in correctional facilities is expanding, few templates exist for how to design computing curricula that foster self-efficacy and digital literacy in carceral environments. As a result, formerly incarcerated people face increasing social and professional exclusion post-release. We report on a 12-week college-accredited web design class, taught virtually and synchronously, across 5 correctional facilities across the United States. The program brought together men and women from gender-segregated facilities into one classroom to learn fundamentals in HTML, CSS and Javascript, and create websites addressing social issues of their choosing. We conducted surveys with participating students, using dichotomous and open-ended questions, and performed thematic and quantitative analyses of their responses that suggest students' increased self-efficacy. Our study discusses key design choices, needs, and recommendations for furthering computing curricula that foster self-efficacy and digital literacy in carceral settings.

Abstract. In Roger Howe's Remarks on classical invariant theory, he introduces the notion of a dual pair of Lie subalgebras - a pair \( (\mathfrak{g}_1, \mathfrak{g}_2)\) of reductive Lie subalgebras of a Lie algebra \( \mathfrak{g} \) such that \( \mathfrak{g}_1 \) and \( \mathfrak{g}_2 \) are each other's centralizers in \( \mathfrak{g} \). This notion has a natural analog for algebraic groups: A dual pair of subgroups is a pair \( (G_1,G_2) \) of reductive subgroups of an algebraic group \( G \) such that \(G_1\) and \( G_2 \) are each other's centralizers in \( G \). In this paper, we classify the dual pairs in the complex classical groups (\(GL(n,\mathbb{C})\), \(SL(n,\mathbb{C})\), \(Sp(2n,\mathbb{C})\), \(O(n,\mathbb{C})\), and \(SO(n,\mathbb{C})\)) and in the corresponding Lie algebras (\(\mathfrak{gl}(n,\mathbb{C})\), \(\mathfrak{sl}(n,\mathbb{C})\), \(\mathfrak{sp}(2n,\mathbb{C})\), and \(\mathfrak{so}(n,\mathbb{C})\)). We also present substantial progress towards classifying the dual pairs in the projective counterparts of the complex classical groups (\(PGL(n,\mathbb{C})\), \(PSp(2n,\mathbb{C})\), \(PO(n,\mathbb{C})\), and \(PSO(n,\mathbb{C})\)).

Abstract. Recently, Fici, Restivo, Silva, and Zamboni introduced the notion of a \(k\)-anti-power, which is defined as a word of the form \(w^{(1)}w^{(2)} \cdots w^{(k)}\), where \(w^{(1)}, w^{(2)}, \ldots , w^{(k)}\) are distinct words of the same length. For an infinite word \(w\) and a positive integer \(k\), define \(AP_j(w,k)\) to be the set of all integers \(m\) such that \(w_{j+1}w_{j+2} \cdots w_{j+km}\) is a \(k\)-anti-power, where \(w_i\) denotes the \(i\)-th letter of \(w\). Define also \(F_j(k) = (2\mathbb{Z}^+-1) \cap AP_{j}(t,k)\), where \(t\) denotes the Thue-Morse word. For all \(k \in \mathbb{Z}^+\), \(\gamma_j(k) = \min (AP_j(t,k))\) is a well-defined positive integer, and for \(k \in \mathbb{Z}^+\) sufficiently large, \(\Gamma_j(k) = \sup ((2 \mathbb{Z}^+-1) \setminus F_j(k))\) is a well-defined odd positive integer. In his 2018 paper, Defant shows that \(\gamma_0(k)\) and \(\Gamma_0(k)\) grow linearly in \(k\). We generalize Defant's methods to prove that \(\gamma_j(k)\) and \(\Gamma_j(k)\) grow linearly in \(k\) for any nonnegative integer \(j\). In particular, we show that \(1/10 \leq \liminf_{k \rightarrow \infty} (\gamma_j(k)/k) \leq 9/10\) and \(1/5 \leq \limsup_{k \rightarrow \infty} (\gamma_j(k)/k) \leq 3/2\). Additionally, we show that \( \liminf_{k \rightarrow \infty} (\Gamma_j(k)/k)=3/2 \) and \(\limsup_{k \rightarrow \infty} (\Gamma_j(k)/k)=3\).

Abstract. Given a finite word \(w\) over a finite alphabet \(V\), consider the graph with vertex set \(V\) and with an edge between two elements of \(V\) if and only if the two elements alternate in the word \(w\). Such a graph is said to be word-representable or 11-representable by the word \(w\); this latter terminology arises from the phenomenon that the condition of two elements \(x\) and \(y\) alternating in a word \(w\) is the same as the condition of the subword of \(w\) induced by \(x\) and \(y\) avoiding the pattern 11. In this paper, we first study minimal length words which word-represent graphs, giving an explicit formula for both the length and the number of such words in the case of trees and cycles. We then extend the notion of word-representability (or 11-representability) of graphs to \(t\)-representability of graphs, for any pattern \(t\) on two letters. We prove that every graph is \(t\)-representable for any pattern \(t\) on two letters (except for possibly one class of \(t\)). Finally, we pose a few open problems for future consideration.

Abstract. The special purpose sorting operation, context directed swap (CDS), is an example of the block interchange sorting operation studied in prior work on permutation sorting. CDS has been postulated to model certain molecular sorting events that occur in the genome maintenance program of some species of ciliates. We investigate the mathematical structure of permutations not sortable by the CDS sorting operation. In particular, we present substantial progress towards quantifying permutations with a given strategic pile size, which can be understood as a measure of CDS non-sortability. Our main results include formulas for the number of permutations in \(S_n\) with maximum size strategic pile. More generally, we derive a formula for the number of permutations in \(S_n\) with strategic pile size \(k\), in addition to an algorithm for computing certain coefficients of this formula, which we call merge numbers.

Abstract. In 2007, McNamara proved that two skew shapes can have the same Schur support only if they have the same number of \(k \times \ell\) rectangles as subdiagrams. This implies that two ribbons can have the same Schur support only if one is obtained by permuting row lengths of the other. We present substantial progress towards classifying when a permutation \(\pi \in S_m\) of row lengths of a ribbon \(\alpha\) produces a ribbon \(\alpha_{\pi}\) with the same Schur support as \(\alpha\); when this occurs for all \(\pi \in S_m\), we say that \(\alpha\) has full equivalence class. Our main results include a sufficient condition for a ribbon \(\alpha\) to have full equivalence class. Additionally, we prove a separate necessary condition, which we conjecture to be sufficient.

Abstract. We study a three-player variation of the impartial avoidance game introduced by Anderson and Harary. Three players take turns selecting previously-unselected elements of a finite group. The losing player is the one who selects an element that causes the set of jointly-selected elements to be a generating set for the group, with the previous player winning and the remaining player coming in second place. We describe the winning strategy for these games on cyclic, dihedral, and nilpotent groups.