# Marisa Gaetz

I'm a first year math graduate student at MIT. Previously, I was an undergraduate at MIT, where I majored in Course 18: Mathematics and minored in Course 24-1: Philosophy. Within math, I am particularly interested in representation theory, Lie theory, and complexity theory. In addition to math, I am very passionate about criminal justice reform. I'm a fellow at The Educational Justice Institute (TEJI) at MIT, which seeks to create sustainable solutions to mass incarceration, social injustice, and barriers to reentry via education and technology. You can find my CV here.

## Research

1. Dual pairs in complex reductive groups.

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 $G$. We present substantial progress towards classifying the dual pairs of 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 their projective counterparts ($PGL(n,\mathbb{C})$, $PSp(2n,\mathbb{C})$, $PO(n,\mathbb{C})$, $PSO(n,\mathbb{C})$). The classifications of dual pairs in $Sp(2n,\mathbb{C})$, $GL(n,\mathbb{C})$, and $O(n,\mathbb{C})$ are known, but lack a unified explicit treatment; we provide such a treatment. Additionally, we classify the dual pairs in $SL(n,\mathbb{C})$ and $SO(n,\mathbb{C})$, and present partial progress towards classifying the dual pairs in $PGL(n,\mathbb{C})$, $PSp(2n,\mathbb{C})$, and $PO(n,\mathbb{C})$.

This research was done under the mentorship of David Vogan and Lucas Mason-Brown.

2. Anti-power j-fixes of the Thue-Morse word.

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 $\limsup_{k \rightarrow \infty} (\Gamma_j(k)/k)=3$.

This research was done at the 2018 University of Minnesota Duluth REU program (see more below).

3. Enumeration and extensions of word-representants. (with C. Ji)
Discrete Appl. Math. 284 (2020), 423-433.

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.

This research was done at the 2018 University of Minnesota Duluth REU program (see more below).

4. Support equalities among ribbon Schur functions. (with W. Hardt and S. Sridhar)
Electron. J. Combin. 26 (2019) 3, P3.52.

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 α has full equivalence class. Our main results include a sufficient condition for a ribbon α to have full equivalence class. Additionally, we prove a separate necessary condition, which we conjecture to be sufficient.

This research was done at the 2017 University of Minnesota Twin Cities REU program (see more below).

5. Quantifying CDS sortability of permutations by strategic pile size. (with B. Flanagan, M. Scheepers, and M. Shanks) Discrete Math. Algorithms Appl. 12 (2020) 1. arXiv

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.

This research was done at the 2016 Boise State University REU program (see more below).

6. A q-player impartial avoidance game for generating finite groups. (with B. Benesh)
Internat. J. Game Theory, 47 (2018) 2, 451-461.

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.

## REU (Research Experience for Undergrads) Work

### Duluth REU

I participated in Joe Gallian's REU program at the University of Minnesota Duluth. I worked on both independent and collaborative projects in the field of combinatorics on words. For the former, I proved results relating to anti-powers of the Thue-Morse word, generalizing previous results of Colin Defant. For the latter, we proved results regarding the representability of graphs by words.

Summer 2018
arXiv
arXiv

### Twin Cities REU

I participated in Vic Reiner's REU program at the University of Minnesota Twin Cities. Advised by Pasha Pylyavskyy, my research group worked on an algebraic combinatorics problem regarding the Schur supports of ribbon Schur functions.

Summer 2017
arXiv

### Boise State University REU

I participated in Liljana Babinkostova and Marion Scheepers's REU program at Boise State University. I collaborated on an enumerative combinatorics problem relating to quantifying the "fixed points" of a particular sorting algorithm.

Summer 2016
arXiv

## Prison Education Work

• I am currently president of the MIT Prison Education Initiative. We are committed to participating in and developing educational programming in the Massachusetts criminal justice system, as well as educating the MIT community about relevant experiences and issues.
• I am also a fellow at The Educational Justice Institute (TEJI) at MIT, a nonprofit that is co-directed by Lee Perlman and Carole Cafferty. As a fellow, I have helped with a variety of TEJI's initiatives. During Summer 2020, I worked with TEJI through the support of an ESG-PKG Fellowship to help organize and lead the Summer of HOPE program for system-involved young people in Boston and to help enable remote learning opportunities in correctional facilities during the COVID-19 pandemic.
• I have also frequently been a teaching assistant for Lee Perlman's Nonviolence as a Way of Life "inside-out" philosophy course. These "inside-out" courses are comprised half of MIT students and half of incarcerated people. This nonviolence course is featured in the New England Emmy Award-winning documentary below, in which I make an appearance!