### Columbia University

**Title**: Non-Left-Orderable Surgeries on Twisted Torus Knots

**Speakers**: Kat Christianson and Justin Goluboff

**Abstract**: Boyer, Gordon, and Watson have conjectured an equivalence between L-spaces and spaces with non-left-orderable fundamental group. The goal of our research has been to verify this conjecture for surgery on twisted torus knots. In our presentation, we will introduce the concepts of left-orderability, L-spaces, and knot surgery, and then we will briefly describe our methodology and results.

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**Title**: Reeb Dynamics of Lens Spaces

**Speakers**: Leonardo Abbrescia and Irit Huq-Kuruvilla

**Abstract**: The lens space L(n+1,n) is obtained as the quotient of S^3 by the cyclic subgroup Z_{n+1} of SU_2(C). We can also view this lens space as the link of a hypersurface singularity M:=f^{-1}(0) \cap S^5 in C^3, where f(z_0,z_1,z_2)=z_0^{n+1}+z_2^1+z_2^2. This talk will explain some contact geometric properties of M. We will introduce the notion of a contact structure and see how it gives rise to a globally defined vector field, known as the Reeb vector field. Using M as the model of the lens space we will explain some particular dynamics of the Reeb vector field that we studied this summer.

**Title**: Numerical and Experimental Studies of Oceanic Overflow

**Speakers**: Thomas Gibson, Fred Hohman, Theresa Morrison

**Abstract**: Overflows in the ocean occur when dense water flows down a continental slope into less dense ambient water. These density driven plumes occur naturally in various locations in the global ocean, but it is important to study idealized and small-scale models which allow for stronger confidence and control of parameters. The work presented here is a direct qualitative and quantitative comparison between physical laboratory experiments and lab-scale numerical simulations. Physical parameters are varied, including the Coriolis parameter, the inflow density anomaly, and the inflow volumetric flow rate. Laboratory experiments are conducted using a rotating square tank and high resolution camera mounted on the table in the rotating reference frame. Video results are digitized in order to compare directly to numerical simulations. The MIT General Circulation Model (MITgcm), a three dimensional, full physics ocean model, is used for the numerical simulations. These simulations are run under the full range of physical parameters corresponding to the specific laboratory experiments.

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**Title**: Sums on some Abelian Groups

**Speakers**: Abraham Bekele, Katie Rosenberg, Ben Wright

**Abstract**: By the fundamental theorem of finite abelian groups, every finite abelian group can be written as a (unique) direct product of cyclic groups (whose lower orders all divide all the higher cyclic orders). The plus-minus weighted Davenport constant of a finite abelian group $G$ is defined as the least integer $k$ such that any sequence $S$ of length $k$ in the group $G$ has a nontrivial zero sum of elements in $S$ with weights either $-1$ or $1$. Marchan, Ordaz, and Schmid found the plus-minus weighted Davenport constant for all groups up to order $100$ (except one), and they provide general bounds for the plus-minus weighted Davenport constant in many other cases. In most cases, the Davenport constants achieved their upper bound, and all achieved either the upper or the lower bound; we find groups of order greater than $100$ whose Davenport constants do not. In this talk, we will present improvements of lower and upper bounds on groups of the form $\mathbb{Z}_3^r\oplus\mathbb{Z}_2^s$, and provide the plus-minus weighted Davenport constants that we found for some of these groups.

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**Title**: Homotopy equivalence in graph-like digital spaces

**Speakers**: Jason Haarmann, Meg P. Murphy, Casey Peters

**Abstract**: For graph-like digital topological spaces, there is an established homotopy equivalence relation which parallels that of classical topology. Some classical homotopy equivalence invariants, such as the Euler characteristic and the homology groups, do not remain invariants in the digital setting. This paper develops two numeric digital homotopy invariants and begins to catalog all possible connected digital spaces on a small number of points, up to homotopy equivalence.

### Mount Holyoke College

**Title**: Explorations into the Discrete Logarithm Problem

**Speakers**: Abigail Mann, Anne Waldo, Adelyn Yeoh, Caiyun Zhu

**Abstract**: How secure are our encryption methods? Current encryption schemes include the ElGamal signature scheme, which was described by Taher Elgamal in 1985. It is thought that the functions that such schemes use have inverses that are computationally intractable. In relation to this, we are interested in counting the number of solutions to two particular equations: the Welch equation and the Discrete Lambert problem, using p-adic interpolation and Hensel’s lemma.

**Title**: Splines mod m

**Speaker**: Nealy Bowden

**Abstract**: A Generalized Spline is a set of vertex labels on an edge-labeled graph that represents a solution to a system of congruence equations. This talk will touch upon work in Generalized Splines with a specific focus on the special case of splines mod m, presenting new findings in the latter and discussing how they relate to and inform work in the former. Minimal generating sets are a very helpful tool when investigating splines mod m and these generating sets will be an important component of the talk.

**Title**: Large gaps between zeros of GL(2) $L$-functions

**Speakers**: Owen Barrett, Patrick Ryan, and Karl Winsor (joint with Brian McDonald, Steven J. Miller, Caroline Turnage-Butterbaugh)

**Abstract**: The distribution of critical zeros of the Riemann zeta function $\zeta(s)$ and other $L$-functions lies at the heart some of the most central problems in number theory (under the Generalized Riemann Hypothesis these zeros are of the form $1/2 + i\gamma$ with $\gamma$ real). The Euler product of $\zeta(s)$ translates information about its zeros into knowledge about the distribution of the prime numbers; similar arithmetically important results (such as the ranks of elliptic curves and the size of the class number) is encoded for other $L$-functions. A natural question to ask is how often large gaps occur between critical zeros of $L$-functions relative to the normalized average gap size. A striking connection to random matrix theory suggests that the spacing distributions between zeros of many classes of $L$-functions behave statistically similarly to the spacing distributions of eigenvalues of large Hermitian matrices. In particular, it is believed that gaps as large as arbitrarily many times the average gap between zeros occur infinitely often. However, few nontrivial results in this direction have been established.

Through the work of many researchers, the best result to date for $\zeta(s)$ is that gaps at least 2.69 times the mean spacing occur infinitely often, assuming the Riemann hypothesis. In the present work, we prove the first nontrivial result on the occurrence of large gaps between critical zeros of $L$-functions associated to primitive holomorphic cusp forms $f$ of level one. Combining mean value estimates from Montgomery and Vaughan and extending a method of Ramachandra, we develop a procedure to compute shifted second moments, which are of interest to other questions besides our own. Using the mixed second moments of derivatives of $L(1/2 + it, f)$, we prove that there are infinitely many gaps between consecutive zeros of $L(s, f)$ on the critical line which are at least $\sqrt{3}$ times the average spacing. Our techniques are general and promise similar results for other primitive $GL(2)$ $L$-functions such as $L$-functions associated to Maass forms.

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**Title**: Complex Ramsey Theory

**Speakers**: Andrew Best and Jasmine Powell (joint with Karen Huan, Nathan McNew, Steven J. Miller, Kimsy Tor, Madeleine Weinstein)

**Abstract**: Optimal bounds in Ramsey theory usually require the construction of the largest sets not possessing a given property. One problem which has been investigated by many concerns the construction of the largest possible subset of $\{1, 2, \dots, n\}$ (as $n\to\infty$) that are free of 3-term geometric progressions. Building on the significant recent progress in constructing such large sets, we consider higher dimensional analogues.

Specifically, let $\Z[i] = \{a + i b, a, b \in \mathbb{Z}\}$ (with $i = \sqrt{-1}$) be the Gaussian integers. We consider analogous questions in this setting. The construction is more involved now as there are significantly more ways to form a progression. We derive sets of Gaussian integers that avoid 3-term geometric progressions by utilizing the relationship between norms and powers of primes in the norms’ factorization. We analyze how our choice of allowed ratios affects the problem – pure integer ratios lead to a projection of the original problem into two dimensions, whereas Gaussian integer ratios create rotation and dilation in the complex plane. Motivated by Rankin’s canonical greedy set over the integers, we construct a set of Gaussian integers avoiding Gaussian integer ratios in a similar fashion, and compute its density using Euler products and the Riemann zeta function. This density turns out to be significantly lower than the density in the case of the integers. We also establish upper and lower bounds on the maximum upper density of such sets. Finally, we discuss our extensions to other quadratic number fields, and the dependence of the density on the structure of the number field (in particular, on the norm and class number).

**Title**: Expected Gonality of Random Graphs

**Speakers**: Jenna Kainic and Dan Mitropolsky (joint with Andrew Deveau)

**Abstract**: In recent years, a graph theoretic analog to classical algebraic geometric Brill-Noether has developed, and can be interpreted in the simple and fun language of chip-firing games! Given a graph to play on, it is intriguing to ask how many chips we need so that one can ‘fire’ them to any spot on the graph. Our talk will focus on understanding this number- known as gonality- in the context of other graph invariants, and ultimately on establishing tight bounds on the expected gonality of random graphs. In particular, this expectation turns out to be bounded linearly in the number of vertices. This theory has connections to multiple areas in mathematics, including abstract algebra, combinatorics, probability theory, and computational complexity.

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**Title**: Sequence non-squashing partitions

**Speakers**: Youkow Homma, Jun Hwan Ryu, Benjamin Tong

**Abstract**: Over the past 100 years, binary partitions, and more generally, m-ary partitions, have been a topic of study by many mathematicians, including Andrews, Churchhouse and Mahler. More recently, Hirschhorn, Sellers, and Sloane, among others, have explored m-non-squashing partitions due to their connection with m-ary partitions. In this talk, we define a generalized version of m-non-squashing partitions called sequence non-squashing partitions. We explore combinatorial interpretations of sequence non-squashing partitions, and discuss several congruence relationships and asymptotic properties that these functions satisfy. In particular, we provide asymptotics and congruences for m-ary partitions, which have been widely studied, as well as for factorial partitions, which have not been well understood.