REU Project Summaries

In the subject of mirror symmetry, there is believed to be a correspondence between Fano varieties and Laurent polynomials. In their paper Maximally Mutable Laurent Polynomials, Coates et al. proved that a family of Fano three-manifolds correspond one-to-one with certain mutation classes of Laurent polynomials. Here, a mutation is a change of variables which changes a Laurent polynomial \(f\) into another Laurent polynomial \(f\). The mutation graph is the graph whose vertices are all Laurent polynomials produced in this way, and edges are the mutations. In our project we will examine the mutation graph of the Laurent polynomial \(x+y+z+\frac{1}{xyz}\) corresponding to projective three-space \(\mathbb{P}^3\).

The deletion channel is a simple model of a communication channel, yet its analysis has proved surprisingly complicated over the past half a century. This project aims to develop efficient codes for communicating over the deletion channel. The goal of this project is to explore redundancies in the English language to create novel binary-English codes that are resilient to deletions. We utilize the power of LLMs to correct deletions in English texts.

Deoxyribonucelic Acid (DNA) follows a particular biological and mathematical structure. It constantly interacts with enzymes, such as DNA gyrase, which plays an essential role during transcription. There are many mathematical quantities that allow us to understand the topology of a DNA structure. With recent development in chemical and biological modeling software, a program called emDNA allows us to visualize DNA at both the base-pair step and global level. With this software and previously published gyrase-bound structure, we explored the effects of gyrase-binding on the topology of DNA minicircles. We concluded that gyrase generally causes distortion for certain minicircle lengths and with particular structural features, gyrase can cause minicircles to supercoil where high levels of torsion stress are involved.

We worked on large cardinals and compactness theorems in abelian group theory. More specifically, we generalize two classical compactness results for free abelian groups to the broader context of direct sums of cyclic groups.

A prediction market over a set of outcomes enables participants to trade based on their beliefs, with market prices reflecting the perceived probabilities of each outcome. Market designers often employ such markets to elicit information about the underlying probability distribution of an event of interest. We introduce a market design that allows the designer to allocate liquidity selectively across different subsets of outcomes. This targeted allocation causes prices in selected regions to be more or less sensitive to trading activity, enabling finer control over how beliefs are aggregated. Compared to the traditional LMSR, our approach achieves improved probability elicitation in regions of interest, while preserving the same worst-case loss guarantee

Topological data analysis (TDA) provides tools for studying time-varying data, but many existing methods struggle to distinguish data that are isometric at each time but differ qualitatively. Kim and Memoli's spatiotemporal persistent homology addresses this shortcoming but produces multiparameter modules that are computationally challenging. To mitigate this, we introduce dynamic curvature-set persistent homology, extending Gomez and Memoli's curvature-set persistence to dynamic metric spaces. The resulting modules are interval-decomposable and admit efficient algorithms for computing interleaving distances. Our construction is stable with respect to the dynamic Gromov-Hausdorff distance introduced by the same authors, and we implement both the erosion distance $d_E$ (due to Puuska) and a new Hausdorff-inspired distance $d_2$ with cubic and quadratic runtime, respectively. This enables a robust computational pipeline for distinguishing dynamic data, as demonstrated in experiments with the boids model, where we successfully detect parameter changes.

We studied the distortion of randomized voting rules under mild utility assumptions, focusing on settings where utilities are bounded below by 1 and above by a parameter $d$. We first prove a lower bound showing that the distortion of the uniform distribution over candidates can be as large as $Omega(sqrt{d})$, even when preferences are cyclic. We then match this with an upper bound: in the committee selection setting, any $alpha$-approximately stable committee of size $k = lfloor sqrt{d} floor$, when paired with a uniform lottery over its members, achieves distortion at most $O(sqrt{d})$. Additionally, we explored two extensions based on dice-induced utility models, where one candidate receives utility from a discrete distribution (interpreted as a die roll) and the others receive uniformly random utility. In the first direction, we investigated how to minimize distortion with respect to the Nash social welfare under such dice-generated utilities. In the second, we tried to construct families of dice whose induced utility distributions yield a uniform ranking over the special candidate. Our work hope to open new paths for bounding distortion in probabilistic social choice.

This paper addresses the compatibility of four fundamental axioms in social choice theory: anonymity (A), neutrality (N), resolvability (R), and Pareto optimality (PO). Building upon the seminal characterization of Bubboloni and Gori, which completely determines when an anonymous, neutral, and resolute (ANR) social choice rule exists on the domain of total orders, we examine whether Pareto efficiency introduces additional constraints. We show that the divisibility condition identified by Bubboloni and Gori - namely, that the number of alternatives m must be strictly smaller than the smallest nontrivial divisor d of the number of voters N - is both necessary and sufficient for the existence of a rule satisfying all four properties simultaneously. Our argument demonstrates that Pareto optimality can be accommodated without further restrictions, and that in the feasible case any voter's ranking can serve as the social outcome while preserving all desired axioms. We conclude by discussing the theoretical significance of this finding, its relation to symmetry considerations in social choice, and how the result may extend to domains beyond total orders, including partial orders and restricted preference classes.

The Moduli Space of ASD connections is an object of great importance in the study of smooth 4-manifolds. Specifically, it produces invariants which can distinguish between smooth structures over 4-manifolds. We provide a revised and reorganized account of the gluing construction for Anti-Self-Dual connections found in sections 7.2.2 and 7.2.3 of The Geometry of 4-Manifolds by Donaldson and Kronheimer, restricted to the case G = U(n), SU(n) and assuming the second cohomology groups of the deformation complex associated to the ASD connections A_i vanish.

We present a simple proof that any constant-approximation for clique requires (n^{Omega(log n)}) time under the Exponential Time Hypothesis. In particular, our proof avoids the highly non-trivial PCP Theorem, the foundation for all known hardness-of-approximation results for clique at the time of writing.

Szegedy and Yu introduced the concept of Rubiks tables which provide an abstraction of object rearrangement problems. In this project, we showed several problems related to Rubiks tables are NP-hard through a reduction from minimum vertex cover.

In learning theory, realizability is a property a distribution holds if there exists a hypothesis. We say that a distribution ( X x Y ) where X is a set of attribute vectors and Y ) is realizable if there exists a hypothesis within a class of hypotheses H such that the true error of the selected hypothesis is 0. When we aim to determine whether a specific distribution is realizable, using property testing can help us tolerantly test for realizability. In this paper, I introduce two different algorithms that explore efficient low-memory strategies in both the strict realizability testing case (aiming to accept only when true realizability is met) and the tolerant realizability testing case (accepting when hypothesis classes contain a hypothesis that is within ε from realizable, and rejecting all hypothesis classes that only contain hypotheses that are at least ε-far from realizable). I also provide additional future directions and conjectures about low memory realizability testing.

Alon-Edmonds-Luby (AEL) codes are combinatorial objects designed from bipartite spectral expander graphs: these are concatenated codes that offer powerful unique decoding properties with small alphabet sizes and extremely robust distance amplifications (while preserving the rate property of concatenated codes). When studying AEL codes, we follow the Hamming model: there are polynomial-time algorithms to uniquely decode an AEL codeword when e≤δ/2 (δ is the distance of the code). However, the list-decodability of AEL codes is an emerging area: following the work of Srivastava and Tulsiani, we addressed the problem of proving that when e = kδ/(k+1) as k → ∞, the singly exponential (in k) list size bounds hold but that for the realistic case of constant values of k, we can have much smaller, tight list size bounds (as opposed to the singly exponential list size bounds previously known even for local k values).

The Plackett-Luce ranking model is one of the most widely used models for dealing with ranking and preference problems. Although it simplifies work with probability distributions of individual preferences, the parameters are quite complicated to learn from the available preference data. While several methods for doing so are developed and guaranteed to converge over time, some real-world applications require a rather fast but not as precise solution. To fulfill this need, I tried to investigate the possibility of learning Plackett-Luce parameters using simple neural networks and loss functions. This work shows that as the number of voters increases, neural networks are much faster than standard algorithms in determining suboptimal solutions, which are enough for testing and highly loaded services like search engines or non-playable character behavior computation units. These models also introduce a way to initialize less random parameters, which potentially can improve performance of iterative approaches.

The Discrete Schwarz-Pick Lemma is a discrete analogue of the classical result from complex analysis, arising from the connection between circle packings and conformal maps established by Thurston. Previous works by Beardon-Stephanson and Van Eeuwen proved this lemma for circle packings where circles are tangent or intersect at non-obtuse angles, corresponding to inversive distances $I in [0,1]$. This paper extends the investigation to circle packings with obtuse intersections ($I in (-1,0)$) and disjoint packings ($I>1$). We prove that the Discrete Schwarz-Pick Lemma holds for the full range of intersecting circle packings with inversive distances in $(-1,1]$, provided an additional condition on the weights of each triangle is satisfied. The proof relies on a variational principle for circle packings with inversive distances. Conversely, we show that the lemma fails for disjoint circle packings where I≥1. This is demonstrated by constructing a specific counterexample on a triangulated disk with four vertices and providing a numerical analysis to confirm its validity beyond computational error.

We study the problem of fairly allocating indivisible goods when limited sharing is allowed, that is, each good may be allocated to up to k agents, while incurring a cost for sharing. While classic maximin share (MMS) allocations may not exist in many instances, we show that allowing such controlled sharing can restore fairness guarantees that are otherwise unattainable in some scenarios. We additionally propose the Sharing Maximin Share (SMMS), a natural extension of MMS to the k-sharing setting. We show the impossibility of the universal existence of an SMMS guarantee in the k-sharing setting. We further design an algorithm that guarantees a (1 - C)(k - 1)-approximate MMS allocation, where C is the maximum cost of sharing a good. Notably, when (1 - C)(k - 1) ≥ 1, our algorithm recovers an exact MMS allocation. Finally, we establish a connection between SMMS and constrained MMS (CMMS), yielding approximation guarantees for SMMS via existing CMMS results.

Using a proof technique pioneered by Poincare in pursuit of the uniformization theorem, we prove a transversality result which partially implies a major conjecture of Eremenko. This proof requires additionally information about the solutions to a particular Laplace equation, but we conjecture that this information is attainable in the quite general case where our metric is not co-axial.

This paper introduces a stochastic observation model for liquid democracy that accounts for uncertainty in delegation decisions, addressing limitations of existing deterministic models. By analyzing key properties such as Positive Proxy Gain (PPG) and Delegated Nash Harmony (DNH), alongside node influence metrics, we evaluate system performance. Theoretical analysis and simulations validate the model's generalizability and PPG property, though DNH may not hold in certain configurations. We incorporate machine learning and heuristic methods to predict node influence, providing robust analytical tools for liquid democracy systems.

We consider the problem of approximate Maximum Inner Product (MaxIP). While inapproximability results for high dimensions exist using Distributed PCP, this problem has not been well-studied when the dimension is small--particularly d = O(log n). In this paper, we study the application of threshold graphs for hardness of approximation of MaxIP. Specifically, we describe a reduction that would follow through if a threshold graph did exist, and then explore a potential workaround. Additionally, we discuss applications of the same threshold graph to the problem in the Hitting Set Conjecture, and future directions for this problem.

In the sequential learning problem, a network of agents make decisions, informed by a noisy private signal and the predictions of neighboring agents before them. We explore the properties of networks where agents make decisions in a uniformly random order. We characterize necessary conditions for such networks to achieve asymptotic truth learning and introduce various graph constructions that learn in different ways. We also develop an algorithm to transform arbitrary graphs into random-order learning networks using few edge/vertex modifications, with provable approximation guarantees. Finally, we analyze the robustness of learning networks, demonstrating that those achieving asymptotic truth learning under random orderings are resilient to a bounded number of adversarial modifications. Our findings reveal structural properties in networks that achieve random learning and offer algorithmic tools for engineering strong social networks.

Verifiable sources of randomness are a ubiquitous resource for modern algorithms, like machine learning and cryptography. Existing methods like distribution and statistical testing can be used to check the randomness of a source. While the distribution testing is more theoretically sound, statistical tests are often the only computationally feasible choice. This report analyzes ways to bridge techniques from both fields, reframing the NIST test suite in the language of distribution testing and using randomness extractors to empirically validate these tests. This includes technical optimizations needed to complete the experiments in a reasonable amount of time. Additionally, we explored using compression as a measure of randomness, although our results demonstrate that it is unlikely to be stronger than statistical tests. More importantly, the framework used to test the compression algorithms involve notions of epsilon-closeness, which could be used to design more robust uniformity tests.

This project focused on clock auctions, which are auctions where the price is gradually raised according to a clock and the item is awarded to the last bidder who stays in the auction. We studied how clock auctions perform on graphs, where the agents are the vertices, and our goal is to serve an item to an independent set on a graph with the aim of maximizing the total welfare. The three types of graphs we focused on were the K_1,2, star graph, and serving a matching on a graph with weighted edges rather than weighted vertices. For the K_1,2 case, we gave a randomized auction with an approximation ratio close to the current best upper bound. We also lowered the upper bound on the approximation ratio of a randomized auction over a star graph. Finally, we gave a deterministic auction for the matching case, which improves the current best auction in the literature.

Coupling from the past is a variation of traditional Markov Chain Monte Carlo methods and is used to create a perfect simulation of a model. Many MCMC algorithms sample from a close approximation of the stationary distribution. As a result, we end up with undesired convergence issues because we want to sample from the exact distribution. CFTP solves this issue and produces perfect samples from the desired stationary distribution. In this paper, we show how Coupling from the Past differs from ordinary MCMC and optimize it for a perfect simulation of a statistical mechanics model, the Ising model of ferromagnetic materials. We anticipate that our work may be a starting point for the simulation of a broader range of more sophisticated models with real-world applications relating to phase transition phenomena using this perfect sampling.

This research extends the work of Chaudhuri et al. on terminal cut functions in multiterminal networks by developing a system of necessary inequalities for networks with six terminals. We delve into the mathematical foundations of flow networks, thoroughly review the contributions for the cases with three, four, and five terminals, and generalize these findings to six terminals. Additionally, we integrate insights from recent advancements by Zihan Tan and explore the implications of submodularity and laminar families in this context.

We consider the problem of identifying the densest subgraph, one with many applications, in a novel computational setting in which we have access to only a noisy graph signal and limited queries to a strong oracle. Our algorithms solve this problem when the density of the densest subgraph is asymptotically larger than the noise of the weak signal. When the noise in the weak signal is small and the maximum density is Ω(log n), we show an algorithm that achieves a constant approximation guarantee. Alternatively, when the noise is large, we are only able to design an algorithm whose additive approximation guarantee deteriorates with the magnitude of noise. We conjecture that no algorithm can be designed to circumvent this difficulty without excessive use of the strong oracle.

The conformational preferences of B-DNA sequences hold significant research value, particularly in the context of gene expression and DNA replication processes. Equally important are the interactions between protein structures that bind to these sequences and impact the conformational preferences of the complex. However, identifying the ideal binding site to which the protein should be bound on a sequence can be challenging, as differing locations may lead to less favorable conformations that impact DNA processes. We present a method that utilizes sequence-dependent force field models derived from high-resolution structures to predict the ideal binding site for a protein structure on a DNA sequence. We enhance existing energy minimization protocols within the emDNA software and conduct a case study using the DNA gyrase enzyme and a 601 base pair DNA minicircle. Through iteratively energy minimization procedure across a sequence with varying protein binding sites, we are able to calculate and derive low-energy states of the protein-DNA complex which may reveal ideal binding locations and thus make educated inferences that may assist in solving crystal structures.

Consider a network N consisting of n agents, where all of them with ability to learn, want to correctly determine the value, i.e. state of the world they live in. Knowledge of each of the agents consists of his own private information and of actions that the other agents in the agent's neighborhood made before him. The actions of agents are being made in a sequential setting following an ordering σ. The goal is to achieve so called asymptotic truth learning on the whole network, meaning that almost all agents make a correct prediction of the current state of the world. To illustrate, when the number of agents n → ∞, then the probability of predicting the correct ground truth approaches one. In this paper we investigate and dive further into the potential relation of the truth learning and the field of statistical physics, and aim to perceive the problem from the dynamics point of view.

We are interested in counting isotopy classes of curves on compact, topological, orientable surfaces of the form Σ_g,n, where g is the genus of the surface and n is the number of punctures. We are interested in surfaces which can be assigned a hyperbolic structure; namely, this requires the Euler characteristic χ(Σ_g,n) = 2-2g-n to be negative. In particular, we are not interested in spheres with at most 2 punctures or the genus 1 torus, as (1) those are not hyperbolic surfaces, and (2) those only contain boundary-parallel or nullhomotopic curves, which are not very interesting.
Starting in the late 2000s, M. Mirzakhani produced some major results concerning geodesic counting on hyperbolic surfaces. More recently, methods have been developed so that computations may not rely on hyperbolic geometry but rather a strictly topological and combinatorial point of view, even though the underlying structures may be hyperbolic. The notion of length that we use--called combinatorial length--is based on the number of intersections with special kinds of triangulations called ideal triangulations. Our goal is to find the number of curves on a given surface under a certain combinatorial length. A known and important fact is that isotopy classes of simple closed curves have a bijective correspondence with closed hyperbolic geodesics. Furthermore, while our definition of ideal triangulations is strictly topological, there is a construction of ideal triangulations in the language of hyperbolic geometry. One result of this is that combinatorial length is proportional to the geodesic length with respect to a hyperbolic metric. Therefore, our results will serve as estimations for geodesic counting problems.

As we continue to use large public datasets for studies and models, there are increasing concerns over the risk of sharing public data. Many studies have shown that potential malevolent agents can cross-reference public datasets to find out sensitive information about individuals in public datasets. One such theoretical remedy that has been suggested is to use differentially private synthetic datasets in place of the original dataset. However, no studies so far have evaluated these synthetic data generators on actual real-life studies and measured how effective they are at replicating results. In this work, we look at the current differentially private synthetic data generators and evaluate them on recent social science studies that use large public datasets. We focus on the synthetic data generator called DataSynthesizer and analyze how faithful it is in replicating results from the study using the Panel Study of Income Dynamics. We find that Datasynthesizer struggles with complex queries that consider more than just marginal distributions. Additionally, we examine how the quality of the synthetic data varies across different settings to give future guidance on how to provide quality differentially private synthetic datasets.

Large Language Models (LLMs) are increasingly pivotal in enhancing human-AI interactions, yet their behavior in competitive environments remains underexplored. This study investigates whether LLMs can develop cooperative strategies in the Prisoner's Dilemma, knowing only the game rules and receiving no external decision-making support. We simulate an Axelrod's Tournament with a limited set of predefined strategies and one LLM agent, assessing its behavior based on successful strategy characteristics and tournament outcomes. Our findings reveal that the LLM adapts its decision-making process to each opponent, demonstrating an ability to balance cooperation and competition effectively. This adaptability suggests significant potential for deploying LLMs in complex, dynamic environments such as economics, diplomacy, and social governance. While further validation in real-world scenarios is necessary, these results provide a promising foundation for understanding how LLMs make choices within different environments and contexts.

Triple product L-functions are of great interest in number theory and quantum physics. Trilinear forms have deep connections with these L-functions, and these forms are the main objects of study in this work. This project explored the relation between two separate trilinear forms of the group GL(2, K), for some finite field K. We first used Sage to gather a large amount of experimental computations of these trilinear forms for small values of q = |K|. Then using the computational results as insight, we proved these results for general q.

Cells in the body are continuously multiplying and dying, a process that releases nucleosomal free-floating DNA into the bloodstream. This DNA, originating from various tissues, offers valuable insights into the state and progression of diseases, particularly cancer. By analyzing this circulating DNA, researchers can pinpoint the origin of the DNA and track the evolution of tumors. Our project focuses on leveraging computational algorithms and advanced mathematical techniques to detect and interpret patterns in this free-floating DNA from blood tests. These patterns can provide critical information about the patient's condition, including the emergence of drug resistance. Specifically, we aim to develop a method that not only identifies the presence of drug resistance but also constructs a timeline of its development. Early detection of drug resistance is crucial for timely intervention and effective treatment planning. Through this project, we hope to refine our understanding of the dynamics of drug resistance and improve the precision of personalized medicine. By combining computational tools with biological data, our goal is to enhance the early detection and management of cancer, ultimately contributing to better patient outcomes.

Sequential learning models situations where agents predict a ground truth in sequence, having access to their private, noisy measurements, and the predictions of agents who came earlier in the sequence. We study a generalization of this model to networks, where agents only see a subset of the previous agents' actions - those in their own neighborhood. We consider a setting where agents' rationality is bounded, only basing their decisions on a simple majority rule. The fraction of agents who predict the ground truth correctly depends heavily on the ordering in which the predictions are made. An important question in this area is whether there exists an ordering, under which the agents predict the ground truth correctly with high probability. In our project, we show that it is in fact NP-hard to answer this question for a general network for agents with bounded rationality.

What rules dictate the wrinkle-patterns in general, simply-connected curved sheets? This project aimed to expand on results in the case of simply-connected sheets to non-simply connected sheets. The overall goal was to find a relation between the upper and lower convex envelopes of admissible convex functions (ones that satisfied certain requirements in order to successfully model wrinkle patterns), which we achieved and wish to further understand and simplify. We found concrete functions that solved the optimization problem in the case of an annulus, and found leads into the cases of a shifted annulus (where the hole is not centered on the disk) as well as have some idea as to how to solve the case of a disk with concentric regions of differently-signed curvature.

We study the furthest pair with outliers problem, where the goal is to identify a given number of outlier points such that the diameter of the remaining pointset is minimized. This is closely related to various clustering problems such as the Max-k-diameter with outliers. We show that the furthest pair with outliers is NP-complete by a reduction from the independent set. Further, we share found limits of (in)approximability for the furthest pair with outliers for the usual ell-p metrics. We report that 4 + ε approximation is efficiently computable and that no PTAS exists for the problem.

Researchers are interested in using data to demonstrate the results of health and human behavior of individuals. In sharing data, researchers have begun to shift to utilizing more data visualization techniques, and while these charts and graphs are visually more aesthetically pleasing, it is important to not violate the privacy of individuals who have reported their information. These data points must be protected using different techniques. While many focus on both qualitative support and quantitative results, the focus on the outputs must use algorithms to ensure that the data privacy remains secure and protected. I experimented using a linear correlation formula to understand how correlation works with different pieces of data from the US Census. Our goal for this project is to create differentially private correlation estimates.

Proving the amount of storage necessary to learn under memory-constraints has important applications to machine learning and bounded-storage cryptography. In this project, we study memory-sample tradeoffs on the learning parity with noise problem (LPN) in the streaming model. The longterm goal is to prove a lower bound on memory-sample tradeoffs for LPN of Omega(n^2/eps^2) or an exponential number of samples. n parameterizes the size of the LPN secret, while epsilon parameterizes the sample noise. In this report, we describe intermediate observations, obstacles, and results toward proving the tradeoff within our modified branching program model, similar to prior work. Finally, we provide a conjecture and some intuition toward a weaker tradeoff, which instead tolerates a polynomial number of samples.

Essentially all of cryptography requires the use of unproven assumptions, and much work is dedicated to relaxing these assumptions as much as possible. Digital signatures were thought to require public-key cryptographic primitives until 1989, when Naor and Yung proposed the class of Universal One-Way Hash Function families, which can be constructed from injective one-way functions, a private-key primitive. This assumption was later relaxed again to arbitrary one-way functions. Naor and Yung showed these hash functions are sufficient to construct efficient cryptographically secure digital signature schemes, however, in their original paper, many details and proofs are omitted for this claim. Most of this project is dedicated to analyzing and replicating Naor and Yung's result with full exposition, and we worked towards improving their scheme in different settings.

In this project we studied properties of linear extensions of a poset and worked on proving/disproving conjectures related to them.

Property testing is a notion of approximation for decision problems, where given a property, the task is to distinguish whether a given instance has this property or is "far" from any instance having the property. A key complexity measure for property testing algorithms is its query complexity, which is the maximum number of input elements queried to determine whether the graph satisfies a given property. While there are established bounds for the query complexity of various graph properties, there are hardly any results on the impact of memory constraints. This project investigates property testing on graphs, studying how memory constraints affect the complexity of property testing in both deterministic models and scenarios involving random queries.

To make better decisions, agents often try to aggregate information from others. This phenomenon has been studied extensively through the lens of social networks. However, in real-world settings, one cannot always assume that others intend to make the best decision, or share their information honestly. Such agents act as adversaries for the task of social learning. This research investigates whether social networks can be used to aggregate information in a way that is robust against such adversaries. We present results on the robustness of two specific social networks: the Celebrity network and the Butterfly network. These results improve our understanding of how information can be aggregated effectively in social networks, both in networks with high-degree nodes and networks where all nodes have a constant degree.

Recent years have witnessed the rise of online marketplaces (e.g., Amazon, App Store) that create great convenience to facilitate matchings and improve economic efficiency. While these platforms use third-party sellers data to facilitate better matchings, they have also leveraged those sales data to inform the decisions of entering the marketplace with their own products. We analyze historical data of Amazon products provided by Keepa.com in order to understand Amazon's entry strategy and analyze the impact on third party sellers.

A hypergraph ℋ on {1 ... n} is just a collection of subsets of {1 ... n}. We say that there is a "k-Venn diagram" in ℋ if there are k sets ( A₁, ... , A_k ) in ℋ such that all 2^k intersections ( B₁ ∩ ... ∩ B_k ) are non-empty where each B_i can either be A_i or the complement of A_i (equivalently, if we draw the Venn diagram associated with (A₁, ... , A_k), all the regions of this picture are nonempty). Our goal is to try to understand the following question: how many sets can our hypergraph ℋ have if it fails to contain a k-Venn diagram?

Ergodic theory is an area of mathematics which studies the statistical properties of dynamical systems. The Birkhoff ergodic theorem is a classical result of ergodic theory, which establishes pointwise convergence almost everywhere for a specific average under certain conditions. In this project, we worked towards proving an ergodic theorem with some similarities, but involving fractional powers of primes. With techniques from analytic number theory and Fourier analysis, we studied exponential sums involving these fractional powers of primes to obtain certain bounds. Using these bounds, we use harmonic analysis tools to prove a sequence of theorems, eventually leading to our desired result.

Nonparametric statistics is a sub-field of statistics involving minimal assumptions about the distribution of data, making it applicable to the analysis of real-world phenomena. The test of Prentice is a non-parametric statistical test for the two-way analysis of variance using ranks. The null distribution of this test is approximated using the Chi-square distribution. However, the exact null distribution deviates from the Chi-square approximation in certain cases commonly found in applications, motivating adjustments to the distribution. This summer, we presented adjustments to this null distribution, and that of related tests with non-polynomial scoring systems, correcting for continuity, skewness, and kurtosis in the multivariate case.

We present the first (non-trivial) Ω(n²) lower bound on streaming algorithms for tournaments on n vertices, demonstrating that (exactly) solving the minimum feedback arc set (FAS) problem on tournaments is hard. This complements recent work on upper bounds for the (approximate) FAS problem on tournaments, and shows that even though acyclicity testing can be solved in near-linear memory on tournaments, some important related questions remain hard. We also investigate a number of fundamental graph problems on tournaments: we prove new upper and lower bounds for s-t distance and reachability, and settle the streaming complexity of acyclicity testing. Finally, we settle the streaming complexity of sink finding in general directed graphs.

For each positive integer n, let S_n be the symmetric group on { 1,2, ..., n }. Then a group G is said to be locally embeddable into finite groups if for every finite subset F ⊂ G, there exists an injection φ : F → S_n for some n ≥ 1 such that whenever g, h, gh ∈ F, then φ(gh) = φ(g)φ(h). In this case, we say that G is an LEF group.

In the group theoretic literature, LEF groups are usually characterized in terms of embeddings into ultraproducts of finite symmetric groups. It is natural to ask whether there is a characterization in terms of the more concrete notion of a reduced product of finite symmetric groups. In more detail, let P = ∏_{n ≥ 1} S_n be the full direct product and let N be the normal subgroup of elements (π_n) ∈ P such that π_n = 1 for all but finitely many positive integers n. Then the reduced product is the quotient P₀ = P/N.

We have shown that it is neither provable nor disprovable using the classical ZFC axioms of set theory that if G is a group such that |G| ≤ 2^ℵ₀, then G is an LEF group if and only if G embeds into P₀. We have obtained results concerning characterizations in terms of nonprincipal ultrafilters. Certain subtle questions about the nature of the independence remain. We have obtained analogous results for sofic groups.

Satellite knots are a method of producing new knots from existing ones, utilizing a pattern knot P inside a solid torus and a companion knot K. Two knots are said to be concordant if they form the boundary of an embedded annulus in S³ x [0,1]. P. Ozsvath and Z. Szabo defined an invariant of the concordance class of a knot, called the tau-invariant, and in 2014, A. Levine computed tau for satellite knots with Mazur patterns, which loop twice clockwise around the torus before turning around and looping once counterclockwise and joining back. We determined a formula for tau for generalized Mazur patterns with m clockwise turns and n counterclockwise turns by computing the box tensor product of the CFA- and CFD modules associated to the pattern and exterior of the companion knot, respectively.

Powder Bed Fusion (PBF) is an extremely lucrative additive manufacturing technique that allows for highly customizable parts, increased resource efficiency, and reduced cost. However, PBF is prone to overheating which results in part defects. One possible solution to this deficiency is to employ a machine learning based, data driven method to predict overheating before it occurs. Such a method must predict process behaviors with high accuracy, provide explanations and clarification for the prediction, and obey physics principles. In this paper, methods of meeting these three conditions are explored, and a resulting technique is proposed.

Logistic regression is a supervised classification algorithm that predicts the probability of an event given a set of features. This technique relies on maximum likelihood estimation (MLE) to estimate the regression parameters for the log-odds of the observed data. The MLE, however, does not always exist. In particular, if there exists a linear decision boundary separating the classes, then the MLE will not converge. In 2018, Candes and Sur proved a theoretical phase transition curve for the existence of the MLE in binary logistic regression parametrized by the ratio of features to number of observations p/n and the norm of the regression coefficients. In our project, we are interested in finding an empirical and theoretical phase transition curve for the existence of the MLE in unregularized multinomial regression for K classes (K>2).