REU Project Summaries

We examined a combinatorial game called Division game. There are 2n coins and two players, Alice and Bob, who want to split the coins so that each player has exactly n coins. In the first turn, Alice picks a coin and Bob decides who gets it. Then, the roles swap and the game continues until one of the players gets n coins. The other player then obtains the remaining coins and the player with the larger sum of coins wins the game. We conjecture that Bob always has a non-losing strategy which we proved for a small number of coins and for special classes of games.

We study vertex sparsification for distances in planar graphs: given a weighted planar graph G and a subset of k terminal vertices, the goal is to construct an emulator, which is a smaller (weighted) planar graph G' that contains the terminals and exactly preserves pairwise distances between the terminals. We construct exact planar emulators of size O(k^2) in the special where all terminals lie on exactly 2 faces.

Grid Multi-Robot Path Planning (MRPP) has been extensively studied, but much less is known about the corner-following constraint (CFC) variant, which has applications ranging from automated garages to grocery packing warehouses. On an m₁ x m₂ grid, the problem has a tight upper and lower bound for the makespan objective when there are Θ(m₁m₂) escorts, but there is a gap when there are only Θ(m₁+m₂) or Θ(1) escorts. In addition, nothing is known about the intractability of this problem or its related variants. In this work, we close the gap for an arbitrary number of escorts, finding an expected constant factor algorithm when there are at most min(m₁,m₂) escorts and a high probability constant factor algorithm when there are more. We also show that the problem is NP-hard under the makespan objective, and by the same reduction, standard grid MRPP is NP-hard also, unifying the two grid MRPP makespan hardness results. In addition, we show that when there is a single escort, the two-colored variant and partial variant are both NP-hard, which significantly impacts the design of efficient systems using the MRPP with CFC paradigm. These results contribute to our understanding of the design of efficient algorithms from both an upper and lower bound perspective.

Tumors grow from somatic cells inside the human body, often undetected without any major symptoms. But tumors shed cell-free tumor DNA (cfDNA), as well as proteins, intact tumor cells, and other molecules into the blood and other bodily fluids. As cfDNA travels through the bloodstream, it is degraded by various nucleases, salt, etc. such that when blood is drawn from patients for liquid biopsy for diagnostic purposes, the cfDNA has been fragmented into smaller pieces. The methylation, length, and molecular signatures of these cfDNA fragments have the ability to provide information about the location and type of cancer. However, the methods through which different types of cfDNA are degraded remain a mystery, and there is currently no broad, all-encompassing cancer diagnostic tool that utilizes this data. We utilize a mathematical technique called non-negative matrix factorization (NMF) to identify predominant characteristics of end-sequences of cfDNA from cancer patients with multiple types of cancer.

We consider the problem of approximate clustering with the diameter objective function. When the number of clusters is allowed to be unbounded, the inapproximability ratio is fairly well understood. However, this problem has not been well-studied when the number of clusters is constant. In this work, we show improved hardness bounds when there are a fixed number of clusters. Specifically, we show that the problem is hard to approximate within a factor of 1.5 in Hamming space, and within 1.304 in Euclidean space. Additionally, we prove several barrier results to known methods of proving hardness, and we provide a polynomial time algorithm for the problem when the number of dimensions is constant.

In this article, we study the problem of classifying graphs that admit `KT orientations'. A~directed graph $G$ is said to have a KT orientation if there is at most one directed path between any pair of vertices of $G$. We extend the known examples of classes of graphs that admit a KT orientation and those graph families which do not. We show that the problem of determining whether a given graph admits a KT orientation is NP-complete, in particular this also applies if we restrict ourselves to planar graphs. Moreover we provide an algorithm to decide if a digraph with max-degree at most 3 has a KT orientation, while for graphs with max-degree $4$, this remains NP-complete. Finally we construct a graph family with small independence number (sub-linear in the number of vertices), and thus has unbounded fractional chromatic number, that admits a KT orientation.

Given a Legendrian submanifold Λ of a circle-fibred contact manifold Z over a symplectic base, one can ask whether Λ admits a Lagrangian filling L. In this paper, we disprove the existence of such a filling for the Legendrian given by the n-fold power of the Riemman Sphere CP¹. Wehrheim and Woodward proved a dimension requirement of the image of H(Λ) in H(L). We use this obstruction and a combinatorial count of relations between elements in H₁(Λ) and H_n-1(Λ) induced by the filling to show an obstruction for fillings of (CP¹) ⁿ for n>2.

In this project, we studied Steiner minimal trees for the points given by the corners of the regular simplex. We provide an explicit formula for the coordinates of the Steiner points of a conjectured Steiner minimal tree for simplexes of dimension 2k-1. Also, we explore the function APTC(T), a notion used to characterize trees with respect to the sum of all distances between the terminals, and proved that the conjectured topology for Steiner minimal trees for the regular simplex minimizes this function over all tree with n terminals and n-2 interior points.

Finding the minimum cut of a graph has been studied extensively on its own as a fundamental graph problem and as a tool for understanding other important problems such as connectivity and network flow. We studied the minimum-cut problem in the insertion-only semi-streaming model, where edges of G are received one at a time in an input stream and space usage is confined to Õ(n) = O(n poly log n) bits. As such, we can't afford to store the entire graph at once, so we are forced to compute an approximation to the minimum cut. We present improved upper bound and lower bound results for the (1+ε)- approximation to the minimum cut problem in the insertion-only semi-streaming model. We achieve new results that are optimal in their dependence on ε.

Logistic regression is used to make a prediction between two different outcomes based on a data set. Typically, when the sample size is much larger compared to a fixed dimension, maximum likelihood estimation is used to estimate the parameters of the model, in which case the estimate has many convenient properties such as being unbiased. It was shown by Sur and Candes (2018) as well as Salehi et al. (2019) that these assumptions do not hold in the high dimensional regime where the sample size and dimension are proportional. While these two groups were able to find adequate methods of characterizing model performance in high dimensions, there is an absence of work on the performance and impact of bagging on high dimensional logistic regression models. In our case, bagging refers to the method of dividing the larger data set into two or more overlapping subsets and fitting a logistic regression model on each subset before aggregating the parts together. This work aims to show a single scalar that would be able to predict the correlation between the two unaggregated estimates. Drawing from previous results in linear regression and the unbagged setting, we were able to successfully infer this result.

In this study we focused on the prevalence of ARID2 mutations in publicly available datasets and their association with known melanoma genetic features. Leveraging publicly available data from cBioPortal, we conducted comprehensive mutation analysis and statistical tests to explore the associations between ARID2 mutations and other melanoma-associated mutations, including BRAF, NF1, NRAS, KRAS, and HRAS. Our findings shed light on the potential functional significance of ARID2 mutations in melanoma biology

A fake projective plane is a complex surface with the same Betti numbers as CP² but not biholomorphic to it. We studied the fake projective plane P²_fake = (a = 7, p = 2, ∅, D₃2₇) in the Cartwright-Steger classification. In this summer, we exploit the large symmetries given by Aut(P²_fake) = C₃xC₇ to construct a simpler embedding of this surface into CP⁵ as a system of 56 sextics with coefficients in Q(\sqrt{-7}). For each torsion line bundle T ∈ Pic(P²_fake), we also compute and study the linear systems |nH + T| with small n, where H is an ample generator of the Neron-Severi group. Finally, we constructed explicit equations of the fake projective plane $(a = 7, p = 2, ∅, D₃X₇), a closely related fake projective plane to P²_fake.

This project was focused on flag varieties and computations in cohomology rings. I studied symmetric polynomials and covered different basis for this family of functions, as well as computations in one of these basis, the set of Schur functions. I also studied root systems, Weyl Groups and the classification of irreducible root systems. Finally, I worked on some actual cohomology rings and how they connect to root systems and multiplication of Schur functions.

The 1/3−2/3 Conjecture states that in every finite partially ordered set that is not totally ordered, there exists a pair of elements x and y with the property that at least 1/3 and at most 2/3 of the linear extensions of the partial order place x earlier than y. This conjecture has become one of the more prominent conjectures in set theory, and while no one has succeeded to prove such a result for all partially ordered sets, there are partial results for posets with certain properties. Our goal this summer was to prove that this conjecture holds for a specific set P_3,n. which consists of {x₁, y₁, z₁, x₂, y₂, z₂, ..., x_n, y_n, z_n} where x_i≤x_i+1, y_i≤y_i+1, z_i≤z_i+1, and x_i≤y_i≤z_i, for all i.

We are interested in designing social networks which support the spreading of true information while limiting the ability for disinformation to spread. Having access more information doesn't always lead to discovering the truth as phenomena known as "information cascades" can lead to herd mentality propagating false information. Representing social networks with graphs, we studied two areas: high degree and low degree graphs. High degree graphs have the ability to quickly aggregate information and decipher truth into a few high value nodes with many neighbors. One example we studied was the preferential attachment model by Albert-Barabasi. In low or constant degree graphs, a more nuanced approach is needed to allow truth learning to succeed. We studied two examples of these: an adjusted Connected Caveman graph and a grid structure. Moreover, in large-scale networks such as news portals or content-sharing platforms, having a solid grasp of the underlying network structure and ordering helps optimize the process of dissemination and improve the accuracy of information spreading. In modern society, misinformation is a pervasive issue that can have large societal consequences. As we strive to combat the spread of false information, studying the ways information spreads in a variety of networks becomes crucial.

The ribosome is responsible for creating proteins. Understanding how the ribosome itself folds into its final 3-dimensional structure is critical to deepening our understanding of the mechanisms behind proteins, and thus life itself. There is much research that has been done in understanding the 3-dimensional structure of ribosomes, however many of these studies have fallen short in giving an intuitive understanding of the 3D structure. In this project, we used computational models to simulate large subunit ribosomal RNA of Escherichia Coli.

In data-driven research fields like healthcare and technology, access to sensitive individual information is crucial for generating valuable insights. However, privacy regulations pose challenges for publishing such data, hindering researchers' access. Differential privacy has emerged as a promising solution, allowing researchers to learn from sensitive data while protecting individual privacy. This research focuses on generating differentially private scatterplots, a common data visualization tool, while retaining visual utility. The approach combines strategies of partitioning data, adding calibrated noise, and post-processing to suppress noise. The study explores factors like ε (privacy level), algorithms, bin size, data distribution, and sample size to understand their impact on visual utility. Two small datasets are used, derived from the US Census and World Health Organization, with direct application for Differential Privacy. The results highlight the trade-offs between privacy and visualization integrity, aiding researchers in making informed decisions to balance privacy and data visualization accuracy. Future directions include exploring more sophisticated point regeneration techniques and integrating differentially private heatmaps techniques to improve visual accuracy

Differential privacy is a promising mechanism for privacy protection, providing a way to calculate the maximum amount of privacy risk when an individual's data is included in a study. The US Census Bureau recently incorporated differential privacy to protect census data using the Disclosure Avoidance System (DAS). The DAS essentially consists of 2 steps: noise injection and post-processing. Researchers have conducted analyses on the DAS to examine its effectiveness and found numerous biases, which they specifically credited to the post-processing step. However, the US Census Bureau did not release the census data following the noise injection but before the post-processing, so there is no way to differentiate the effects of each step. Through the production of noisy data and replication of research by Kenny et al. (2021, Science Advances), we found that the biases found were due to post-processing. More importantly, we found that the analyses we replicated presented misleading results. When analyzing the error of the DAS, the researchers used a fitted error, which they calculated using a generalized additive model (GAM). The model's predictor variables included parameters plotted on the x-axis of their figures, essentially creating the illusion of stronger trends.We found that using the fitted model resulted in deceptive images that exaggerate the effect of the biases. By replicating their study, we found that the biases are still present, though less intense. We hope that our findings will provide a more accurate depiction of the biases of the DAS.

Consider the following problem: we are given a graph with $n$ vertices and $n$ distinct tokens. Given two arrangements of the tokens on the vertices, we wish to know the length of the shortest possible sequence of "edge-swaps" needed to get from one arrangement to the other. Here, an edge-swap refers to the act of swapping tokens that lie on adjacent vertices. This problem, known as token swapping, is APX-hard, and a constant-factor approximation has been known since 2016. This summer, I studied the approximability of token swapping on general graphs.

The Johnson-Lindenstrauss lemma shows that for any n points in R^d and ε>0, there is a map into O(\log(n)ε^-2)-dimensional Euclidean space that distorts the pair-wise ℓ² distances of S by a multiplicative factor of at most (1+ε). This result is known to be optimal for ℓ₂, whereas our understanding of dimensionality reduction in Hamming space is far from complete. In this project, we looked for dimensionality reduction methods in Hamming space, with a focus on an approach using the 7-4-3 Hamming code.

Nervous systems are organized for efficient integration of information. It has been shown that, in neuronal connectomes of the C. Elegans worm, high degree nodes are highly connected to form what is called a "rich-club" structure. This rich-club structure allows different neurons to reach one another in relatively few edge hops. The rich-club organization is a global architectural feature of the C. Elegans connectome. It has yet to be explored how the local organization of the connectome is supporting integration integration. By using the notion of course Ricci curvature (a generalization of Ricci curvature) and a path motif analysis, we found that the C. Elegans connectome exhibits local weak-bridge structures. More importantly, our analysis highlights some of the critical pitfalls in hypothesis testing graph properties. These pitfalls motivate the need for a principled investigation into statistical methods for determining the significance of graph properties.

In this paper, we lay the groundwork for a formal justification of the plane wave assumptions underlying Arthur Compton's scattering calculation. Using a recent relativistic formulation of Bohmian mechanics, we start by examining the behavior of a single free photon and a single free electron. We offer significant evidence that the electron wave function becomes asymptotically plane wave-like, meaning the particle's momentum and energy approach fixed values as assumed in Compton's calculations. We examine the effects the wave function's initial parameters have on the behavior of particle trajectories, and we explore different ways to visualize trajectories and this asymptotic behavior prior to interaction.

Graph Cities are a new framework for visualizing, exploring, and interpreting graphs that have the order of billions of edges. Graphs of this size are simply too much data to be meaningfully processed by humans, from the macro scale down to the actual semantics. Graph cities break such "huge" graphs into metaphorical buildings and meta-nodes that can be individually explored. In this manner, large graphs can be explored at different levels of detail to extract patterns at both the macro and micro scales. The focus of this paper is on speeding up the computation of the meta-graphs within each building, allowing the graph city to be explored interactively rather than needing to be precomputed.

DNA minicircles of base pair length 336 have recently been used to inject genes into mice. The step-parameter model describes a DNA strand's configuration in space and is packaged into the software emDNA, which currently finds the minimal energy state for an input sequence. This project expands upon emDNA and adds the emDNA-move tool, allowing for the thermal fluctuation of constrained linear or circular sequences of DNA. This tool was developed in C++ and allows for the customization of the degree of random movement and fluctuation length. This project also compares the deformability of each unique tetramer and the coupling of step parameters.

Public transit in America in recent years has suffered from attacks and crimes since they are very vulnerable to terrorist/mass casualty attacks. These vulnerabilities are due to the lack of strict screening and content policing, unlike security at airports. Although current public transit is designed to efficiently allow a way that it allows for passengers to quickly travel as needed, there is not a strong security system in place. Utilizing metro station security check systems (SCS) can achieve great scrutiny by transit authorities and the public due to their high throughput, risk factors, and a demand for safety. Modern SCS achieves safety through many layers of active and passive security checks. In order to implement strong security around public transit around America, it is important to understand the different types of transit stations and how they are operated, and the types of passengers that utilize them. This paper aims to develop an understanding of the current state of security check systems as applicable to high-traffic subway stations and what must change in the near future to establish effective security check systems in metro systems. By working toward creating a proof-of-concept risk analysis model using crime and other types of publicly available data on the NYC and NJ regions was used to make predictions on how transit stations are more at risk than others. With these predictions, it would be up to local governments and stations to implement more appropriate security measures.

We worked on a method which allows for human visualization and exploration of large graphs (i.e. billion edge graphs). Our method relies heavily upon the core decomposition algorithm developed by Abello and Nakhimovich. Using this method, we create interactive Graph Cities which makes human interpretation of exponentially larger graphs possible. The Graph Cities interface also makes it easier to catalogue patterns and subgraphs of the parent supergraph. We also studied compelling avenues in which this work could continue.

The 1971 Fortuin-Kasteleyn-Ginibre (FKG) inequality for two monotone functions on a distributive lattice is well studied and has shown to have many applications in statistical mechanics and other fields of mathematics. In 2008 Sahi conjectured an extended version of this inequality, called (E_n) for all n>2 monotone functions on a distributive lattice. We considered a special version, namely (F_n) and examined its properties on the space on n monotone Boolean functions. We showed that (F_n) does not satisfy the quasi-concave property by formulating a counter-example. We proved that (F_3), over one-dimensional monotone Boolean functions, does not possess non-zero minima and we examined a method to prove a more general version.

Decomposition of a graph into sub-graphs that are invariant with respect to degree peeling, i.e. fixed points, is a useful method of community detection in networks and visualisation of massive graphs. However, with the analysis of massive graph comes the problem of memory size limitations. Currently, present algorithms for fixed point decomposition cannot be used in the scenario where the whole graph doesn't fit into memory,since those approaches would result in too large I/O time. We present two methods for the computation of these decompositions and we also propose a different decomposition that is not based on fixed points but can be computed more efficiently.

We formalized essential theorems in Elementary Analytic Number Theory into the Lean 3 Theorem Prover, with the eventual goal of contributing the work to the already vast library or formalized math in Lean called Mathlib. We focused on Dirichlet's Hyperbola Method as applied to the Divisor function beginning with a mathematical description of the theorem.

Our goal was to introduce L-functions and explain their relevance to the Langlands conjectures. Along the way, we defined Godemond-Jacquet L-functions, which generalize all other known L-functions.

In this project, I mainly explored how the hardness of one way functions is preserved. In particular, given a certain circuit C that computes a one way function that has certain hardness, what can one say about the existence of circuits that compute one way functions that are harder? We investigated a possible approach to prove lower bounds on the hardness of one way functions whose existence is implied by the existence of the circuit C.

This project primarily focused on the class NISZK_L, introduced in a paper from 2020 done with REU students and Eric Allender. Little was known about this class beyond the results in that paper and results from REU students the following summer, mainly comprising of two complete problems, hardness results relating to MKTP, and equivalence to a weaker subclass of NISZK, NISKZ_AC^0. This summer, we expanded these results to primarily show equivalences between NISZK_AC^0, NISZK_L, and NISZK_PM, among an array of other related results.

In this project, we attempted to construct a rate-1 compiler for a non-malleable code with respect to tampering from functions that can be computed by circuits of polynomial size. We extend the ideas presented by Ball, Dachman-Soled and Loss who presented such a code with rate 0. Ourc ompiler does not preserve the non-malleability properties of the code and we present an attack that shows this. Finally, we present some possible improvements in this direction.

We consider a manipulability of tournament rules on $n$ teams, where the rules select (possibly randomly) a single winner based on the result of all $\binom{n}{2}$ matches. A tournament rule is said to be $k$-SNM-$\alpha$, if no $k$ teams can increase their joint probability of winning by fixing the $\binom{k}{2}$ matches between them; hence, $k$-SNM-$\alpha$ is a measure of manipulability of a tournament rule. Previous works show that among all Condorcet-consistent (i.e., undefeated teams always win with probability 1) monotonous (i.e., no team can increase its probability to win by throwing matches) rules the best we can hope for are $k$-SNM-$\frac{k-1}{2k-1}$ rules and show several examples of rules achieving this bound for $k=2$. In the case of $k=3$, there exists only one rule achieving non-trivial non-manipulability, specifically $3$-SNM-$\frac{31}{60}$. Our main result is an existence of a new rule that is Condorcet-consistent, monotonous, $2$-SNM-$\frac{1}{3}$, and $3$-SNM-$\frac{1}{2}$. Additionally, the analysis of this rule is tight and uses a new technique with the possibility of finding similar rules achieving even stronger non-manipulability. Our second result shows that rules for tournaments on $n$ teams satisfying a stronger version of Condorcet-consistency can be extended to rules for any number of teams with only slightly worse manipulability. Finally, for every $d \geq 3$, we generalize a~random single binary elimination bracket rule to a~random single $d$-ary elimination bracket (RS$d$EB) rule (the complete binary tree is replaced by the complete $d$-ary tree). We show that, for every $k \geq 3$, the rule RS$k$EB is Condorcet-consistent, monotone, and $k$-SNM-$\alpha$ for $\alpha < 1$ depending on $k$.

Multi-item revenue-optimal mechanisms are known to be extremely complex, often offering buyers randomized lotteries of goods. In the standard buy-one model it is known that optimal mechanisms can yield revenue infinitely higher than that of any "simple" mechanism, even for the case of just two items and a single buyer.We build off of the manuscript of Assadi and Schvartzman, who first defined the notion buy-$k$ mechanisms, which smoothly interpolate between the classical buy-one mechanisms and the recently studied buy-many mechanisms. Buy-$k$ mechanisms allow the buyer to buy up to $k$ many menu options. Our main results are that the revenue gap with respect to bundling, an extremely simple mechanism, is bounded by O(2ⁿ·n²) for any arbitrarily correlated distribution D over $n$ items for the case of buyer's with arbitrary monotone valuations. This is a generalization of the known O(n²) bound for additive valuations. In addition to this, we also conjecture that there exists a distribution D over two items such that Buy(k)Rev(D) > Buy(k+1)Rev(D). We make partial progress towards proving this conjecture by providing a candidate distribution D and a candidate optimal mechanism M_k which we believe witnesses such a gap.

The Markoff equation is the equation x² + y² + z² = 3xyz: we are interested in the solutions of this equation over Z or F_p, for a prime p. The Strong Approximation Conjecture states that, for each prime p and nonzero solution (x₁; x₂; x₃) to the Markoff equation modulo p, there is a corresponding solution (x₁'; x₂'; x₃') to the same equation over Z such that x_j=x_j' (mod p). We worked on the properties of these solutions.

Sale of independent items is an often studied problem. Myerson's lemma provides an efficient way to realize an optimal trade of a single item. However, in a multi-item setting, the results can be non-intuitive. It has been shown before that in some cases, buyers having a higher valuation for sold items can lead to lower revenue of the seller. This non-monotonicity is not yet understood. In this paper, we provide conditions which guarantee monotonicity and an algorithm to generate non-monotone examples. We show that such examples are rare, even though the revenue can change significantly in some cases.

Machine learning, specifically neural networks, are at the heart of many important everyday jobs such as image classification, and natural language processing tasks. Advances in today's neural networks, have led to very impressive results in varying tasks, with the impressive results comes the disadvantage of very large neural network models, with parameters in the range of hundred millions to billions. There has been a lot of ongoing research to be able to provide solutions to the problem of compressing large neural network models, to perform as well as the original model or even better. The goal currently is to be able to compress deep neural networks to be able to deploy and use on edge devices, failure to deploy large models is inevitable due to memory and computational constraints on edge devices. We explores the trade-offs between compression and accuracy in neural networks using a neural network compression method 'Pruning Filters for Efficient ConvNets'. Neural network compression methods are used to reduce the memory space and computational costs of neural networks, and these methods include but are not limited to pruning, quantization, and knowledge distillation. The tests were conducted using the MNIST Fashion data-set, as well as the CIFAR10 data-set, both of which are image classification data-sets. The model SCNNB is a convolutional neural network that was used for conducting tests.

We consider the Steiner tree problem in Hamming space. While this problem is known to be APX-hard, no hardness of approximation factor is known. We show that the Hamming Steiner tree problem is NP-hard to approximate within a factor of 104. In showing this, we derive a structural correspondence between vertex covers of 4-regular graphs and optimal Hamming Steiner trees of particular point configurations in Hamming space. As a corollary of our results, we show that the Rectilinear Steiner tree problem is NP-hard to approximate within a factor of 104. We also show analagous results for discrete variants of the Steiner tree problem. That is, the Hamming and Rectilinear Discrete Steiner tree problems are NP-hard to approximate within a factor of 104.

Developing a proof of Soddy-Gossett's Theorem that uses inversive geometry is a useful step towards advancing the Lean library and allowing more complex theorems to be proven in Lean. In this project, we produced a concise proof of Soddy-Gossett's Theorem using inversive geometry and began the implementation of the proof in Lean. Thus far, we have defined important concepts that are necessary in the proof such as the definitions of the co-radius, inversive coordinates, the inner product, and the inner product matrix. However, the more complex preliminary lemmas and the main statement of Soddy-Gossett's theorem are still in the process of being proved. A future goal after the conclusion of this project would be to complete the proof of Soddy-Gossett's Theorem in Lean and eventually extend the ideas of this proof to other theorems whose proofs can be simplified using inversive geometry.

Fake projective planes are smooth complex surfaces of general type with Betti numbers equal to that of the usual fake projective plane. Recent explicit constructions of fake projective planes embed them via their bicanonical embedding in P⁹. In this paper, we study Keum's fake projective plane (a = 7, p = 2, {7},D₃2₇) and construct an embedding of the fake projective plane in P⁵. We also simplify the 84 cubic equations defining the fake projective plane in P⁹.

Topological Data Analysis is a promising approach for analyzing brain connectome data due to its ability to abstract underlying geometric structures in complex networks. The research presented in this study investigates a proper filtration that can be used to generate meaningful barcodes in order to represent the functional brain network of humans. Filtration using curvature was also used to explore the neural network of C. Elegans. Analysis of barcodes established a pattern among brain networks. Based on these visualizations, it was determined that differences in connectivity can be observed among various connectomes.

Intratumoral heterogeneity is a feature in the majority of human cancers. The capability to observe the spatiotemporal clonal dynamics in the tumor of a patient is severely limited, raising the need for a mathematical model to further the study of ITH. We present STEDISH, a stochastic mathematical model for tumor growth incorporating the concept of spatiotemporal heterogeneity. STEDISH allows examining the patterns of spatial ITH under different models of tumor evolution and other constraints.

The visibility graph recognition problem asks to determine if for a given graph G there is a polygon P having G as its visibility graph. It is not known to be in NP. Here we show that persistent graphs having a block of consecutive blocking vertices can be realized as a polygon.We conjecture that this technique can be extended to realize a larger subclass of persistent graphs.

In this project, we consider the well-known Weighted Max-Cut problem in the context of two-party communication. We establish a two-way randomized communication complexity lower bound of Ω(n^2) for Weighted Max-Cut on graphs with n vertices, which differs from the trivial upper bound by only a logarithmic factor. To establish the desired lower bound, we construct a chain of communication reductions: Disjointness to 3-Coloring, 3-Coloring to NAE 3-SAT, and finally NAE 3-SAT to Weighted Max-Cut. The reductions we present are non-arbitrary, as the size of constructed instances and the amount of communication used in the reduction impact the strength of the resulting lower bound. We thus apply reductions in a white-box manner, establishing the communication complexity lower bound of Ω(n^2) for Weighted Max-Cut and additional lower bounds for the intermediate problems of 3-Coloring and NAE 3-SAT in the communication reduction chain.

As the COVID-19 pandemic progresses across the nation and the world, data suggests the virus disproportionately impacts marginalized communities. Specifically, disparities exist in infection rates and disease outcomes by race and ethnicity in the United States. However, it is unknown to what degree these differences have impacted specific communities in the past year. During the course of this research project, I intend to research various avenues related to COVID-19 including but not limited to long term care facilities and education systems.

The COVID-19 pandemic has a global impact due to its high infectivity and due to human travel behaviors. Focusing on geographic patterns to analyze the evolution of the pandemic, we can determine which areas are at risk of an outbreak. By analyzing the spatial correlations of new active cases in the US at the county level we can show the extent of these correlations at different times. Our results show that the epidemic was largely not uniform but formed clusters. We found that in the first few months long-range spreading was mainly located in cities rather than rural areas. In addition, there exists a percolation transition in November 2020, and a smaller transition in January 2021, both corresponding to peaks of the epidemic.

Many processes such as the diffusion of information or fake news over a social media network or the transmission of disease may be modeled with an infection process. We investigate the problem of finding the patient zero using noisy observations of a set of infected nodes. Given a set of detected infected nodes and the background graph, we present a MCMC algorithm to calculate the probability a node is the patient zero. We also prove that for an infinite lattice Z^d and a set of n infected nodes, for any ε-level, there exists a credible set C_ε such that there are O(√ n log^(2d)n) nodes in C_ε.

This paper is an extension of the work done by Xintong Wang, David Pennock, and others on combinatorial options markets. Their original paper focused on the design and time complexity of an exchange that accepts call and put options on multiple stocks in a single bundle. These options give traders greater flexibility and precision in their trading. The original work focused on determining the optimal match in single-instance auctions. We extend this in two ways: providing a system for transferring the surplus from these trades back to the traders and designing a continuous combinatorial auction. One of the key principles of modern exchanges is that their profits come from a small fee on every trade, not from any surplus that occurs during the course of trading. If one trader is willing to buy a stock at a given strike price for $10 and another is happy to sell for $7, the $3 surplus is given to whichever trader placed the second bid. In a normal market, the surplus is always a flat amount since identical options are being bought and sold. However, in a combinatorial market, the surplus potentially also contains a variable surplus that depends on the relative successes of the various options. In this paper, we propose a method of transferring the surplus in a combinatorial market from the exchange to a trader by providing them with a bundle of options. In addition to dealing with the surplus, this paper extends the idea of a combinatorial market to the continuous-style auctions found in real-world exchanges. Wang's work focuses on a single-instance auction where all offers are collected and a single optimal match is returned. In practice, most markets are running continuously and decide on trades when new bids are placed. This style of auction allows for faster trading and benefits quick traders over those with better offers. In this paper we introduce a mechanism for a continuous combinatorial market and make some progress on an algorithm for running one.