REU Project Summaries

In the nuclei of healthy human cells, DNA is compacted into chromosomes. However, in the nuclei of cancerous cells, there are also small closed loops of DNA called extrachromosomal DNA. These loops of DNA are believed to be responsible for the incredible adaptability of cancerous cells. Here we provide methods for analytically modeling loops of extrachromosomal DNA through the use of B-spline and Bezier curves. In each case, curves are first determined, local reference frames are attached, and then idealized base pairs corresponding to a desired chain sequence are plotted in each of these reference frames. Through the use of these models, we have studied various phenomena related to the structures of these closed loops of DNA.

We study sum edge coloring for multigraphs, motivated by the bi-processor scheduling problem; the jobs must be scheduled in such a way that no two jobs that share a processor may run at the same time and the average completion time is minimized. If edges are jobs and vertices processors, then the scheduling problem is precisely edge coloring with the intention to minimize the sum over all colors used. We investigate two variants of sum coloring: the Minimum Edge Chromatic Sum (MECS) problem and the Optimum Cost Chromatic Partition (OCCP) problem. The former aims to find an edge coloring that minimizes the sum of the maximum color on each edge while the latter the sum of all colors used. Despite similarity of the two objectives, they are quite different in complexity. This difference is most observable when restricted to trees; we show that the former remains NP-hard, while the latter is polynomially solvable.

Security enclaves are sections of chips that run trusted and verified code that cannot be accessed by other parts of the computer. If code running on an enclave would like to use a key-value store that is stored on the disk, it will have to go through the untrusted main OS. We investigated how to verify integrity and maintain secrecy with regards to this key-value store that the enclave is using. We first showed how to do it in the case when our data structure is a tree. Then, we used this idea to construct a faster set membership proof on disk. Finally, we made this construction write optimized using ideas drawn from B^ε-trees.

The celebrated Brooks' theorem in graph theory states that every connected graph which is not a clique nor an odd-cycle can be Δ colored; here Δis the maximum degree of the graph and we use n to denote the number of vertices. We investigate the possibility of obtaining a Δ-coloring via a space-efficient streaming algorithm. That is, an algorithm that makes one pass over the edges of the input graph while using a limited memory - much smaller than the input size which can be as large as Θ(n²) - and at the end of the stream, outputs a coloring of the input graph. Our main result is a randomized algorithm for this problem that uses O(n^7/4) space.

The ball-recycling problem, a variant of the common balls-and-bins problem, has m balls thrown into n bins, typically according to some probability distribution p. Repeatedly, one bin is selected, and its balls are removed and rethrown according to p. We want to maximize the expected number of balls rethrown at every step. I investigated worst-case performance for online ball-recycling algorithms, that is, ball-recycling when the balls are rethrown not according to p, but according to some adversary. The goal was to find an online algorithm that gets as close as possible to the offline optimum algorithm over all possible adversaries.

The field of neuroscience has long been challenged in quantifying the nature of the neurological disorder of schizophrenia. With the recent introduction of computer modeling, researchers have been able to more formally examine the mechanisms which have been proposed to explain the symptoms and behaviors found in those with the disorder. A variety of biophysical models, often centered around certain neurotransmitter systems such as the dopamine or glutamate systems, have been produced and studied over the past few decades. While these models do often align with certain symptoms found in individuals with schizophrenia, their predictions often contradict certain other observed symptoms or behaviors. This offers a challenge to researchers to either find a different holistic representation of the entire disorder or focus on a particular behavior found frequently in the disorder and explain it in a way that corroborates the relevant preexisting scientific literature. In this work, we opt for the latter by examining the proposed mechanism of apical amplification and its potential role in schizophrenia through the use of a dendritic compartment model architecture. We demonstrate that a neuron model with dendritic compartments is able to mimic the behavior of a neurotypical population in a visual classification task involving distinguishing a contour made up of specifically oriented gratings from a surround of randomly oriented gratings. This neuron model utilizes apical amplification to learn and replicate when individuals in the target population are able to distinguish the contour from the surround and when they are not. This suggests that apical amplification is a plausible mechanism for understanding the underlying neuronal mechanisms used in visual classification tasks. We hope to further this hypothesis in later research by perturbing the network parameters trained on the neurotypical population to then mimic the behavior of the population of individuals with schizophrenia. The new parameters of the model and their distinct configuration in contrast to that of the neurotypical population should provide a biologically plausible insight into whether aberrant functioning of the apical compartment is a reasonable hypothesis for explaining the impairments observed in individuals with schizophrenia in visual classification tasks.

Graphs are everywhere and are growing increasingly compact with data. Understanding the data found within requires a summarization process to dwindle the information down to a human digestible amount. When summarizing text, translating how humans build connections and realize the semantic meaning of phrases to a computer is far from trivial. Some methods involve computing scores for each word based on how often they occur in a given text and computing word vectors to determine similarity between words. However, none of these processes by themselves work well to accurately summarize text. The result of my process shows that combining techniques can produce promising summarized representations of information. These results demonstrate the potential that finding the perfect combination of techniques has. I anticipate my process to be a starting point for more sophisticated combinations of methods to develop.

In Razborov's 1995 paper "Unprovability of Lower Bounds on Circuit Size in Certain Fragments of Bounded Arithmetic", we are told that a certain connection exists between communication complexity and circuit complexity: we are told that if we have a DAG-protocol of size S for the monotone mKW game of f, then there exists a monotone fanout circuit of size S for f, and vice versa. In this work, we begin by discussing a restriction we must place on DAG-protocols for the first direction of this claim to be true; we call this restriction being dead leaf free. We then explore protocols that have what we call reducibility at merges, and we explore what a protocol having reducibility at merges can tell us about the communication problem underlying the protocol. We then think about the relationship between protocols that have reducibility at merges and protocols that are dead leaf free, and we introduce a few open problems about this relationship -some computational in flavor, some combinatorial. Finally, we introduce a new kind of protocol called tie breaking protocols, and we show that tie breaking protocols that are dead leaf free and what we call fair characterize monotone comparator circuits. We end by discussing how we anticipate this protocol can be useful to finding a separation between monotone formulas and monotone comparator circuits.

Our work focused on a variant of the circuit minimization problem (MCSP), denoted MKTP, which studies resource-bounded Kolmogorov complexity in place of circuit size. These problems have gained attention as promising candidates for NP-intermediate problems. We show that MKTP and Graph Isomorphism (GI) are hard for the class DET via nonuniform projections. Previous results have proven hardness under logspace reductions and nonuniform NC0 reductions, our paper strengthens these results.

During pancreatic tumor progression, tumor cells typically disseminate from the primary site to other organs including the liver. These disseminated tumor cells may remain dormant for months, years, or decades before entering metastasis. Single cell research on dormant tumor cells and tumor cells can uncover how gene expression and copy numbers vary within these cell populations. While comparing single cells for both pancreatic tumor and dormant liver cells, two distinct clusters/cell types were distinguished. One cluster consisted of tumor cells, and some dormant tumor cells mixed in. These dormant tumor cells are most likely tumor-like cells. The other cluster identified only consisted of dormant tumor cells which means it may be a group of normal cells or a subpopulation of tumor cells. In addition a copy number analysis was ran for each sample, and showed that in comparison to a reference genome, both the tumor and dormant cells did not have many differences in copy numbers. These results demonstrate that the tumor and dormant tumor cells are very similar in gene expression, and additional analysis is needed on the small changes in gene expression that account for their differences. Furthermore, if specific genes expressed are responsible for the dormancy of a tumor, anti-cancer drugs can be developed that regulate the gene expression of a tumor cell to look like that of a dormant cell.

We studied the Glauber Dynamics for the ferromagnetic Potts Model. We attempt to prove that the initial coloring of all blue has highest probability of ending all blue for any fixed number of updates. For two colors, a solution is provided. For three colors, we prove the result for a restricted class of graphs, and disprove the existence of a monotone coupling for many other graphs.

We studied a new method for simulating T(n)-time-bounded Turing machines by unbounded circuits with low depth and size polynomial in T. Our construction offers lower depth circuits than the ones by Ryan Williams in the case that the total number of reversals of every tape head is less than O(T=logT). This continues a research direction of Borodin to study the connections between simultaneous Turing machine time and space complexity and simultaneous circuit size and depth complexity, which is heavily connected to the research of Hopcroft, Paul, and Valiant on time and space. Finally, reducing the depth of circuits for Turing machines has applications to parallelizing sequential computations, as discussed in Ryan Williams' aforementioned paper.

The use of technology has spread in many industries, including the law enforcement sector. An example of how technology applies in The Department of Homeland Security (DHS) and local law enforcement services is the use of digital forensics to collect and present evidence in court proceedings. The such use of technology has improved the storage and accessibility of critical data in law enforcement sector. However, digital forensic also have some downsides, such as changing laws and technology. As a result, homeland security and law enforcement personnel require continuous training and issuance of certification to enhance their skills in the collection of digital evidence and protecting the public.

Digital forensics is a branch of forensic science involving the recovery and investigation of data from digital devices ("Law Technology" 2018). The increase in cyberattacks and the wide-spread use of technology across the nation has made digital forensics an important component of the criminal justice system. To help avoid challenges when recovering data, digital forensics professionals should be properly trained and receive relevant certifications. The purpose of this project was to analyze digital forensics training and certification requirements for the Department of Homeland Security investigative units and State and Local Law Enforcement, work with the Federal Law Enforcement Training Center (FLETC) to identify opportunities and gaps in digital forensics training, and recommend digital forensics training and certification pathways to standardize training and certification across all of Homeland Security. Two analyses were completed regarding the cybersecurity, digital forensics, cyber forensics, and computer forensics fields to address this problem. The first was a labor analysis focusing on the number of job openings, job titles, certifications, specialized skills, and National Institute of Standards and Technology (NIST) skills. The second was a course training analysis, which includes a comparison of the cost, proficiency level, duration, investigation step applicability, forensic category, and delivery method of each course. This is an ongoing project, but perhaps the discoveries from the analyses will contribute to the Criminal Investigations Network Analysis (CINA) project team's research regarding the certification and training requirements for the Department of Homeland Security and State and Local Law enforcement for the Federal Law Enforcement Training Center (FLETC).

Our work focuses on building preparedness and resilience within vulnerable communities, specifically religious communities and houses of worship, against violent extremism. In today's society, the threat of mass casualty attacks and targeted violence is a global issue, with a notable increase in anti-Islamic and anti-Semitic attacks, as well as an increase in attacks by white nationalists. Many houses of worship have been targets of these attacks with numerous lives lost and many others left physically and emotionally impaired. With anti-Semitic and anti-Islamic incidents occurring globally, it is important for houses of worship to have the tools and guidance to prepare for and prevent further damage; however, there is no widely available, comprehensive resource of best practices or methods for religious communities. The R.E.S.I.L.I.E.N.C.E. model was created as a basis for our study. We researched case studies that highlighted best practices that can educate vulnerable communities about resources and methods to better protect themselves. We have found many strategies to do so, such as building connections between the public and enlisted guardians, sharing information through constant communications, implementing the best practices and plans for a situation, providing sufficient resources for said plans, and neutralizing negative mindsets through encouragement and empowerment. Previously, these strategies were not compiled into a single resource that would be available to the public and vulnerable communities, but with this research, that can change. There are many roles and responsibilities community members can assume to build readiness. While analyzing these roles and responsibilities, we also found many exemplary ways to utilize the aforementioned strategies and build resiliency against any form of violence within any vulnerable community.

Ephemeral messaging is messaging to a public phone number or email account, where all messages are publicly displayed on a website but disappear after a short period of time. We have collected and analyzed over 200,000 text and email messages to explore the uses of ephemeral messages and the security and privacy issues that they present. Unlike private and permanent messages, very few ephemeral messages are part of personal conversations, and most of them contain codes to verify accounts. From these messages, we find that many companies may have a bug or bias in their random number generator, leading to their reuse of verification codes. We also find that many messages contain very specific and personal information that strangers may use to conduct social engineering or access personal, private accounts.

Sea level change is an important issue that will impact millions of people living on United States coastlines by the end of this century (Sweet et al, 2017). This project seeks to develop a model for sea level variability that can make more appropriate predictions with respect to the different components. Using data collected from tide gauge stations located along the United States coastlines, we created a model that predicts monthly local sea level variability. This was accomplished using Gaussian Process Modeling, a machine learning technique that allows us to capture the different components of sea level change. These components include the trend, seasonal cycle, and interannual variability. Using this method of modeling also allows us to create more accurate predictions, which leads to a more compelling visualization of sea level change than simply using linear regression. Analysis with our model allows for comparisons between different aspects of sea level variability among the different stations.

Ribosomes are present in all cells, and are the site of protein synthesis. These molecules are therefore crucial to all biological functions in which proteins play a role. The structures of ribosomes from many organisms have been solved to a high resolution, giving a near complete description of their complex geometry. However, the functions of many regions of the ribosome remain unknown, as does the relationship between structure and function of certain regions. We study the geometry of a motif found near the point at which peptides exit the ribosome. In particular, we study the writhing number of this motif, and we show that this region of the ribosome differs between prokaryotes and eukaryotes, as well as between different species.

We consider the problem of maintaining a maximum weight matching in the online setting with small recourse. The servers are known in advance, and clients come in one at a time with all of their weighted edges. We consider doing this both exactly and approximately. In the exact case, we show an upper bound that differs from the lower bound by only logarithmic factors. The upper bound is proved by a reduction to the problem in the unweighted setting. We also address solving the problem approximately and show that we can use O(1) amortized recourse to maintain a 2 + ε approximate solution.

We reviewed the work of Kiessling, Lienert and Tahvildar-Zadeh on relativistic interacting dynamics of a two-body quantum-mechanial system consisting of one electron and one photon in one space dimension. We conducted numerical experiments to explore the role of various parameters in the theory and their influence on the trajectories of the two particles, and we studied the possibility of applying this theory to the analysis of Compton scattering.

During the 3D printing process, there is a risk of defects such as pores being formed in the part body. These defects weaken the part, so there is a need for methods to predict the presence of these pores quickly during the printing process to take rapid corrective action. We have developed a deep learning model for this purpose, which uses thermal images of the melt pool and part body to generate a prediction. Two separate convolutional neural network classifiers take in input from nearby time steps and produce two outputs, which are then synthesized via an adaptive decision-level fusion process. The combined program offers an initial accuracy between 90% and 93% with minimal computational expense, allowing it to be used as a heuristic to determine problem cases for further observation via interpolative boundary-finding methods.

Phylogenetic trees have been used in biology to graphically represent hierarchical relationships between genes. Many methods used to construct these phylogenetic trees often produce trees that are too large for easy visualization, or statistical analysis. Hence, reducing the dimensionality of large phylogenetic trees is a key to better understanding large data sets. We study what methods are best suited for tree dimensionality reduction and how we can use those methods to reconstruct trees that suit our data set of influenza from 1993-2017. Through analyzing the topology of reduced trees, we concluded that mutations rate and reassortment reduces the effectiveness of vaccines, resulting in individuals being unprotected against the new strains in circulation. Thus, as vaccine effectiveness increases, branched evolution in influenza decreases. Most importantly, the methods we used are beneficial in visualizing and analyzing any type of clonal genomic data.

Traditionally coding has been associated with error correction of data that has been transmitted across a noisy channel. Recently coding has been applied to increased fault tolerance for distributed storage systems. Data storage is extremely important to companies such as Amazon and Google who wish to deliver content efficiently and in a timely manner.However, an aspect that is often neglected is understanding how many users that the system can support. This is the goal of the service capacity problem. The problem has been studied in a continuous setting, however, our goal is to study some of the connections of the service capacity problem to problems in graph theory and combinatorial optimization. Overall these connections may be able to give us insight into a codes' load-balancing properties in distributed storage systems.

We surveyed recent clinical results to compare reproducibility of results given by p-values as compared with other statistical summaries. The aim was to study the stability of the p-values relative to their corresponding parameter estimate. We found that there the variability among p-values was larger than the variation among estimates, supporting the skepticism of the reliance on p-values in statistical inference.

For hospitals one of the main problems in the process of generating mortality and morbidity data that is common to all countries, this requires both statistical and medical expertise. Medical doctors give a list of billing quotes, called ICD, however, this part is manually input, it is very time consuming, and increases the possibility for errors. This makes it difficult for the medical team to clearly assess the data. Which is why the creation of a web application with the utilization of the IDE "RStudio" and its many packages, it would facilitate the use of ICD-9 codes for medical teams. The app would help this process by helping the doctors get a better diagnosis and being more efficient in every possible way, by eliminating the possibility for manually input data errors, being a fast resource, and a trusted app.

The Apollonian circle packing, generated from three mutually-tangent circles in the plane, has inspired over the past half-century the study of other classes of space-filling packings, both in two and in higher dimensions. Recently, Kontorovich and Nakamura introduced the notion of crystallographic sphere packings, n-dimensional packings of spheres with symmetry groups that are isometries of Hⁿ⁺¹, which can be related to configurations of planes in Hⁿ⁺¹ for various quadratic forms in n+2 variables. When applied in conjunction with the Koebe-Andreev-Thurston Theorem, Kontorovich and Nakamura's Structure Theorem guarantees crystallographic packings to be generated from polyhedra in n=2. The Structure Theorem similarly allows us to generate packings from the reflective extended Bianchi groups in n=2 by applying Vinberg's algorithm to obtain the appropriate Coxeter diagrams. In n>2, the Structure Theorem when used with Vinberg's algorithm allows us to explore whether certain Coxeter diagrams in Hⁿ⁺¹ for a given quadratic form admit a packing at all. Kontorovich and Nakamura's Finiteness Theorem shows that there exist only finitely many classes of superintegral such packings, all of which exist in dimensions n<21. In this work, we systematically determine and enumerate crystallographic sphere packings arising from polyhedra on up to seven vertices, all crystallographic packings arising from Bianchi groups, and all known examples of crystallographic packings arising from higher dimensional quadratic forms.

To move a group of agents over a graph from some initial to arbitrary target positions, I implement previously proposed algorithms for formation control and agent path scheduling. The arbitrary graph simple and connected, with unit edge lengths between adjacent vertices. Moreover, agents in the system are indistinguishable and allowed to travel to the best position in the target formation as determined by the control policy. These algorithms ensure that over all agents, a minimum distance is traveled, give a definite amount of time for agents to achieve the target formation, and eliminate the possibility of agent collision.

Through a combination of neuroscience, robotics, and computer science, neurorehabilitation robots (NRs) can help patients with neurological conditions recover through therapy. Many times, patients may have impaired ability in moving body parts such as their arms. The Bionik InMotion ARM is used as the NR platform for this project. The NRs give optimal results when patients have cognitive engagement (CE) while they perform highly repetitive tasks involving the affected body part. An efficient and effective approach to NR therapy while maintaining CE is performing these highly repetitive tasks in the form of serious games. The NR adapts to the patients' behavior or ability in order to maintain active participation from the patient and to encourage improvement. A key to improving the effectiveness of NR therapy is understanding the neurophysiological components to movement of specific body parts that can help brain plasticity and lead to better motor recovery, so another aspect of this project was developing a built-in measure of these components into the robotic system. This is done through the 128-channel Biosemi Active electroencephalography (EEG) system, along with motor behavior metrics of gameplay performance. So, the focuses on developing comprehensive representations of CE when performing neurorehabilitation tasks during therapy and creating new adaptive rehabilitation strategies that can further help patients in their recoveries.

Pulmonary embolism affects hundreds of thousands of people across the United States and can cause complications such as hemoptysis or cardiac arrest. Pulmonary embolism is characterized by an embolus becoming lodged in a pulmonary artery, where the origin of this embolus is often attributed to deep vein thrombosis. Atrial fibrillation often causes emboli to develop in the heart and propagate through the body. Through the use of the Myocardial Infarction Dara Acquisition System (MIDAS), a database of discharge data on myocardial infarction patients in New Jersey, we seek to study the relationship between atrial fibrillation and pulmonary embolism, in order to understand if there exists a relationship between the diseases.

Schoning's algorithm and the break algorithm are SAT heuristic based Markov chains that have performed exceptionally fast compared to a brute force exponential approach. These Markov chains have been known to work well on random SAT instances over industrial SAT instances. In the first half of this paper, we discuss some sufficient conditions and examples in which one algorithm would perform better than the other in an effort to answer the general question of which SAT solver works better given a random SAT formula. In the second half of this paper, we construct a set of operations on Markov chains that preserve mixing time. Two operations in particular have been stated and proved here - the splitting of a state and the merge of two states.

The idea of matroids dates back to the 1930s where mathematician Hassler Whitney introduces the abstraction of linear independence. After the subject was introduced several other mathematicians started contributing to the field; it soon expanded into different areas of interest like: coding theory, topology, and graph theory. For what we're doing, the reader doesn't require any background knowledge of matroids; only the basics of abstract algebra and graph theory. Matroids have a lot of interesting properties and also have many different ways of being defined. They can be defined from matrices, basis of different vector spaces, graphs, projective planes, and etc. Even though many of these objects don't seem to have a clear relantionship, they all are considered matroids. This lead us to believe that studying some more trivial cases of matroids might possibly reveal some unknown nuances of more complicated matroids.

With the rise of big data and machine learning, computers are used frequently to process and compute large quantities of data. An incoming job is parallelized, divided among different servers. Parallelizing introduces the issue of stragglers, the slowest workers. It is known that MDS coding schemes can be utilized so that the last servers to finish do not slow down the entire job. The known hierarchy of job scheduling policies from most efficient to least efficient is JSQ (join shortest queue), power-of-d, round robin, and random. The service time of balanced and incomplete block design (BIBD) is compared to the service time of other schemes in the hierarchy. BIBD is also tested with other models for which the hierarchy is unknown. At first, simplifying assumptions are made and then, a more realistic model of service time including both job specific randomness and server specific randomness is discussed. As a result, in certain scenarios, BIBD outperforms the other schemes in the hierarchy.

Graphs are an excellent form for the visualization of data because they clearly show how individual data points are connected. However, for large sets of data, a direct visualization of a graph is indecipherable to the human eye. Separating sets and k-connected components are interesting structures in graphs that highlight critical data points and clusters of highly connected points respectively. We developed an algorithm that uses minimum separating sets to decompose a graph into a hierarchy of k-connected components. The complexity of the algorithm depends linearly on the number of k-connected components in the graph. For each k-connected component K=(V,E), however, the complexity of finding its minimum separating set is $O(nm\binom{n}{2})$ where n=|V| and m=|E|. By using different approximate procedures this complexity can be improved to O(nm) or at more cost to accuracy to O(n+m). Performing the separating set decomposition creates a tree-structured map of the decomposed graph that more easily shows the connectivity of the graph and consequently the data it represents.

Cancer is the second leading cause of death in the United States and yet it only has two main treatments, radiation and chemotherapy. A more efficient way to eliminate cancerous cells is with a targeted approach. In order to create more effective precision medication, there exists a need to understand how cancer develops and and to determine which cancerous mutations are most frequent in patients. The optimal way to answer these questions is by sequencing cancer tumors and tracking mutations over time with the help of mathematical trees. We use two genetic distances, Hamming and Nei to help structure the trees. We conclude that Nei's distance does a better job of accurately reflecting the changes in mutations over time and thus can be used in the future to track the evolution of cancerous cells.

The structure of Γ, the bialgebra of stable Grothendieck polynomials, is considered and a basis is shown to contain various classes of of polynomials, particularly hook shapes, 2-row shapes, and certain 3-row shapes.

The problem of learning the mean of a distribution from samples with at most an ε-fraction of adversarial corruptions has been the subject of much recent work. Several current breakthroughs are polynomial time algorithms for high dimensional data in specific classes of distributions. We consider the problem of robust mean estimation for "resilient" distributions and show that with an additional preprocessing step and different analysis, the current best polynomial time algorithm will only need O(log n) iterations of outlier removal in one dimension, as opposed to O(n) in the worst case. This is significant due to the costly outlier identification process involving principal component analysis or semidefinite programming in these algorithms. Our work could provide insight for future approaches on improving the practicality of robust mean estimation for non-gaussian distributions.

Approximate computing is a field of statistical inference techniques that can be used to produce estimates given complex sets of data. Approximate Bayesian Computing (ABC) and Approximate Confidence Distribution Computing (ACC) are two such methods that do not require the specification of a likelihood function, and hence can be used to estimate posterior distributions of parameters for simulation-based models. We look to apply these methods to a large data set, namely one containing pedestrian entrance and exit data for Madison Square Park in New York City. Given the technical and financial restraints of the counting procedure, much of the data is either missing or otherwise prone to error. Therefore, we propose several models to characterize the pedestrian traffic flow throughout the park. Then, simulation techniques are applied to replicate the missing data and produce distributions of simulated data. This leads to the application of ABC and ACC methods, which are used to ultimately produce accurate estimates of the total number of park users over a given time period.

Nucleosomes, the first level of compaction for DNA, consist of DNA wrapped around histone proteins. My research involved reading about how nucleosomes are arranged in crystal structures. Crystal structures are identified with space groups, and so every nucleosome crystal is described by its space group. These crystal lattices and space groups are defined by the symmetry operations that produce them, but these symmetry operations do not lend themselves to analysis through other means. I came up with methods of writing symmetry data in terms of rigid body parameters. In addition I came up with different methods of visualizing the crystal structures that make the patterns easier to look at.

Understanding the computational difficulty of the Minimum Circuit Size Problem (MCSP) dates back to the 1950s and has wide-ranging implications in theoretical computer science. Since MCSP is not known to be NP-hard and is believed not to be in P, it is seen as a promising NP-intermediate candidate. However, despite extensive study of MCSP, there is little evidence that it is not NP-hard. In a recent development, Murray and Williams show that MCSP is not NP-hard under TIME(n^0.49) projections. Murray and Williams also conjecture that MCSP is not NP-hard under logtime-uniform AC⁰ reductions. We show that any super-constant multiplicative approximation of MCSP is not hard for NP (in fact, not hard for PARITY) under even non-uniform AC⁰ many-one reductions. This result is surprising (and nearly tight) because Allender and Hirahara show that there is a constant-factor approximation to MKTP, a problem closely related to MCSP, that is hard for PARITY under non-uniform NC⁰ many-one reductions. To much frustration, this PARITY hardness result is not known for MCSP, and we show that the natural way to extend it to MCSP is actually impossible.

I investigated the minimality of a construction for the fourth tangent circle to three tangent circles (aka the Apollonian circle). Baragar and Kontorovich (my mentor) discovered a general 7 step construction and I showed that under certain assumptions about arbitrary points it is indeed minimal. I also found a 5 step construction for when the circles' centers form a right triangle and showed it and known constructions for isosceles (5 steps) and equilateral (3 steps) triangles are minimal. I am pretty sure the assumption I made about arbitrary points, which is that only a certain subset of them actually need to be considered, is correct, but have not formalized the argument.

Matroid Theory is one of the current hot topics in the mathematics world. My project is to gain a better understanding of them so that I may help solve some of the still unsolved conjectures. My goal for this summer was to become fluent in understanding the correlation constant of a matroid, a field, or graph. The correlation constant explains how the edges in a graph or the column vectors in a vector space configuration are correlated. We study different matrices and matroids and find correlation constants greater than 1 to improve the lower bound of the correlation constant for any field F.

Computer scientists have already unleashed a bevy of technical definitions of fairness. It can be confusing for a newcomer to the field to make sense of the various definitions, their implications, and their shortcomings. In this article, we review previous and contemporary definitions of fairness. Earlier work in the field generally focuses on assessing and learning classifiers with respect to so-called observational notions of fairness. However, more recent work pointed out inherent limitations in these fairness definitions. That a classifier is fair with respect to a fixed notion of fairness is often not sufficient to address all intuitive unfairness in its predictions or the situation in which the algorithm is deployed. The upshot of this review is that "fair" algorithms must incorporate more information into the decision that is assessed merely by observational notions of fairness. Our review surveys three kinds of additional information: the causal structure between variables, the intersection of arbitrarily many protected categories, and the long-term effects of different predictions.

This study focuses on the analysis of 3D surface measurements and quality prediction of 3D-printed dome-shaped objects using multimodal data fusion. Dimension, profile, and surface roughness were measured and represented in image data. Dimension reduction techniques were employed for extracting spatial patterns from the measurement images. Quality metrics were developed using profile deviation and surface roughness. In the end, classification and regression models were built to predict quality. The results propose feature extraction from high-dimensional image data as a promising technique for efficient and automated quality inspection.

The curve-shortening flow is a process that modifies curves in the plane according to a certain partial differential equation. With certain starting conditions, the flow will shorten the perimeter and cause the curve to collapse to a circle, and then to a point, in finite time. The flow in the plane has been extensively studied, but we extend the notion of the flow to surfaces. We investigate how the evolution equation must be adapted so that the flow is well-defined, and then analyze certain surfaces and families of curves that provide a solution on those surfaces. We then pose certain open questions that will be studied further.

Cancer is a genetic disease which begins from a single aberrant progenitor cell, which successively divides into pairs of daughter cells with similar reproductive capacities. As the tumor grows, many distinct genetic lineages may proliferate throughout the neoplasm, resulting in significant intratumor heterogeneity. Such high levels of genetic diversity within tumors is associated with low survival rates for patients, as genetically-distinct cell variants have differential sensitivities to current therapies. Despite its importance, how intratumor heterogeneity develops is not well understood. However, mathematical models can provide insight into potential causes. This work applies a stochastic operator-splitting method developed for simulating chemical reaction-diffusion processes to model neoplastic growth. It is shown that high differential birth-rates, high diffusion rates and early onset of mutations can result in substantial levels of intratumor heterogeneity.

The chromatic polynomial is shown to be a characteristicof the graph that demonstrates a relation. Then this idea isdeveloped more with matroids and a diversity of graphs. Amatroid can take up different forms from matrices to graphsto simple sets. There are multiple proofs of each graph'schromatic polynomial using the Deletion-Contraction Relation,also presented in Huh's article. The deletion-contractiondevelops an interesting relation between graphs that can thenbe applied further to matroids. The chromatic polynomial of agraph has and continues to expand beyond 2-D graphs and even3-D graphs, which is where work continues to develop.

Anomaly detection methods play an important role in keeping Internet operating without interruptions. While many methods rely on centralized detection, there are many benefits in implementing distributed methods where Internet nodes can decide whether there is an ongoing attack using local information only. In this work, I extend a distributed anomaly detection algorithm, DIAMoND, for the case of attacks on both layers of a two-layer network system. The nodes in each layer, which may represent the Internet and the power-grid, communicate with each other and can share information on their status, without sharing sensitive information such as the amount of traffic they handle. The main contribution of this work is to implement a system of trust among layers, so that a node weighs differently information received by nodes in different layers. I show that changes in the trust parameter can influence the accuracy of attack detection.

Schubert calculus studies Schubert structure constants for the cohomology rings of a homogeneous space X. In many cases, combinatorial rules involving jeu de taquin on posets associated to these spaces have given positive formulas for these constants. Schutzenberger first proved such a rule for X a Grassmanian. Chaput and Perrin extended this to products of Λ-minuscule classes of spaces under a Kac-Moody group. This includes all Schubert classes for X a minuscule variety. Meanwhile, Buch, Samuel, Thomas, and Yong extended the jeu de taquin theory to the K-theory of all minuscule varieties. We begin to develop the analogous K-theoretic jeu de taquin theory for Proctor's d-complete posets, which are the associated posets for Λ-minuscule Schubert varieties.

The holonomy group of the tangent bundle of the two dimensional unit sphere is precisely the special orthogonal group. Let V denote the tangent space at some fixed point p on the sphere. We seek to consider the invariant space of . Every tensor defines globally a differential operator on . This is of interest since forms a noncommutative vertex operator algebra. The Beltrami-Laplace operator Δ sits inside some . The focus of this paper is to consider the eigenfunctions of Δ and characterize the set of functions in the orbit of differential operators in acting on an eigenfunction. Such a result will allow us to mathematically construct a 2-d nonlinear σ-model.

In this paper, we begin by introducing the concept of differential privacy and why it is a useful notion of privacy compared to other approaches. We also introduce the locally distributed differential privacy model. We then introduce various recommendation problems and formulate them mathematically. We then introduce some theoretical tools useful in differential privacy and recommendation algorithms. We then survey some existing literature in differential privacy to get the reader more familiar with how it is applied in various settings and continue to survey various intersections of differential privacy and recommendation algorithms. We end by identifying and precisely stating two open problems and discuss approaches to solving them.