REU Project Summaries

Project Year:

Atrial Fibrillation and Onset of Pulmonary Embolism

Information: Project Year: 2018; Student(s): Christopher Espana | Rutgers University-New Brunswick NJ; Mentor(s): Javier Cabrera - Statistics and Biostatistics; Project Abstract
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.

Chow Rings of Matroids

Information: Project Year: 2018; Student(s): Froylan Maldonado | San Diego City College CA; Mentor(s): Nicola Tarasca - Mathematics; Project Abstract
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.

Codes for Storage with Queues for Access

Information: Project Year: 2018; Student(s): Dalton Burke | University of Colorado Denver/Anschutz Medical Campus CO , Elise Catania | University of Rochester NY; Mentor(s): Emina Soljanin - Electrical and Computer Engineering; Project Abstract
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.

Curve-shortening Flow and Surfaces

Information: Project Year: 2018; Student(s): Scott Harman | Rutgers University-New Brunswick NJ; Mentor(s): Christopher Woodward - Mathematics; Project Abstract
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.

Elucidating Tumor Evolutionary Patterns using High-Depth Molecular Data

Information: Project Year: 2018; Student(s): Caitlin Guccione | University of Rhode Island RI; Mentor(s): Hossein Khiabanian - Rutgers Cancer Institute of New Jersey; Project Abstract
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.

Expressing Grothendieck Polynomials Using Rectangles

Information: Project Year: 2018; Student(s): Heman Gandhi | Rutgers University-New Brunswick NJ , Adam Jamil | Rutgers University-New Brunswick NJ; Mentor(s): Anders Buch - Mathematics; Project Abstract
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.

Faster Convergence of Robust Mean Estimation in One Dimension

Information: Project Year: 2018; Student(s): Ryan Rice | The University of Texas at Austin TX; Mentor(s): Pranjal Awasthi - Computer Science; Project Abstract
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.

Generating Nucleosome Crystal Visuals and Reference Frames

Information: Project Year: 2018; Student(s): Timothy Stavetski | University of Notre Dame IN; Mentor(s): Wilma Olson - Chemistry and Chemical Biology; Project Abstract
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.

Implementation of Formation Control and Path Scheduling in Multi-Agent Systems

Information: Project Year: 2018; Student(s): Aubrey Hormel | Missouri State University-Springfield MO; Mentor(s): Jingjin Yu - Computer Science; Project Abstract
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.

Matroids and the Correlation Constant

Information: Project Year: 2018; Student(s): Pejmon Shariati | Rutgers University-New Brunswick NJ; Mentor(s): Nicola Tarasca - Mathematics; Project Abstract
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.

Moving Beyond Observational Notions of Fairness

Information: Project Year: 2018; Student(s): Michael Yang | Minerva Schools at Keck Graduate Institute CA; Mentor(s): Anand Sarwate - Electrical and Computer Engineering; Project Abstract
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.

Multimodal Data Fusion in 3D Printing Quality Prediction

Information: Project Year: 2018; Student(s): Xiaotong Gui | Pomona College CA , Xinru Liu | Wheaton College MA; Mentor(s): Weihong 'Grace' Guo - Industrial and Systems Engineering; Project Abstract
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.

Shiny ICD-9 web app for the Cardiovascular Institute

Information: Project Year: 2018; Student(s): Evaristo Rodriguez | San Diego City College CA; Mentor(s): Javier Cabrera - Statistics and Biostatistics; Project Abstract
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 Chromatic Polynomial of a Graph

Information: Project Year: 2018; Student(s): Ruby Ortiz | Muhlenberg College PA; Mentor(s): Nicola Tarasca - Mathematics; Project Abstract
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.

The Non-Hardness of Approximating Circuit Size

Information: Project Year: 2018; Student(s): Neekon Vafa | Harvard University MA , Rahul Ilango | Rutgers University-New Brunswick NJ; Mentor(s): Eric Allender - Computer Science; Project Abstract
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.

Visualization of k-connected Components and Minimum Separating Sets of Fixed Points of Degree Peeling

Information: Project Year: 2018; Student(s): Daniel Nakhimovich | Cooper Union for the Advancement of Science and Art NY; Mentor(s): James Abello - Computer Science; Project Abstract
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.

Advanced Mathematics Note-Taking Study

Information: Project Year: 2017; Student(s): Jerel Roane | Morgan State University; Mentor(s): Keith Weber - Graduate School of Education; Project Abstract
This study is about students' note taking in their advanced mathematics lectures. We wonder why students were not able to identity the content the professor was convey during lectures. We found out that (I) students' notes were only copy of what the teacher wrote on the blackboard, (II) students did not record information that was orally stated, and (III) students did not show evidence of being actively engaged during their lectures. We record 7 80 minutes mathematics lectures and photographed of 69 students' notes from these lectures. We found that in the lectures, much of the important information was stated orally, but students typically did not record what was orally stated in their notes.

Anomaly Detection in Multilayer Networks

Information: Project Year: 2017; Student(s): Gianna Schwarz | Rutgers University; Mentor(s): Lazaros Gallos - DIMACS; Project Abstract
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.

Constant Factor Optimal Multi-Robot Path Planning on the Plane

Information: Project Year: 2017; Student(s): Edgar Jaramillo-Rodriguez | University of California, Berkeley; Mentor(s): Jingjin Yu - Computer Science; Project Abstract
The multi-robot path planning problem (MPP) on the plane has long been studied both as a topic of research interest and because of it's numerous applications in numerous fields. While many effective algorithms and heuristics exist for discrete MPP on a graph, up until there have been very few results for the continuous version on the plane. In this work we provide a complete algorithm that solves the MPP problem on the plane with a guarantee of constant factor optimality in expectation. We hope that our work may be a good first step into exploring this problem extensions of it in special cases that reflect real world applications.

Constructing a 2-d nonlinear sigma model

Information: Project Year: 2017; Student(s): Arthur Wang | Rutgers University; Mentor(s): Anders Buch - Mathematics; Project Abstract
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 $(V^{\otimes k})^{SO(2)} = \{x \in V^{\otimes k} \mid gx = x, \forall g \in SO(2)\}$ . Every tensor $(V^{\otimes k})^{SO(2)}$ defines globally a differential operator on $C^{\infty}(S^2)$ . This is of interest since $\coprod\limits_{k \in \mathbb{N}}(V^{\otimes k})^{SO(2)}$ forms a noncommutative vertex operator algebra. The Beltrami-Laplace operator Δ sits inside some $(V^{\otimes k})^{SO(2)}$ . 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 $\coprod\limits_{k \in \mathbb{N}}(V^{\otimes k})^{SO(2)}$ acting on an eigenfunction. Such a result will allow us to mathematically construct a 2-d nonlinear σ-model.

Differential Privacy and Recommendation Systems

Information: Project Year: 2017; Student(s): Akilesh Tangella | University of Pennsylvania; Mentor(s): Muthu Muthukrishnan - Computer Science; Project Abstract
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.

Effect of modularity on local and global synchronization in complex networks

Information: Project Year: 2017; Student(s): Kayla Cummings | Pomona College; Mentor(s): Lazaros Gallos - DIMACS; Project Abstract
Network science recasts real-life systems as graphs to study diverse dynamical processes in technological, biological, and physical contexts. More specifically, heterogeneous network structure leads to novel phenomena during synchronization. A network's transition from disorder to coherence can present different forms depending on specific network properties. Modularity in particular can strongly influence the outcome of a synchronization process. This work investigates the effect of modularity on synchronization. I show that the degree of modularity may severely delay global synchronization even though local order emerges after a stable coupling strength threshold. As a result, measurements of global synchronization may be misleading when evaluating coherence in modular networks. These results can be useful for studying the propagation over time of ideas, disease, or information since social, biological, and communication networks often have modular properties.

FineTravel: Fine-Grained Travel Time Estimation for Multiple Transportation Networks

Information: Project Year: 2017; Student(s): Aaron Zhang | Brown University; Mentor(s): Desheng Zhang - Computer Science; Project Abstract
Estimating passenger travel time in public transportation systems provides useful information to commuters and improves their travel experience. Most existing work on travel time estimation has focused on estimating riding time for a single mode of transportation. However, passengers often use different modes of transportation, e.g. subways and buses, and a significant portion of travel time is spent walking and waiting. Therefore, estimating riding time for a single mode of transportation underestimates actual travel time. Using vehicle GPS and passenger fare transaction records from existing transportation infrastructures, we propose a novel unified model that integrates multiple transportation networks in a city. In the model, travel time is divided into components by passenger status and transportation modality. Estimators for each component are constructed from historical data. We evaluate the performance of our travel time estimation model on large-scale real-world data from multiple sources in four transportation networks: subway, taxi, bus, and private vehicle. We show that our model is more accurate than baseline estimates in all modalities by at least **%. In addition, our model is elastic to support new transportation modalities. Finally, we discuss potential applications based on the travel time estimates from our model.

Genomic Data-Guided Mathematical Modeling of Cancer

Information: Project Year: 2017; Student(s): Roxanne Casio | Rutgers University-Newark; Mentor(s): Subhajyoti De - Rutgers Cancer Institute of New Jersey; Project Abstract
The margin between cancer survivors and those who receive cancer treatments continues to widen despite modern-day medical efforts. Although there is a great variability of the effectiveness of a treatment from cancer patient to the next, within a single patient and a single tumor,which is composed of thousands of cancerous cells, there is an even greater level of cellular variability, denoted as intra-tumor heterogeneity. Intra-tumor heterogeneity arises as time progresses because cancerous cells continually evolve and acquire mutations that aid in their own survival. Common techniques that analyze the composition of a tumor, such as biopsies, are not accurate measures of a tumor's true composition. It is possible that these treatments are targeted for only a small subset of cells, of the same type, within a single tumor population. By utilizing an open source Java program, it is possible to observe the growth of all individual subpopulations of cancerous cells within a single tumor. This code, may be used with varying parameters that affect tumor growth, would aid in providing more accurate diagnoses, which might help to remedy the ineffectiveness of treatments due to intra-tumor heterogeneity.

Hyperspectral Image Classification of Liquids: The HyL12 Dataset

Information: Project Year: 2017; Student(s): Andrea Burns | Tulane University; Mentor(s): Waheed Bajwa - Electrical and Computer Engineering; Project Abstract
As demand rises for machine learning tasks such as classification and regression in many fields, state-of-the-art algorithms and new data collection techniques are necessary to continue to improve performance. Tools such as hyperspectral imaging have been introduced to increase available information, and therefore increase prediction accuracy as well. This creates motivation to curate more hyperspectral datasets, as improved image classification can be achieved through introducing additional spectral bands. Here we introduce a newly curated Hyperspectral Liquid 12-Band dataset (HyL12) that consists twelve classes including eleven liquids and an additional empty class. This dataset can be expanded for security uses to help liquid classification in contexts of safety concerns such as airport security, concerts, and politic events.

Informing Guessing Attacks on Publicly Performed Secrets

Information: Project Year: 2017; Student(s): Laura South | Colorado State University; Mentor(s): Janne Lindqvist - Electrical and Computer Engineering; Project Abstract
Unlike traditional protocols, mobile authentication is frequently performed in a public setting, so a certain amount of information leakage is inevitable. A common attack called "shoulder surfing" takes advantage of this inadvertently leaked information for various harmful purposes. The objective of this project is to determine whether it is possible to obtain enough information from observing the full-body action of a user unlocking a mobile device to improve the guessing attack accuracy rate.

New upper bounds for the tensor ranks of certain Lie algebras

Information: Project Year: 2017; Student(s): Terence Coelho | Rutgers University; Mentor(s): James Lepowsky - Mathematics; Project Abstract
In 1969, Volker Strassen published a revolutionary paper showing that one can multiply two 2 x 2 matrices with only 7 multiplications, instead of the natural 8; in formal algebraic terms, this means the "tensor rank" of matrix multiplication is at most 7. Since then, there has been extensive literature on the upper and lower bounds of the tensor ranks of various algebraic structures. Despite the computational usefulness and long history, very little is still known about the precise tensor ranks of interesting algebras. In this paper, we provide a method to reduce the natural upper bounds on the tensor ranks of semisimple Lie algebras and lay the framework for improving our bounds on the tensor ranks of semisimple Lie algebras of type A, D, E. In doing so, we not only show how to construct efficient algorithms for computing the Lie bracket in these algebras, but also reveal some important underlying symmetry. In this work we invoke a result of Robert McRae.

On the Structure of Fixed Points of Degree Peeling

Information: Project Year: 2017; Student(s): Seth Bartynski | University of Pennsylvania; Mentor(s): James Abello - Computer Science; Project Abstract
Degree peeling is frequently used to analyze large networks. In particular, previous work has focused on using degree peeling to decompose networks into cohesive subgraphs. These decompositions allow the researcher to focus in on particularly important structures within the network at hand. However, the properties of such structures have not been studied extensively and may be quite large themselves. Here we examine the properties of the structures generated by the iterative edge decomposition of Abello and Queyroi, which are fixed points of degree peeling. We describe how fixed points of degree peeling can be combined to form larger fixed points. Furthermore, we investigate the effects of graph homeomorphism on degree peeling, which can be very drastic. We provide novel methods for minimizing this influence. Combined with the edge decomposition of Abello and Queyroi, our work can be used in network analysis, community detection, and graph visualization.

Pan-Cancer Precision-Oncology Analysis of Tumor Mutational Signatures

Information: Project Year: 2017; Student(s): Simon Bird | Rutgers University; Mentor(s): Hossein Khiabanian - Rutgers Cancer Institute of New Jersey; Project Abstract
Given the results of analysis of a tumor sample, can we determine whether a mutation is hereditary or it developed during tumor progression? In this study, we use statistical analysis and tumor simulations to determine in which kinds of tumors the mutational status can be inferred. We found that if we know the ploidy of the sample, we can predict the model with over 95% accuracy in 68% of cases. Otherwise, we can predict the correct model only 11% of the time, where the rest of the time we can make no conclusion. This implies that in many scenarios, doctors do not require additional tumor sequencing in order to know the mutational status. This is useful for both treatment and preventative purposes.

Providing an Alternate Version of the Logspace Hierarchy to Recreate the Polynomial Hierarchy

Information: Project Year: 2017; Student(s): Marina Knittel | Harvey Mudd College; Mentor(s): Eric Allender - Computer Science; Project Abstract
The polynomial hierarchy is a central topic in computer science today, particularly because of its close relationship with the infamous question of whether or not P=NP. For instance, we know that P=NP implies that the polynomial hierarchy collapses. Therefore studying the polynomial hierarchy, for instance finding new ways to define its structure, is an important part of research in theoretical computer science. This project aimed to discover a new hierarchical structure that has a strong correspondence with the polynomial hierarchy. In the end, we discovered a structure based off of the logspace hierarchy that is equivalent to the polynomial hierarchy when given an additional quantifier. The main consequence of this result is that we now have a new way to consider the polynomial hierarchy. Additionally, we were able to define a new set of complete problems for various levels of the hierarchy. More generally, we hope this work can act as a stepping stone for further study in formulations of complexity classes that have somehow relate to the polynomial hierarchy. The results also provide a strong framework for the discovery of new natural complete problems for the polynomial hierarchy phrased in terms of the properties of our model as opposed to the standard model. The ultimate goal is to generally improve our understanding of the various components of the polynomial hierarchy, especially P and NP, in the hopes of answering broader questions such as whether or not P=NP.

Research for Stadium Security on WTMDs and Drones

Information: Project Year: 2017; Student(s): Bishwa Silwal | Howard University , Mustapha Olokun | Morgan State University; Mentor(s): Christie Nelson - Masters of Business and Science in Analytics and CCICADA; Project Abstract
Stadium Security has increased in recent years due to stadium attacks and harmful activity, being a threat to large crowds. Within this security, we examine Walk Through Metal Detectors as they are widely used and an important part of large crowd venue locations. Security in general, is examined to understand the process and activities taking place during real environment security processes through data collection and security software testing.

Some Properties of Quantum Cohomology Determined by Positivity

Information: Project Year: 2017; Student(s): Nalinpat Ponoi | Prince of Songkla University (Thailand); Mentor(s): Anders Buch - Mathematics; Project Abstract
We study the multiplicative of a (small) quantum cohomology ring which is encoded by Gromov-Witten invariants. The multiplicative structure of the cohomology ring is rather complicated. In this work, we use the Pieri formula to work on the special cases. Since Gromov-Witten invariants are positive integers, a conjecture of Buch states that this fact determines some properties in quantum cohomology. He also proves that it is true for the Grassmann variety Gr(2,n). We try to prove that this is also true for Gr(m,n) when m>2.

Spatiotemporal Big Data Analytics for Osteoarthritis Knee

Information: Project Year: 2017; Student(s): Tram Pham | University of California, Berkeley; Mentor(s): Weihong 'Grace' Guo - Industrial and Systems Engineering; Project Abstract
Osteoarthritis (OA) is the most prevalent disease amongst knee joints which mostly affects the cartilage in elderly and overweight people. Articular cartilage is simply defined as a soft connective tissue at the end of bones which prevents the bones from erosion and allows smooth glide bones in the joint. OA is characterized by the gradual degeneration of cartilage in knee joint. In Osteoarthritis, cartilage seizes up, erodes away, and eventually disappears in some regions causing the bones to rub again one another with severe pain during motions. Measurement of cartilage loss and 3D visualization can help to quantify the severity of osteoarthritis. Nowadays, Magnetic Resonance imaging (MRI) is extensively used to image the knee joint due to its high resolution and contrast displacement between the bones and cartilage. Though a profound work has been done to detect boundary, there is lack of discussion on cartilage measurement of severe OA knee and the advantages of image pre-processing have not been fully exploited. Encouraged by initial research, this paper proposes a semiautomatic method to improve clinical evaluation of osteoarthritis disease. It emphasizes on improving image pre-processing techniques and the treatments to severe osteoarthritis cases. We explore the image processing techniques applied on MRI in the progression of osteoarthritis diagnosis. Namely, we will identify the clinical biomarkers for early detection of OA by (1) pre-processing MR images, (2) detecting boundary of cartilage by modified radial search method, (3) 2D and 3D cartilage visualizing and (4) calculating volume and thickness of cartilage for OA quantification.

Unique rectification in d-complete posets

Information: Project Year: 2017; Student(s): Rahul Ilango | Rutgers University , Michael Zlatin | Rutgers University; Mentor(s): Oliver Pechenik - Mathematics; Project Abstract
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.

Using Mathematical Tools to Analyze Chromatin Structures

Information: Project Year: 2017; Student(s): Ondrej Maxian | Case Western Reserve University; Mentor(s): Wilma Olson - Chemistry and Chemical Biology; Project Abstract
The packing of DNA into nucleosome core particles (NCPs) is vital to the field of epigenetics and gene expression. Until now, the packing of NCPs has been studied physically via X-rays and cryogenic electron microscopy (cryo-EM), with few to little mathematical tools available. We here develop tools to quantitatively evaluate the amount of interaction between two nucleosomes on their faces and sides, as well as to give biological meaning to the regions of interaction. We apply our methods to X-ray crystal and cryo-EM structures, in the process confirming much of the existing observations while developing new insights into some atypical structures. We also deploy our algorithm to analyze configuration outputs from Markov Chain Monte Carlo (MCMC) simulations, which we found to closely resemble those from cryo-EM and X-ray.

Visibility Graphs of Staircase Polygons: Algorithm for Building Staircase Polygons from Slope-Ranking Balanced Tableaux

Information: Project Year: 2017; Student(s): Yulia Alexandr | Wesleyan University; Mentor(s): James Abello - Computer Science; Project Abstract
There exists a relationship between staircase polygons, persistent graphs, and balanced tableaux. While many aspects of this relationship had been rigorously studied, the problem of recovering a simple staircase polygon whose visibility graph is isomorphic to the skeleton of a given slope-ranking balanced tableau remained open and was previously known to be PSPACE. In this paper, we demonstrate a deterministic polynomial-time algorithm for constructing a staircase polygon with desired properties and prove that certain conditions required by the algorithm hold for any balanced tableau.