# Honors Research Projects Options

## Please select your top three project choices below by using the drop down boxes provided. The projects are listed by number. Do not select a project more than once.

**2021 Research Projects**

**1. Stabilization index of permutations; ****Mentor: Suho Oh**

Jeu-Da-Taquin is a game where you have boxes with numbers on it and slide according to a certain rule. This game is a fundamental tool in studying representation theory and symmetric groups. Stabilization index comes from plaything this game on a board coming from permutations and measuring a certain statistic. We will try to come up with an efficient way to measure this index directly from the permutation.

**2. Covid-19 Dynamics; ****Mentor: Young Ju Lee**

We shall develop and apply a quantitative method to understand Covid-19’s dynamics. Such quantitative method is crucial to push toward eradication of human infectious diseases. Taking Covid-19 data collected in Korea, we apply the recent data analysis tool, so-called the dynamic mode decomposition to understand how Covid-19 spreads and what causes the slowing down of the spread of such an infectious disease.

**3. Using Process Mining Techniques on Classroom Data; ****Mentor: Kate Melhuish**

Process mining encompasses a set of techniques from data science used to abstract the structure of processes based on event logs of timestamp data. In this project, we will work to convert the techniques developed to analyze timestamp data from an online setting for use with live classroom data. The goal is to create models to better understand teacher and student activity in math classes. Example:

4. **How students summarize and explain the key ideas of proofs; ****Mentor: Kate Melhuish**

Reading and understanding proofs is a notoriously challenging, yet important of advanced mathematics. In this project, we will qualitatively analyze how undergraduate students summarize and distill the key ideas in proofs found in an introductory real analysis class. We will explore what students attend to in this process and how their attention may differ than that of experts.

**5. Prime graphs of finite groups with a focus on solvable groups; ****Mentor: Thomas Keller**

The focus of this project is to study prime graphs of finite groups with a focus on solvable groups. There are a number of open questions on these graphs which are purely combinatorial in nature. An important notion here is the one of minimal prime graphs. A simple graph is called a minimal prime graph if it is connected, has at least two vertices, and its complement is triangle-free and has chromatic number 3, and removing any edge results in a graph whose complement has a triangular chromatic number 4. For example, what are bounds for the diameter of a complement of a prime graph of n vertices? What do these bounds look like for minimal prime graphs? It would also be nice to find more examples of what are called super-minimal prime graphs. A thorough introduction on what is known on these kinds of questions and information on the group theoretical background will be provided.

**6. Graph Theory / Algebraic Topology Project - ****Exploration of Topological Implications of Asymmetric 2-Colorings of graphs;**** Mentor: Tim Chase**

In their paper from 2016, Flapan, Rundell and Wyse showed that the edges of every 3-connected planar graph except *K*4 can be colored with two colors in such a way that the graph has no color-preserving automorphisms. Also, they characterized all graphs that have the property that their edges can be 2-colored so that no matter how the graph is embedded in any orientable surface, there is no homeomorphism of the surface that induces a nontrivial color-preserving automorphism of the graph. The study of asymmetric 2-colorings of graphs embedded in R3 was originally motivated by the desire to classify the symmetries of nonrigid molecules. Such complex molecules can be represented by graphs in R3, where different colored edges represent different types of molecular chains or different types of bonds. Thus, results about topological symmetries of colored graphs in R3 have potential applications to the study of symmetries of nonrigid molecules. I propose we look through this characterization, and see if there are either ways to strengthen the characterization of 2-colored graphs, or if the theory can be extended to other colorations / graph types (3-colored, 2-connected, etc.)

**7. Knot Theory / Algebraic Topology Project -****Topological descriptions of protein folding; ****Mentor: Tim Chase**

Self-tying of proteins was long considered as impossible, as it requires the threading of a chain through a twisted loop. However, similar structures have been known to exist in other biopolymers (such as DNA) for many years. In the case of proteins, the situation began to change approximately 20 years ago, and nowadays, every week, new structures are deposited in PDB, resulting in many deeply-knotted and slipknotted proteins discovered. In 2019, Flapan and others in their paper *Topological descriptions of protein folding *proposed an interesting but far from comprehensive characterization and explanation for the creation of knot and knot-like structures in extremely long protein chains. I propose we springboard from their research in this paper, and see if we can extend the results any further.

**8. Set Theoretic Topology Project - ****Extensions of Monotone Orthocompactness;** **Mentor: Tim Chase**

Monotone versions of many covering properties have been defined and studied in recent years by a number of authors. But different characterizations of the same covering property may generate different monotone versions of that covering property, and these different versions in many cases can be radically different. While monotonically paracompact, monotonically metacompact, and monotonically Lindelof have all been looked at rather extensively, not much has been explored in the property of monotonically orthocompact. I propose seeing if we can characterize this property (especially in regards to metrization theory).

**9. Hypergraphs, signed graphs, and the Tutte-Polynomial; ****Mentor: Lucas Rusnak**

The Tutte-polynomial encodes the structure of the graph and can be used to count various things from spanning trees, to acyclic orientations, to colorings, and even flows. We will examine hypergraphic and sign graphic analogs of the Tutte polynomial to generalize various graph concepts. Topics will initially cover deletion/contraction variants, acyclic orientations, alternatives to Dhar’s Burning algorithm and chip-firing, and more, before focusing on topics that the group is most interested in pursuing. Students will be expected to work with concepts from combinatorics, graph theory, and matrix algebra.

**10. Generalizing Kirchhoff Laws for Hypergraphs; M entor: Lucas Rusnak**

Tutte's demonstrated how algebraic graph theory can reclaim Kirchhoff's Laws as a specific configuration of spanning trees and 2-forests in the graph. We will develop generalized versions of these concepts for oriented hypergraphs, and show that in the energy potentials vary with the chosen spanning tree and give rise to non-conservative Kirchhoff-type analogs. To do so we will examine a larger collection of hypergraphic combinatorial objects that what Tutte originally examined, and demonstrate that in a graph only the set of 2-forests are non-cancellative. Students will be expected to work with concepts from combinatorics, graph theory, and matrix algebra.

**11.Matroids and the Algebraic Frame Completion Problem; Mentor: Cameron Farnsworth**

A finite unit norm tight frame may be seen as a generalization to an or- thonormal basis. The frame completion problem has been studied by numerous researchers usually asking: ‘If r many, n-dimensional vectors are determined, how many additional vectors must be added to complete a frame?’ A recent paper asked a slightly different question: “Given an n × r matrix with some real entries fixed, what criteria must the entries satisfy in order to complete the matrix so that its columns form a finite unit norm tight frame?” The authors’ results characterized the algrebraic matroid underlying the finite unit norm tight frame variety in the dimension 3 case. Combinatorial criteria related to connec- tivity of a graph associated to the known entries were found which characterized the matroid. In this project we hope to study the methods used in this paper to extend the results to 4 dimensions or higher.

**12. Scientific Workflows;**** Mentor: Rodion Podorozhny**

**Scientific workflows are used to automate various research projects: Ecological studies, Genome project, Medical procedure studies and others. They can be viewed as "programming in the large" systems. Quite often scientific workflows are written by different people and are applied to the same environment. They also must conform to predefined properties. In this project we will study application of program analysis algorithms such as model checking to representations of combined parallelized scientific workflows formalized in a Little-JIL workflow language.**

**13. Runtime Verification of Concurrent software systems; Mentor: Rodion Podorozhny**

**Reliable and efficient concurrent software systems are key to utilization of multiple core CPUs. There is already a noticeable number of existing concurrent implementations of basic data structures. It is important to verify that these implementations are reliable. The project will focus on methods of efficient verification of concurrent data structures by runtime verification, in particular combination of slicing (automatic reduction of the analyzed code relevant to the property verified) and model checking algorithms.**

**We can use existing slicing tool by IBM, Wala and some existing model checking tool (SLAM by Microsoft, Java Pathfinder by NASA Ames Center etc.)**

**14. Formal verification by proof of software systems; Mentor: Rodion Podorozhny**

**In this project we will evaluate an existing formal proof system, such as Isabelle, for verification of components of a multi-agent control system (e.g. distributed negotiation component)**

**15. Robotic distributed mapping and localization; Mentor: Rodion Podorozhny**

**In this project we will design and evaluate a distributed version of Simultaneous Localization and Mapping algorithm on a pre-exesting prototype**

**16. Data Compression and Encryption Tandem;**** Mentor: Dan Tamir**

Generally, data- compression and encryption operations are applied independently (serially) and the compression has to precede the encryption since encryption negatively affects redundancy. Very few lossless compression methods can fit into a framework of combining the encryption and the compression into a single operation. Several of these methods employ dictionary techniques where permutation of dictionary elements is feasible. Nevertheless, one of the most appealing tandems uses the Shannon Fano Elias (SFE) compression method, which is known to be inferior to optimal compression techniques such as Huffman coding. Furthermore, the resilience of the tandem to cryptanalysis is challenging. Dr. Tamir has developed a new method for performing SFE which provides results that are in par Huffman code compression ratio. Furthermore, when applied along with Arithmetic Coding it provides superb compression and encryption. In this project we will study several lossless data compression methods concentrating on Huffman, SFE, and arithmetic coding. Additionally, we will study the principles symmetric encryption and the basics of cryptanalysis for symmetric coding. We will evaluate, compare, and contrast, the compression ratio and resilience of SFE encryption and arithmetic coding encryption tandems as well as, ways for evaluating other performance criteria such as throughput and latency. In addition, we will develop variants of SFE and perform experiments with current methods and these variants.

**17. Construction and Evaluation of Stream Cyphers;**** Mentor: Dan Tamir**

A stream cypher generates a sequence of pseudo random numbers and applies these numbers through operations such as “switching boxes” and Exclusive-OR (XOR) to the message to be encrypted. In this project, we will study several methods for generating pseudo random numbers (PRN), as well as methods for evaluating the “randomness” of the PRN sequences and their suitability for providing strong encryption. In specific we will evaluate PRNs generated as derivatives of the Collatz Recursion (https://en.wikipedia.org/wiki/Collatz_conjecture). The Collatz Conjecture / recursion is a problem that draws the interest of many of the most prominent mathematicians including Erdos, (https://en.wikipedia.org/wiki/Erd%C5%91s_number) and is known as one of the problems for which an Erdos prize is yet to be claimed.

**18. Gaming VR AR & 3D to 3DS; ****Mentor: Dan Tamir**

3DS is the name of a specific computer program for generating 3D models to be used in gaming and virtual/augmented reality environments. In this paragraph it is used as a generic name for any application that is capable of generating 3D environments. In this project, we will explore ways to take 3D images, a pair of 2D images, or a stream of images (e.g., the result of taking videos through a 360-video camera) and automatically convert these images into 3D computer generated images (CGI) and models. While several computer programs for enabling part of the process of 3D to 3DS exist, there is no end-to-end solution. The potential of such a solution is enormous. For example, it can be used to convert movies and videos into virtual reality environments. Specifically, we will study the basic principles of CGI and use existing software packages for CGI. We will augment these packages by adding our own software modules. The goal is to produce an end-to-end 3D to 3DS system.

**19. Combining Perception Considerations with Artificial Intelligence in Underwater Threat Detection Systems Employing the Art Gallery Problem with Mobile Guards Approach; ****Mentor: Dan Tamir**

Often, Human reasoning involves computation that is using natural language constructs and is referred to as Computation with Words (CWW). This is in contrast to the computing employed in numerous tasks of autonomous and intelligent computer-based control systems, where large amount of “numbers” obtained from sensors, images, and videos is used to perform the control operations via Computations with Numbers (CWN). For example, consider the task of identifying human and vehicle activities, such as “two cars are diverging close to school,“ based on videos obtained from street surveillance cameras and drones. Human beings would have “no problem” identifying behaviors such as “vehicle X is following vehicle Y,” or “vehicle V is circling the secured area” from direct observation or from inspection of surveillance cameras content. In doing so the humans are employing CWW. The detection process can be divided into two stages, 1) data acquisition and 2) inference. The state of the art in Artificial Intelligence (AI) might approach the data acquisition task using an image processing-based Convolutional Neural Network (CNN) that can reliably detect objects of interest in videos. Several types of agent-based AI inference engines including other types of artificial neural networks (ANNs) have been proposed as candidate solutions for the inference stage. However, evaluation of these approaches yields the observation that even semi-supervised automatic systems, including a “human in the loop,” might miserably fail in executing the reasoning part of the task. Often, due to lack of training data. Consequently, approaches that combine perception based CWW reasoning with CWN AI techniques are currently explored.

In this project, we will study the state of the art of AI inference; concentrating on Automated Reasoning, State Space Search Techniques, Markov Models, and ANNs. Additionally, we will study the state of the art in Perception-based Reasoning and CWW. Furthermore, the problem of activity detection will be explored, specified, and analyzed. The effectiveness and usability of the CWW systems will be evaluated. As a part of evaluating the proposed approach, we will employ the following process: we will adapt the CNN object detector that produces near real-time high-recall object-detection results for underwater swimmers, in the vicinity of marine ports, that can pose a threat to the port. Next, we will use a CWW-based inference system that is fed by the CNN detectors and other sensors’ data, to interpret these activities. The students will: 1. Select a CWW inference approach. 2. Apply the theory of the Art Gallery problem with moving guards (mobile surveillance cameras) to identify an optimal guarding procedure for the guards. 3. Implement a CWW-system for identifying underwater swimmer activities based on information obtained from static and mobile surveillance cameras mounted on patrol devices.

**20. Investigating Security Issues Related to Distributed Ledger Technologies Such as Blockchain; ****Mentor: Dan Tamir**

The term Distributed Ledger Technologies (DLT) refers to a decentralized (distributed) data (base) and resource management system. As such, it introduces well known and thoroughly analyzed set of cost-efficiency tradeoffs. Relevant examples include using a library as a centralized mechanism for providing access to printed material and using a mainframe computer (or cloud) vs. using a personal computer. There are several different forms of Blockchain (BC) and DLT systems. In this project, we refer to a BC as a system that has the main features of the Blockchain used for digital currency, such as Bitcoin. In this sense, a BC is a specific embodiment of a DLT. We are using the term DLT to denote a general Distributed Ledger Technology and BC to denote the DLT used in Bitcoin. BC adds a few tradeoffs to the list of tradeoffs associated with centralized vs. distributed systems. Specifically, the BC provides a unique approach to access, ownership, and trust. We distinguish between two types of use-cases for DLT. Under the external facing use-case, the DLT is open to a wide public of users. Hence, there is a low level of trust in the authentic identity and the intentions of the users. In the internal facing use-case, the use of the DLT is restricted to members of an organization (e.g., members of the Costal Guard). In this case a different level of trust might be justified.

In this project we will explore several security issues in the context of internal and external facing DLT-based systems. We plan to investigate issues such as: 1) double spending, 2) exchange hacks, 3) social engineering (block chain can be vulnerable to phishing, whaling, etc.), and 4) Software flaws that may arise due to the use of legacy software and hardware systems, especially when it comes to external vs. internal information exchange of sensitive information.

**21. Introduction to Multivariate Data Science; ****Mentor: Alex White**

This project/course introduces multivariate statistical methods. The topics include data visualization, data wrangling, basic concepts of statistical inference, statistical methodologies for simple and multiple regression, assessment of model fit, and criteria for selection of optimal models. Working in teams of 3-4, students will “wrangle” and analyze moderately large data and messy data sets to model multivariate relationships. Possible data sets include COVID 19, College Tuition, Spotify User data and others. We will use free statistical software R and R Studio.

R Studio is a user interface built for R, so it requires R to be installed. R is available online at *www.r-project.org*, and R Studio Desktop (Open Source License) can be downloaded from *http://www.rstudio.com/products/rstudio/download*. No prior knowledge of R is required.