REU Project Summaries

Understanding complexity, dynamics, and stochastic patterns in genomic data - concepts native to physics and mathematics - is critical for elucidating how disease states originate and evolve. In this project, we focus on the application of the Tunable Biclustering Algorithm (TuBA) to examine genetic and clinical data of cancer patients, aiming to identify genetic markers that can provide us insight into how cancers evolve. Our ultimate goal is to develop novel statistical platforms for fast translation of genomic data into clinical practice.

DNA folding plays an important role in gene regulation and genome packaging. The ability for a piece of DNA to bend is largely impacted by the elastic energy stored between base pairs in a segment of DNA. There are well-established techniques to mathematically model the elastic energy of a collection of DNA base pairs. It has long been known that the intrinsic elasticity parameters of DNA are sequence-dependent, but past work only models DNA folding using elasticity parameters that are specific to dimers, a sequence of two base pairs. Recent studies have suggested the intrinsic elasticity parameters of DNA are actually unique to tetramers, a sequence of four base pairs. We present software that can simulate DNA folding by optimizing elastic energy using tetrameric steps. With this new software, we optimized the configuration of DNA minicircles and confirmed that increasing the scope of sequence dependence from dimers to tetramers has an effect on DNA conformation.

Our goal is to interpret the dynamics of multi-parameter nonlinear dynamical systems. Such systems can have very complicated behavior. Because of this and the fact that in applications data collected may not accurately represent the generating system, it is desired to gather from the data robust patterns which are consistent with respect to perturbations in parameter values. A method to analyze data generated by such systems, as introduced in previous work, is to discretize the dynamics via a decomposition of the compact phase space into square grids. With a grid decomposition of the phase space, a directed graph which approximates the behavior of the system for a fixed set of parameter values is created. Each node in the directed graph represents a cubical cell in the phase space, and the directed edges denote where each cell maps to under the system. With graph algorithms we can find regions which contain recurrent dynamics, and using Conley index theory we can further identify special properties of the system. In this project we use the method of analysis described above, with the change being that we decompose the phase space using Voronoi cells rather than cubical cells. A Voronoi cell complex is generated by a finite set $P = \{x_1,x_2,\dots, x_n\}$, where each $x_i$ is chosen in the phase space. Each Voronoi cell $V_i$ is defined by $V_i = \{x|d(x,x_i)\leq d(x,x_j) \text{ for all } j\neq i,\ 1\leq j\leq n\}$. Our goal is to determine whether using Voronoi cells to decompose the phase space yields comparable results to the results generated by the cubical complex; in visual quality and accuracy. The goal is to apply these methods to real data collected from systems generated by weather patterns and robotics.

With the pervasion of social media in modern society, evolution of opinions and network dynamics in social networks have gained special interest from industry, academia, and governments. The study of opinion and network dynamics has several applications- social sciences, behavioral economics, game theory- with consequential implications. Adversarial attacks through fake news and troll bots are increasingly common and there is a need for a rigorous understanding of their impact on individuals and the social networks. We study opinion and network dynamics and characterize social networks at their limit states extending recent work in co- evolution of signed networks and opinions to better represent the real world. We build upon intuition of the real world, in which individuals hold multiple opinions on a range of issues for example- healthcare, gun control and seek to better represent complex relationships/social ties between individuals in a network. We formulate these intuitions mathematically and incorporate them in a co-evolution model which combines signed network dynamics based on structural theory and opinion dynamics. We conduct a comprehensive study through numerous simulations and validate the model on the benchmark Zachary's Karate Club dataset to achieve 100% accuracy. We mathematically address divergence of social ties to infinity, which in the real world would translate to limitless polarization, by introducing normalization which keeps the relative magnitude of the social ties intact but prevents divergence.

This paper outlines a project that started in the summer of 2021 with the goal of coming up with a better method than existing ones to solve the Low Rank Phase Retrieval Problem, primarily in terms of sample complexity, but also potentially in runtime. It remains a work in progress.

The applications Graph Cities and Graph Strata are able to decompose graphs of over 1 billion edges into smaller subgraphs that can fit on a screen and be interactively explored without losing the context of the original graph. Observing these local subgraphs reveals a variety of patterns. This report serves to find a way to describe these patterns as well as unpack meaning from these patterns by generating data stories and detecting semantically important vertices through the vertex statistics, activity and diversity.

Although all cells in a tumor come from one original mutated cell, there is genetic, nongenetic, and microenvironmental variation among these cells, referred to as intratumor heterogeneity. Nongenetic heterogeneity, which includes epigenetic changes, is closely related to the cancer stem cell (CSC) model, which states that certain tumor cells are characteristic of regular stem cells, and are likely to contribute to tumor survival and regrowth after treatment. Being able to identify phenotypic features that can classify tumor cells into a stemlike and differentiated state can help to better understand the CSC model, and the characteristics of cancer stem cells. Using the data of cells grown from the T24 cell line, PCA (principal component analysis) and UMAP analysis in R was used to group tumor cells into two different classes, which are thought to correspond with more holoclone (stemlike) and more differentiated cells. This classification can be used to analyze the colony makeup of other tumors, and might help determine the likelihood that a tumor will regenerate after treatment, based on the proportion or density of holoclone and meroclone-like cells.

The identification of higher order bifurcations in combinatorial dynamical systems allows for deep insights into the underlying structures of these systems. The modeling of such phenomena, specifically biological ones, in a combinatorial fashion has been explored; however, a systematized process to which higher order bifurcations may be found through is still nonexistent. The use of multiple paths between given adjacent parameter nodes and applying slow-fast dynamics to represent transitions between these nodes point us toward the discovery of such bifurcations. A process to identify bifurcations therefore allows us far greater insight into these systems, especially as their complexity increases.

Laser metal deposition (LMD) is a type of additive manufacturing (AM) during which metal components are created using a laser beam that fuses metal powder by melting it as it is deposited. Porosity, or small cavities that form in this printed structure, is generally considered to be one of the most destructive defects that can occur in metal AM. Currently, porosity can be measured after printing with CT scans. While this is useful for understanding the nature of pore formation and its characteristics, purely physics-driven models lack real-time prediction ability. Meanwhile, a purely deep learning approach to porosity prediction leaves valuable physics knowledge behind. Here we create a hybrid model that takes advantage of both empirical and simulated LMD data to show how various physics-informed loss functions impact the accuracy, precision, and recall of a baseline deep learning model for porosity prediction. In particular, we find that some versions of the physics-informed model are able to improve upon the precision of the baseline deep learning-only model (albeit at the expense of overall accuracy). This work lends itself to a wide array of applications as LMD is frequently used in the automotive, aerospace, energy, petrochemicals, and medical industries.

In this project we worked with involutive Heegaard Floer homology, developed by Hendricks and Manolescu as a refinement to Heegaard Floer homology. Involutive Heegaard Floer homology introduced two new knot concordance invariants. These invariants are interesting, because unlike other concordance invariants like τ, ε, and ν, they do not necessarily vanish on knots of finite concordance order, like the figure-eight knot. We computed these two new invariants for all non-thin and non-L-space 10 and 11-crossing (1,1)-knots for which they were not yet known. We also studied (1,1)-diagrams, and the relationships between them and the resulting CFK^∞ chain complexes.

In the graph semi-streaming model of computation, the input graph arrives as a stream and the goal is to compute solutions in O(n polylog n) space. With randomization, Assadi, Chen, and Khanna recently showed that the coloring problem for graphs of maximum degree Δ admits a solution in this model using an optimal Δ + 1 colors. However, little is known about this problem in deterministic settings. In this project, we showed that when Δ = Ω(n^{2/3}), no deterministic streaming algorithm can output proper colorings using O(n^{1-ε}) colors, for all ε > 0. In particular, any general algorithm for deterministic graph coloring must use at least Ω(Δ^{3/2 - ε}) colors.

Morse flow trees are a tool to compute Legendrian contact homology, and Ekholm has showed that these flow trees have a correspondence with certain types of rigid pseudoholomorphic disks in T*M if M is a manifold. There are still some open questions regarding the space of these flow trees, some of which we investigated this summer. Our aim here will be to describe some of the background material required to understand the definition of Morse flow trees. First we discuss the elements of Morse Theory and define the Morse complex; then we discuss a little bit of symplectic topology so that we define what a Lagrangian submanifold is; finally, we give the definition of a flow tree and introduce a problem related to it.

Ehrenborg conjectured that, for a simple bipartite graph G, the number of spanning trees of G, is bounded above. The bound is known to hold with equality for a class of bipartite graphs known as Ferrers graphs, and not necessarily with equality in other, specific classes of bipartite graphs. In this paper, we introduce another inequality motivated by the Cartesian product of bipartite graphs - in particular, an inequality that relates the eigenvalues of the Laplacians of the multiplicand graphs, with products of the vertex degrees and the part sizes of the factor graphs. Furthermore, we introduce an inductive framework through which the conjecture can be approached by means of a determinantal inequality, along with some preliminary results that can be proved using such a technique.

Let L⊂J^1(M) be a closed Legendrian submanifold of the 1-jet space of a closed Riemannian manifold. We conjecture that the moduli space of flow trees with at most n edges has a natural compactification given by the moduli space of broken flow trees with at most n edges. In order to solve this one would have to compactify the various moduli spaces of gradient trajectories of local function differences determined by L. We establish a compactification of the moduli space of Morse trajectories - gradient trajectories connecting critical points of a fixed local function difference - similar to the compactification in standard Morse theory.

Graph neural networks (GNNs) in recent years have gained popularity due to the increasing need for machine learning models to accommodate graph structured data. Interdisciplinary fields such as material informatics use graph neural networks to conduct research on materials that are otherwise difficult to analyse. One of these materials, volcanic glass, is used in storing nuclear waste due to their durability in extreme conditions. This durability, measured by its dissolution rate, is difficult to determine in a traditional lab environment due to the extensive time and resources needed to collect and analyse samples. I seek to implement a GNN model that performs regression in order to predict the dissolution rate of ten different types of volcanic glass. In addition to implementing the GNN, I explore varying methods of optimizing the performance of the model. Results demonstrate that unknown glass materials and their accompanying dissolution rate can accurately be determined the GNN in a short period of time. The long term goal for this research is to be able to discover new glass materials based on their dissolution rates.

Internet routers deal with significant amounts of internet traffic in everyday use. This increasing amount of traffic over the internet affects the performance, and speeds, at which these routers operate. The goal of our research project is to understand how routers manage processing packets at such high speeds while remaining flexible and reconfigurable. Through this project, we have come to understanding layers of networking-abstraction, key networking concepts, pipelined systems, along with the history of routing software, hardware, and information on completed P4 programming exercises. With this knowledge, we hope to be able to design improvements for the performance of routers and the management of campus network resources.

The complexity of the minimum circuit size problem (MCSP) - and its many variants, including the minimum Kolgomorov time-complexity problem (MKTP) - is linked intricately to countless other questions in theoretical computer science. For instance, NP-hardness of MCSP or MKTP is known to imply ZPP ≠ EXP, and even if such reductions exist, they cannot be nearly as simple as the standard NP-completeness reductions for other problems. In this project, we study whether various variants of circuit minimization can be complete for NP, or smaller classes, under simple types of reductions. Specifically, we investigate the following questions. Can recent hardness results for MKTP be extended to MCSP? How important is adaptivity in NP-hardness reductions for circuit minimization? How robust is the recent definition of non-interactive statistical zero-knowledge with logspace-bounded verifiers and simulators?

3-SAT is a known NP-Hard problem in theoretical computer science. In other words, it takes exponential time to deterministically solve this problem. However, there is a fairly effective random algorithm that finds the solution quickly most of the time, if there is one. This "Random Walk Algorithm" finds the solution by acting randomly at each timestep, but the results may not be completely random. This leads us to explore what features about an instance of the problem are important and how we can use these features to predict if a solution can be found at the end of a certain number of iterations of the algorithm. We used various machine learning techniques to make predictions using features that we found to be significant.

I formulated a relativistic Lorentz-covariant system of wave equations for a three-body, three-time one-dimensional system consisting of two photons and one electron. These particles interact only upon contact, where a boundary condition is applied to preserve the conservation of probability current. The solutions for various configurations of the initial-boundary value problem were found, and hopefully will pave the way to finding the solutions for a system of one electron surrounded by N photons.

A tournament on n agents is a binary relation that describes the outcomes of the $\binom{n}{2}$ matches played between two distinct agents. Tournaments play an important role not only in sporting events but in any setting where any two agents are comparable and some "most qualified" agent should be selected, such as elections. The winner of a tournament is determined by a tournament rule that maps tournaments to probability distributions over the agents. We want these rules to be fair (i.e., choose a high quality agent) and robust to strategic manipulation. Prior work has shown that under minimally fair rules, manipulations between two agents can be prevented when utility is nontransferable but not when utility is completely transferable. Motivated by the fact that neither utility model reflects reality, we consider pairwise manipulations under partially transferable utility. These situations arise, for example, when agents care about winning themselves but are still willing to collude if joint gains offset perceived individual losses. We show that previously studied tournament rules require high levels of selfishness in order to prevent pairwise manipulations and that for fixed levels of selfishness, there are always two agents who stand to gain a constant positive amount of probability under these rules by manipulating in large enough tournaments. We also show that for stronger notions of fairness, non-manipulable tournament rules are closely related to tournament rules that witness decreasing gains from manipulation as the number of agents increases. A major open question that remains is whether there is a tournament rule that is fair and non-manipulable for a fixed level of selfishness that meets the lower bound we prove. We conjecture that indeed such a rule exists.

We study a problem from mechanism design where we try to find a mechanism which assigns N houses to N players based on their preferences with desirable properties. These aim to unsure fairness and efficiency while limiting exploitation by the players and give incentives to tell the truth. We aim to show that the Random serial dictatorship (RSD) is the only mechanism with these prescribed properties. It has been shown before that RSD in unique for N=3. We provide a computer aided proof of uniqueness for N=4 as well as rigorous for selected preference profiles for arbitrary N.

In the United States alone, over 3.8 million people have contracted the Covid-19 virus. Due to the increased demand for testing, there have been shortages and issues for the tests themselves. Access to testing and the reporting of cases varies by region, so reported numbers are mainly consisting of people who exhibit symptoms of the virus. Taking into account those who are asymptomatic or do not have access to reliable tests, one can safely assume that the actual cases of this virus are actually much larger. Despite rates continuing to rise and testing becoming more scarce, the United States government pushes for the economy to reopen. This may lead to multiple consequences, to the cases of citizens surging to the negative social impact. Which makes research on methods to help protect individuals from infection a necessity. Continuing to see the changes and trends of this virus is crucial to both understanding causation of case spikes and to finding solutions to help flatten the curve as well as find effective ways to distribute resources. To understand the full scope of the Pandemic's impact on the United States, we considered a wide range of information regarding the virus. Focusing on the changing mobility patterns and state government mandates to understand the risk management of state governments as well as the social impact of Covid-19.

The current pandemic as a result of COVID-19 has created a new sense of reality. Across the nation, state governors have held press conferences related to the pandemic. Topic modeling, word frequencies, and sentiment analysis were conducted on press conference transcripts from various states. By incorporating these techniques into the study, weekly themes and word frequencies were revealed as well as monthly analyses of sentiment pertaining to the data. With these results, the researchers are better able to develop a comprehensive resource allocation model.

Although tumors arise as a result of the rapid proliferation of cancerous cells, these cells comprise only a fraction of the tumor. Many types of cells, each of which possesses a specific role in the growth of solid tumors, make up the tumor microenvironment (TME). In recent years, immunotherapy, a treatment that utilizes the significant presence of immune cells within the TME, has proven effective for liquid tumors such as leukemia and lymphoma. However, in solid tumors, the complex relationships within the TME can make the response to immunotherapy challenging to predict and the treatment difficult to use effectively. Thus, we created a mathematical model of pancreatic adenocarcinoma growth which includes cancer cells, T-cells, and macrophages, and used it to study the dynamics of the cellular interactions within the TME over time and throughout treatment. The model behavior indicates that the ability of CAR T-cells to kill cancer cells is more critical than their persistence. However, when used in combination with chemotherapy, the tumor can be eliminated using less effective T-cells than would be needed if using immunotherapy alone. These results indicate that CAR T-cell research should focus on engineering T-cells with high efficacy in killing cancer cells rather than long persistence in the body. Doing so could allow for effective immunotherapy use in solid tumors, either as a lone treatment or used in combination with chemotherapy.

In the recent light of the SARS-CoV-2 pandemic, the United States' response has proven to be incredibly poor with respect to policy decisions made on the local and federal levels. With deaths totaling at 142,755, this is more than the number of battle deaths in the Vietnam War, Korean War, and World War I combined. One of the easiest ways to prevent such an outbreak from reoccurring has shown to be a stronger implementation of policies on all governmental levels. Three categories which require more heavy implementation include Test, Track, and Treat (TTT), Distance, Isolate, Mask (DIM), and Prepare & Organize, Lead & Inform, Coordinate & Yaager results (POLICY). The American response was not sufficient enough to prevent mass infections and fatalities, and analysis of state and federal policies with respect to COVID-19 containment outstandingly point to a need for more drastic policy decisions.

AES is one of the best known and most widely used block ciphers to date. However, the primary argument towards the security of AES is simply the fact that it has not been broken in practice. We present a new way of theoretically analyzing the security of AES and related block ciphers by looking at the expansion of a related Cayley graph. More specifically, we provide a generalization of a single round of AES, which we consider as a generating set for the symmetric group. This then gives a Cayley graph for which one could theoretically explicitly calculate the spectral gap. We explicitly perform this calculation for small security parameters, and we compare these results to related expansion results for the ideal cipher. We hope that this analysis, when continued, will either give some indication of the security of AES or will lead to an explicit attack.

Understanding gene regulatory networks is key to targeting diseases with drugs that don't include dangerous risks. In this research project, we studied the analyses generated by Huang et al. using random circuit perturbation (RACIPE). After gaining a comprehensive understanding of Huang's paper, we employed RACIPE to reproduce these results. We then worked to produce analogous results in DSGRN (Dynamic Signatures Generated by Regulatory Networks). Moreover, we found the DSGRN parameter nodes corresponding to the RACIPE data. We added weights to the parameter nodes based on what percentage of the RACIPE models occurred in each parameter node. Our preliminary low dimensional results lead to enticing conjectures. When including weighting, the DSGRN results are very similar to RACIPE for high Hill coefficients. Thus, weighting in DSGRN with sampling from RACIPE yields comparable results to RACIPE with far quicker computations. Hence, with additional work we hope to show that DSGRN can achieve comparable results to RACIPE for a fraction of the computational cost of RACIPE.

Laser-Based Additive Manufacturing (LBAM) is a promising process in manufacturing that allows for capabilities in producing complex parts with multiple functionalities for a large array of engineering applications. Melt pool is a defining characteristic of the LBAM process and known defects of porosity in the melt pool and LBAM process has prevented the expansive adoption of LBAM. High-speed monitors that can capture the LBAM process have created the possibility for in-situ monitoring for defects and abnormalities. This paper focuses on augmenting knowledge of the relation between the LBAM process and porosity and providing models that could efficiently, accurately, and consistently predict defects and anomalies in-situ for the LBAM process. Two models are presented in this paper, Random Forest Classifier and Early Stopping Neural Network, which are used to classify pyrometer images and categorize if those images will result in defects. Both methods can achieve over 99% accuracy in an efficient manner, which would create an in-situ method for quality prediction in the LBAM process.

Impedance control is a type of interaction control method used to control the force of resistance within a robot from an external manipulator (e.g. a human). Impedance control has motion inputs and force outputs. Impedance control can be used as a means to improve the physical and dynamic relationship between robot and manipulator and has applications in rehabilitation robots and vehicles due to their interaction with humans (fragile manipulators). The goal of this project is to work with the Baxter robot, created by Rethink Robotics, to implement an impedance controller and then a spike-based impedance controller for the limbs of the Baxter robot. The spike-based impedance controller will be executed using a spiking neural network (SNN) to control the robot. An SNN is a type of artificial neural network that is more biologically plausible and its neurons only fire when an action potential reaches a certain value.

We systematically determine infinite circle packings that arise from 8 and 9-vertex polyhedra. We then classify packings where all radii are the reciprocals of integers.

We consider a variant of a problem posed by Lubetzky: given an arbitrary vertex coloring of a graph, if we randomly update the color of each successive vertex in a given sequence, so that at each step, each color has probability proportional to λ^x with x its count among the neighbors, then what initial coloring maximizes the probability of all vertices being the same color, say blue, at the end? If there are only two colors, there is a trivial proof by coupling that the all blue initial coloring is best. For more colors, we show that this is still true under certain assumptions on the graph, but we construct a counterexample in general. We build our counterexample by first appealing to a generalized setting, where the graph and the update function λ^x change at each step, and then repeatedly modifying it until it works in the original setting. The techniques we will use include identifying and exploiting ergodic Markov chains within our process and implementing a digraph within a standard graph by ensuring that the neighbors that we want to be the inneighbors vastly outnumber the neighbors that we want to be the outneighbors for every vertex.

Feder and Subi conjectured that for any 2-coloring of edges of the hypercube Q_n, there always exists a pair of antipodal vertices connected by a shortest path which changes color at most once. Leader and Long proved that there always exists a path between antipodal vertices with at most n/2 changes and Dvorak improved this bound to (3/8+o(1))n. We give some partial results which may lead to further improvements of Dvorak's upper bound.

We build on previous work by Teresa Ngo and Hannah Fell to help construct career pathways and determine the criminal implications of digital forensics. Federal Law Enforcement Training Center needed additional knowledge to understand the training, certification, and skills necessary at different levels of a career. We looked at the certification providers CompTIA and Global Information Assurance Certification to find certifications relevant to digital forensics. We did text analysis of Thomson Reuters Westlaw criminal case documents to determine the relevance of digital forensics. This project has not concluded, so extending the time period of Westlaw document analysis would make viewing trends available and improve understanding of cybersecurity and digital forensics career pathways.

Single cell sequencing is a new field in biology that allows for the extraction of tumor cell populations, and subsequent analysis on this data leads to a better understanding of the differences in cells from these populations. The data in regard to reads and mutant allele coverage is critical to understanding how phenotype of cells are informed. Data extracted from thousands of cells yielded wide ranges of depth of UMI (Unique Molecular Identifier) and mutant UMI reads for single cells and base pairs. The analysis showed similar patterns of depth between the genes studied with low mutant coverage, which is not unexpected with single cell data. The results of having low data coverage results in less statistical power when trying to create confidence intervals for variant allele frequency. Further analysis using known mutation data showed that a previously developed probability function can result in a negative value when variant allele frequency is greater than .5. Thus, future directions of this project will focus on a development of a probability model based on our data.

We studied the problem of deterministic vertex coloring in the semi-streaming model. The (Δ + 1)-coloring problem is easily solved in the classical setting via a greedy coloring, where Δ is the maximum degree. In the streaming model, we are limited to a memory of size O(n) and can only read off edges of the graph one at a time. Although there exist randomized algorithms solving the (Δ + 1)-coloring problem in the semi-streaming model, it is unknown whether there exists a deterministic such algorithm. We establish two deterministic O(poly(Δ))-coloring algorithms towards this; the first produces an O(Δ³)- coloring in two passes, while the second produces an O(Δ²)-coloring in O(log Δ) passes. We also briefly touch upon the related independent set problem, and prove a deterministic Ω(n/Δ²)-independent set single-pass algorithm.

Let X and Y be two finite-area hyperbolic Riemann surfaces. Since X and Y are both hyperbolic, we know that they must both be uniformized by ℍ ; however, the maps ℍ → X and ℍ → Y are not in general of finite degree. In general, guaranteeing that X and Y share conformally equivalent finite covers is too much to hope. The Ehrenpreis conjecture states that we can achieve something close: in particular, for any K>1, we can find finite covers X' → X and Y' → Y such that X' and Y' are K-quasiconformally equivalent; in other words, we can always find finite covers which have little distortion of angles relative to each other. The conjecture was recently settled in the case of closed surfaces in 2015 by Kahn and Markovic, but there has been no such resolution for punctured Riemann surfaces, and in fact the truth of the conjecture remains contested. The modular surface is a model example of a punctured Riemann surface of finite area. In this project, we numerically find and study finite covers of the modular surface constructed by gluing together immersed hyperbolic pants and "degenerate pants", in which we allow one cuff to degenerate into a cusp, subject to similar technical constraints. We hope that the work will help in generating insight towards an angle of attack in a proof of the cusped Ehrenpreis conjecture inspired by the methods used in the proof of the case of closed surfaces.

Most data informatics researchers considered Data visualization to be just the graphic representation. Here we interacted with and represented a data set of Danish folklore by creating human understandable stories which combine topics of the different stories in the data set.

For a classical system, momentum has been understood as the product of mass and velocity, however in a quantum system this is not the only notion for momentum. In the Bohmian interpretation of Quantum mechanics, there exists a guiding equation which helps support the classical understanding of momentum by providing both mass and velocity in a proportional relationship. However for specific particles such as photons, the classical notion of momentum is no longer applicable. To better understand momentum in a 1-dimensional quantum system I analyzed the possible different notions of momentum for systems with a finite number of particles and compared the results to strive for the overall "winner" in momentum measurement.

Our project is inspired by the mathematical model used in the software package DSGRN. We try to find the equilibria in the systems of ODEs containing Hill models using Newton's method and validate them by radii polynomials. Finally, we introduced interval arithmetics to prove the nonexistence of other equilibria besides the ones we have found.

The network interface card (NIC) is the hardware component that connects a computer, such as a laptop, server, or datacenter, to the internet. The NIC sits in the path of the data moving in and out of your machine, allowing it to compute over the data as it transits. For example, a NIC might automatically compress data going in and out of your machine to save bandwidth and reduce download time. With the advent of software-defined networking, along with ever-increasing network bandwidths, NICs have become increasingly flexible in the computations they support, and their architectures more complicated. The Netronome NFP-6000 is one example of these "smart NICs". We present a formal specification of arithmetic and logic instructions for the Netronome NFP-6000, and a programmatic framework for interpreting and formally validating Netronome programs. This work brings us closer to a superoptimizing compiler for Netronome smart NIC programs, and will help further understanding of the capabilities and limitations of hardware acceleration using NICs.

The Riemann Hypothesis is a famous unsolved problem in mathematics first studied by Bernhard Riemann in 1859 which is important for understanding the distribution of prime numbers. In recent years, the development of type theory and automated theorem proving has made possible a new level of rigour in the ability for computers to check mathematical proofs. In this project, we formalize the statement of the Riemann Hypothesis in the Lean Theorem Prover using elementary methods in analysis and study the minimal algebraic requirements necessary to construct the statement of the Riemann Hypothesis.

Zero-knowledge proof systems are of great interest to cryptographers, since they allow for the sharing of knowledge of secret information without divulging the actual secret. Interactive versions of these proof systems contain fundamental cryptographic problems such as discrete log and decisional Diffie-Helman. We analyzed the class of non-interactive versions of these proof systems, NISZK and a variant where the verfier in the system is limited to being a log-space machine, NISZKL. We show that EA for NC0 circuits is complete for NISZKL under AC0-many one reductions. We also show that a problem concerning the minimum time-bounded Kolmogorov complexity of strings, MKTP, is hard for co-NISZKL under P/poly many-one reductions by reduction from EA. MKTP is a candidate NP-intermediate problem and previously had only been shown to be hard under NC0 reductions for a subclass of P.

A knot is an embedding of S¹ into S³. Knots are studied up to the equivalence relation of concordance; two knots are said to be concordant if their disjoint union is the boundary of an annulus in S^3 \times I. Heegaard Floer homology is a suite of invariants of knots and 3-manifolds; the knot version categories the classical Alexander polynomial. The variant we studied assigns to a knot K a filtered and graded chain complex over F₂[U,U^-1]. This complex admits various chain maps which carry geometric information, such as the Sarkar involution associated to a Dehn twist around the knot, and skew-filtered chain map iota_K associated to a natural symmetry on the construction of Heegaard Floer homology. Our project dealt with the Heegaard Floer homology of pretzel knots P(-2,m,n) with m and n positive odd integers. These knots have relatively simple Heegaard Floer knot homology, and we were therefore able to give a combinatorial computation of the chain map iota_K. Since the map iota_K carries geometric information, this led to new computations of invariants of knot concordance.

It is known result in differential geometry that a 3-ellipsoid which is near spherical contains 4 minimal 2-spheres, and that if the 3-ellipsoid is scaled enough along one axis, additional minimal 2-spheres occur. We attempt to show that as this scaling approaches infinity, in other words, as the 3-ellipsoid approaches a higher dimensional cylinder, the number of minimal 2-spheres approaches infinity. The method we use is to take advantage of the rotational symmetry of an ellipsoid to reduce the problem to finding geodesics in a two-dimensional metric space, which can be done by finding the solutions to a system of nonlinear ordinary differential equations.

In the problem of object rearrangement, we have n items in an initial configuration and we want to move them to certain goal positions. In particular, we are considering the case where each item has its own respective goals and the objects may be initially positioned atop each other's goals. It has been shown how by considering the items' relationships as a dependency graph where each item is a vertex and there is an directed edge (u,v) if and only if v's initial position intersects u's goal. Finding the minimum amount of items that need to be moved to temporary locations, or buffers, in order to move all objects to their goals is then shown to be equivalent to finding a minimum feedback vertex set for this dependency graph, as the collisions which are unsolvable without buffers are cycles. The goal of this project was to look at the related problem of the minimum number of active buffers required. In other words, the minimum amount of items that need to be simultaneously stored in temporary locations in order to resolve the graph.

In this project, we explored the notion of polynomial mutation in the context of algebraic geometry. It is conjectured that these polynomials can tell us something about the smoothing components of deformations of toric varieties. We worked on a simple combinatorial proof that an irreducible 0-mutable polynomial is rigid maximally mutable. An algorithm for reducing polynomials was developed, although it was demonstrated that such an algorithm cannot be guaranteed to find 0-mutable polynomials efficiently. Finally, we considered different notions of polytope mutation, and proved an equivalence between different definitions. Using this equivalence, we showed a fact about the number of possible mutations of a polynomial, up to isomorphisms of the mutations' convex hulls.

The software Dynamic Signatures Generated by Regulatory Networks (DSGRN) allows for a nearly instantaneous computational analysis of the global dynamics of (genetic) regulatory networks. There is a great need to further understand biologically relevant regulatory networks, which motivates the development of the techniques employed by DSGRN. In order to expand and enrich the abilities of DSGRN, I consider a generalized regulatory network which gives greater flexibility to the interactions between the nodes of the network. This flexibility arises from the introduction of multiple threshold interactions between species. In order to allow DSGRN to handle these generalized regulatory networks, a new kind of parameter space decomposition is required. This parameter space decomposition consists of calculating which parameter sets are possible for the regulatory network. For my project, I developed code to compute this decomposition and store the results in a database. This database can be used by DSGRN in the future in order to handle regulatory networks with multiple thresholds.

Virus spread can be simulated through computer programming which can help us to better understand how to minimize the rate at which it spreads. The coronavirus pandemic has impacted how everyone lives their day-to-day lives. The goal of this research was to develop a computer simulation to better understand how the virus spreads and to investigate different graphs such as the connected caveman graph that we can implement to reduce the spread. The idea behind the connected caveman graph was to partition the population into groups to limit the spread of the virus and quarantine specific groups if an outbreak would occur.

In my summer work I worked with the P₄ programming language and the Barefoot SDE. Information about the P₄ programming language includes implementing exercises that exhibit network-level use cases of programmable switches through P₄ programs. Information about the Barefoot SDE includes experiences with a real hardware device that can process data at 6.4 Tbit/s.

I surveyd equilibrium arrangements, proper as well as pseudo, of four point particles on the unit circle which interact pairwise with a repulsive power law force. The bifurcation diagram which jointly exhibits all these equilibrium arrangements as functions of s features four obvious "universal" equilibria, which do not depend on s, and many more continuous families of s-dependent non-universal equilibria, some of which were known before but many were not. All equilibria are also planar special cases of the analogous four-particle equilibrium problem on the 2-sphere, some of which are also studied.