(The conference was held virtually via Zoom. )
Advisor: Dr. William Ott
Abstract: The dynamics of physiological systems are significantly impacted by delay. The time-delay caused by the transport and processing of chemical components and signals may be of significant consequence. Biological systems present a challenge to model, analyze and predict. The utilization of machine learning to build mathematical models of complex systems has rapidly grown. For time-dependent series, generally a recurrent neural network (RNN), capable of returning past states, is used. In most common RNN implementations, multiple hidden layers are rebalanced during training to achieve adequate results. However, these implementations can be computationally expensive and may require extensive training data. Here, we utilize a type of RNN called an echo state network (ESN), a static, randomly initialized reservoir of nodes. We study two types of physiological systems exhibiting delay: degrade-and-fire circuits and the insulin-glucose cycle. Manipulating only signal propagation and input smoothing, we model both systems with generated reservoirs. In future works, we will further tune the reservoir, addressing stability and noise. We will also research the use of other machine learning techniques in determining the optimal parameters for reservoir generation.
Advisor: Dr. Stephen A. Fulling
Abstract: We want to analyze the time evolution of a quantum mechanical system with a potential linear to the right of the origin and zero to the left. We can approximate the propagator for a quantum system in terms of paths of a classical particle which starts at $y$ and ends at $x$. Five geometrically distinct classes of these paths were studied in a previous paper. However, this approximation, when based on initial position, $y$, diverges in amplitude at certain places called caustics. Here, we propose an alternative approximation based on initial momentum, $p$, in order to avoid the breakdown. We then build from the position-based analysis of the various classical paths by reformulating them with initial momentum. In order to construct the WKB propagator, we calculate the action and amplitude functions which tell us the contribution of classical paths to the approximation. We can then compare the results with the position based approximation of the WKB propagator. Finally, we use the propagator to construct the approximate time-development of the wave function starting from the Gaussian initial wave packets. Analogously, we construct the $y$-based Gaussian initial wave packets to compare with the $p$-based result in order to see which method best describes the time evolution of the quantum system in various circumstances.
Advisor: Dr. Shawn Walker
Abstract: Since the dawn of Big Data, machine learning has taken flight becoming a part of our daily lives. These algorithms that fall under the umbrella of machine learning can be categorized into supervised and unsupervised learning. In most situations, the data might appear chaotic, but there could be features hiding inside that we are unable to see. This is when one turns to an unsupervised learning algorithm such as self-organizing maps. These maps are considered a dimensionality reduction going from n-dimensional space to 2-dimensions, which allow for visualization of high dimensional data. Using the curvature and torsion of 3-D space curves as the inputs into this n-dimensional space, these curves are then classified in a 2-D space. The end goal of this project will consist of these 3-D curves representing the packing of viral DNA inside the capsid to see if the 3-D curve data forms clusters.
Advisors: Dr. Daniel Onofrei and Neil Jerome A. Egarguin
Abstract: The spring-mass system is an invaluable model with unmatched versatility for studying wave-like physical phenomena or material deformation. We consider linear spring-mass systems and the use of experimental data to locate and characterize defects ("error" masses) somewhere along its length. By taking the Laplace transform of the first body’s trajectory, the system eigenvalues are recoverable if damping is negligibly weak. Encoded within the trace/determinant eigenvalue relations are the masses of the defects; a minimization procedure with the recovered masses can then reveal their locations. For up to two present defects, this scheme is reasonably successful.
One of the most challenging questions a mathematics graduate student must ask is, “Should I continue in academia or go into industry?” I asked myself that question. I also asked relatives, my friends, other graduate students, my advisors, etc. If I took a survey of their responses, I probably would have ended up with 50% suggesting academia and 50% suggesting industry. I followed their advice and took an industrial postdoc position where I worked 50% of the time in academia and 50% of the time in industry. Ultimately, I chose to go into industry. I am now a software engineer on the MATLAB indexing team, where I am responsible for ensuring indexing operations are correct, consistent, and fast. In this talk, I will present examples of how scientists and engineers are using MathWorks products to solve challenging real-world problems. I will also tell you the story about how I made the transition from academia to industry, from programming in MATLAB to programming MATLAB.
Andy received his Ph.D. in mathematics from the University of Utah in 2014. In his dissertation, he made extensive use of MATLAB while studying electrical impedance tomography and cloaking due to anomalous localized resonance. He then accepted an industrial postdoctoral position at the Institute of Mathematics and its Applications at the University of Minnesota. While there, he continued his work on cloaking; he also worked on oil exploration algorithms in collaboration with scientists from Schlumberger-Doll Research Center in Cambridge, Massachusetts. In 2016, he began his position as a software engineer at MathWorks in Natick, Massachusetts.
Advisor: Dr. Wencai Liu
Abstract: We give a proof of unique determination of the finite discrete free Schrödinger operator from its spectrum, also known as Ambarzumian problem, with Floquet boundary conditions, using a Discrete Fourier Transform based approach. We provide a counterexample to this inverse spectral problem if the knowledge of the spectrum is replaced by the knowledge of the spectrum of a rank one perturbation of the free operator. We also proved the following Ambarzumian-type mixed inverse spectral problem: Diagonal entries except the first and second ones and a set of two consecutive eigenvalues uniquely determine the finite discrete free Schrödinger operator.
Advisor: Dr. Anton Zeitlin
Abstract: Teichmuller space is a component of the moduli space of flat $ PSL(2,\mathbb{R}) $ connections. Moduli spaces answer a very general yet useful question: how many ways can something be configured? For instance, Teichmuller space describes the way in which geometric structures can be placed on a surface, up to isotopy. Hyperbolic geometry lends itself to very convenient parametrizations of Teichmuller space, such as Penner's lambda lengths. Lambda lengths allow for a combinatorial view of the Teichmuller space, as we can look at triangulations amongst punctures on our space. A Whitehead flip of two adjacent triangles reveals a formula in the lambda lengths which is reminiscent of Ptolemy's classical formula for cyclic quadrilaterals, while at the same time giving rise to structures such as cluster algebras. All of this stands for one choice of $G$; we can choose any other transformation group and study the resulting moduli space. In my research, I am investigating the choice of $G = GL(1|1)$, the space of $ (1,1)\times (1,1) $ non singular super matrices.
Advisors: Dr. Daniel Onofrei and Neil Jerome A. Egarguin
Abstract: Backscattered cloaking attempts to mask vibrations in the direction of measurement. This research focuses on one dimensional backscattered cloaking. This problem can be described in terms of spring mass mechanics, circuits, acoustics, and atomic lattice vibrations. In this poster, we discuss methods for cloaking both a ten mass system toy problem and a large one hundred mass system. We utilize Plancherel’s theorem to allow optimization of both systems in the Fourier domain. We then proceed to optimize the ten mass system using a Fourier transform polynomial and the hundred mass system using Tikhonov regularization. Simulation results are provided, and results show a large improvement in the residual value from optimizing both the ten and one hundred mass systems. The results also show that the residual value decreases significantly in dissipative media, and an explanation for why this is true is provided. Overall, we find that cloaking can be done well at a large scale using Tikhonov regularization.
Advisors: Dr. Stephen Fulling
Abstract: In the study of vacuum energy in Quantum Field tTheory, it is desirable to quantify the energy gradient in a hypothetical experiment verifying the Casimir effect. When the experimental setup is modeled theoretically, it is instructive to model the boundary using "soft" power potentials, rather than the standard Dirichlet boundary originally utilized. While more realistic, these models are more mathematically complicated. Closed form expressions for the corresponding Green function and stress tensors do not exist in general power wall models. To analyze these models, these quantities are approximated via a Pade' approximant. To verify our results, we compare the resulting stress tensor to a numerical counterpart, and verify that both obey the expected conservation laws.
Advisors: Dr. William Ott
Abstract: We take a kinetic theory approach to the problem of modeling pedestrian dynamics, studying how people move around in a closed environment. We weigh our approach with other models and discuss the benefits of using kinetic theory, which is a statistical approach that allows us to describe the geographical and pedestrian factors that influence an individual's movement. We couple our dynamics (movement) model with a contagion model, which allows us to track the spread of an infectious disease between agents in a room. We run some preliminary simulations, and discuss results and future work.
Advisors: Dr. Andrei Tarfulea
Abstract: The Fast Fourier Transform can be utilized to solve partial differential equations with numerical methods because of its properties related to derivatives. This project attempts to solve several partial differential equations using this method. It concludes by using Tarek Elgindi’s initial data for solutions that cause finite-time singularities in the 3D Euler Equations and applying this data to the 3D Navier-Stokes Equations to observe if numerical blowup occurs.
Advisors: Dr. William Ott
Abstract: Suppose one wants to monitor a domain with sensors each sensing small ball-shaped regions, but the domain is hazardous enough that one cannot control the placement of the sensors. A prohibitively large number of randomly-placed sensors would be required to obtain static coverage. Instead, one can use fewer sensors by providing only mobile coverage, a generalization of the static setup wherein every possible intruder is detected by the moving sensors in a bounded amount of time. Here, we use topology in order to implement algorithms certifying mobile coverage that use only local data to solve the global problem. Our algorithms do not require knowledge of the sensors' locations. We experimentally study the statistics of mobile coverage in two dynamical scenarios. We allow the sensors to move independently (billiard dynamics and Brownian motion), or to locally coordinate their dynamics (collective animal motion models). Fewer sensors are required in the mobile setting. Our detailed simulations show, for example, that the expected time until mobile coverage is achieved decreases more rapidly, as the sensing radius is increased, in the billiard motion model then in the D'Orsogna collective motion model.
Advisors: Dr. Robert Lipton
Abstract: Thin films involving metamaterial surface and optically coupled thickness changing layer can exhibit unique wave properties. In this project, a 180-degree phase tuning device is found possible by homogenizing the metamaterial gold bars and solving the time-harmonic Maxwell's Equations in the first order approximation. Such mathematical model provides machine learning a pathway for deeper investigation.
Advisors: Dr. Matthew Young
Abstract: We present a generalized version of the well-known Dedekind sum which we will call newform Dedekind sum. We then present the interesting patterns obtained by the kernel of the newform Dedekind sum. Then we conclude by presenting our main results that can be loosely described as showing that the kernels are neither "too big" nor "too small".