Great job to all of our Summer 2020 REU students!

## July 30, 2020

**TITLE: **Generating a fractal tree

**PRESENTER: **Jingyi Gao

**MENTOR: ** Mitchell Newberry

**ABSTRACT: ** Being inspired by the beautiful tree pattern on the marble screen of Sidi Saiyyed Mosque, we want to generate such self-similar tree pattern on any arbitrary shape of space. In this talk, we will discuss the concept of fractal, Hausdorff dimension, self-similarity, and scaling relationships. We will then explain why such tree pattern counts for a fractal, how does the fractal dimension of it correspond to its physical properties, and how do we determine the fractal dimension of our tree. Finally, we will explain how are fractals built from recursive relationships, and the future direction of the project.

**TITLE: **Stefan Problem with Internal Heat Generation in Cylindrical Coordinates

**PRESENTER:** Jiashu Han

**MENTOR:** Lyudmyla Barannyk

**ABSTRACT:** In this project, we consider a problem of solid-liquid phase change driven by internal heat generation of the material in cylindrical coordinates. We formulate initial boundary value problems for temperature in solid and liquid phases using non-homogeneous heat equation. The energy balance equation on the interface between two phases closes the system. The method of superposition is used to split the solutions into steady-state and transient solutions. The transient solutions are obtained by the method of separation of variables. The steady-state solution is achieved by direct integration. We find the ordinary differential equation for the interface, which involves Fourier-Bessel series. This equation governs the evolution of the interface with time. We solve this equation numerically. We observe the overheating phenomenon in the solid phase during melting. Our goal is to understand what causes this problem and how we can modify the governing equations to overcome this inconsistency. The problem has applications to a nuclear fuel rod during meltdown.

**TITLE:** Pooled Annuity and Income Stability

**PRESENTER: **Ge Qu

**MENTOR: **Thomas Bernhardt

**ABSTRACT: **This project focuses on the income stability of members in the pooled annuity fund. The income stability is measured by the proportion of fund members who will receive stable incomes for life with a certain probability. The idiosyncratic longevity risk causes the fluctuations in members' income. To mitigate the negative effect of the idiosyncratic longevity risk on the income stability, a possible solution is to diversify the initial account values of fund members. In this project, investment returns and the group size are fixed and the systematic risk is ignored. The quantified effect of composition of initial account values on the income stability will be studied. In this study, fund members hold two different initial account values. That is, high budget members hold account with a certain high initial value and vise versa. One goal is to quantify the effect of the percentage composition of high budget members and low budget members on the income stability. Another concern of the study is to check weather the distance between two initial account values will affect the outcome. An approximated expression that is independent of mortality model will be derived to calculate the numbers of members who will receive stable income for life with a certain probability. The applicable outcome of the study is to calculate the time period in which account members can receive the stable income given the composition of the initial account values.

**TITLE: **Extensions of Irreducible Representations of Quaternion Algebras over p-adic Fields

**PRESENTER:** Alex Wang

**MENTOR: **Karol Koziol

**ABSTRACT: **The Local Langlands Conjectures give a correspondence between n-dimensional mod-p representations of the absolute Galois group of Q_p, the p-adic numbers, with mod-p representations of GL_n(Q_p). In the case where n = 2, we can look to understand representations of quaternion algebras over Q_p, which naturally correspond to GL_2(Q_p). In this talk, we discuss the irreducible mod-p representations of quaternion algebras over Q_p as well as a classification of their extensions.

## July 23, 2020

**TITLE:** Ramsey Theory in Models where the Axiom of Choice Fails

**PRESENTER: **Zekun Jia

**MENTOR: ** Zach Norwood

**ABSTRACT: ** There are a lot of theorems in the Ramsey theory whose proof explicitly or implicitly uses the Axiom of Choice. This project focuses on Ramsey-theoretic statements in models where the Axiom of Choice fails. We are interested in two kinds of models: permutation models with atoms and symmetric submodels of generic extensions obtained by forcing. So far I have investigated the Ramsey’s theorem and the Open Ramsey theorem in the basic Cohen model, the basic Fraenkel model, and the ordered Mostowski model. And I gave a proof of Open Ramsey over omega without using the Axiom of Choice. The next step is to study feeble filter, which is closely connected with Ramsey property, in the Feferman model, as well as other related topics.

## July 16, 2020

**TITLE:** A Low Dimensional Model for Mouse Circadian Rhythms

**PRESENTER: **Carrie Fulton

**MENTOR: ** Victoria Booth

**ABSTRACT: ** Circadian rhythms are the environmentally driven daily cycles of physical and cellular activity exhibited by most all organisms. Disruption of said cycles in humans has been found to negatively impact overall health. The utilization of model organisms, such as mice (Mus musculus), are essential to providing behavioral, physiological and cellular scale analyses that may never be available for humans. This talk will discuss the work done thus far to develop a low-dimensional limit cycle oscillator model for mouse circadian rhythms, which has been fit and validated against a large library of light phase response curve data for mice. The intention is to enable a cross-species comparison to improve integration of multiscale data sets between model organisms and human studies.

**TITLE:** Periodic points on translation surfaces

**PRESENTER:** Rafael Saavedra

**MENTORS:** Christopher Zhang, Paul Apisa, & Alex Wright

**ABSTRACT:** If you you light a candle in a polygonal room with mirrored walls, will it illuminate the whole room? Suppose a billiard ball bounces in a polygonal table. Given points A and B, is there a point C such that all billiard trajectories from A to B pass through C? I will talk about these two problems, called the illumination and blocking problems, and explain how they lead to the study of translation surfaces. I will then motivate the study of periodic points on translation surfaces. I will explain our work to classify the periodic points on regular polygons.

**TITLE:** Entropy and Information Inequalities

**PRESENTER: **Michael Tang

**MENTOR: ** Andreas Blass

**ABSTRACT:** The entropy H(X) of a random variable X describes how much "information" one gains by learning the value of X in any given trial. Shannon's inequality for entropy states that the entropy H(X,Y) of the joint variable (X,Y) is at most H(X) + H(Y); inequalities that can be derived solely by applying Shannon's inequality to different sets of variables are called "Shannon-type." However, not all information inequalities (linear inequalities involving entropies of random variables) are Shannon-type. In this talk, we will analyze some work by Dougherty, Freiling, and Zeger (2011) that produced a long list of non-Shannon inequalities, and describe an application to secret sharing, a topic in cryptography and information theory.

## July 9, 2020

**TITLE:** Stochastic volatility modeling: from rough Heston model to affine Volterra equation

**PRESENTERS: **Zixuan Wang & Yuanchun Ye

**MENTORS: ** Shuoqing Deng

**ABSTRACT: ** In the famous Heston model, where the variance satisfies a CIR process, the characteristic function of the log-price can be calculated explicitly by the Riccati equation. When replacing the Brownian motion by the fractional Brownian motion, one now faces the Rough Heston model and difficulties arise because of the lack of Markovian structure. In this project, we will first study the mechanism of calculating the characteristic function of the Rough Heston model using the convergence of a suitable class of Hawkes processes. Then, we will study the optimal consumption and stochastic control problems in the Heston framework and Affine Volterra framework by some specific examples.

**TITLE:** Minimal stretch factors for punctured non-orientable surfaces

**PRESENTER**: Caleb Partin

**MENTORS:** Sayantan Khan, Becca Winarski, Alex Wright

**ABSTRACT:** It is a natural question to ask how much a map from a surface to itself “mixes up” the surface. Certain homeomorphisms (or continuous bijective maps) locally stretch a surface by some factor, called the “stretch factor”, which measures how much mixing the map does. We call such homeomorphisms pseudo-Anosov, and they provide a rich yet difficult to understand family of maps. One way to gain insight into these homeomorphisms is to ask for a given surface, how small can the stretch factor be for maps on this surface? In other words, what is the minimal stretch factor? For punctured orientable surfaces, Yazdi (2019) showed that the minimal stretch factor of a g-holed orientable surface is bounded above and below by some constants depending on the number of punctures over g. In our project, we aim to generalize Yazdi’s result to prove an analogous statement for punctured non-orientable surfaces.

**TITLE:** Triangular Billiards

**PRESENTER:** Anne Larsen

**MENTORS: ** Alex Wright & Chaya Norton

**ABSTRACT:** Consider a billiard ball bouncing around a triangular table. What can be said about its path? For example, will it ever return to its starting position? Will it spend more time in some areas of the table than others? To approach these questions, it is useful to "unfold" the piecewise linear path on a billiard table to a straight-line path on a different surface, where the dynamics of paths might be better understood. But even for triangles whose angles are rational multiples of pi, it is an open problem to determine which triangles unfold to so-called Veech surfaces, for which these questions can be answered, and which do not. This talk will present the basic definitions and summarize some past results relating to this problem, as well as describing our current research.

** TITLE: **Optimal Stopping of exotic American option by Deep Learning

**PRESENTER:**Shuoqing Deng

**MENTORS:**Qinghao Lin

**ABSTRACT:**The project mainly focuses deep learning method for optimal stopping problems which directly learns the optimal stopping rule from Monte Carlo samples. The introduction to the theory, mathematical proof and appliance of Deep Learning will be discussed. We test the approach on the pricing of a Bermudan max-call option and swing option. Also, deep learning on Longstaff Schwartz algorithm's regression bases will be mentioned.

## June 30, 2020

**TITLE:** Quiver Polynomials and Bumpless Pipe Dreams (Pt. 1)

**PRESENTERS: **Zixuan Xu

**MENTORS: ** Anna Weigandt

**ABSTRACT: ** A quiver is a directed graph; a quiver representation is an assignment of vector spaces to vertices and linear transformations to directed edges. We will explain how isomorphism classes of representations of quivers can be encoded as orbits in the representation space of the quiver. These orbit closures are called quiver loci and give rise to quiver polynomials. Buch and Fulton (1999) conjectured a formula for type A equioriented quiver polynomials as a positive sum of products of specialized Schur functions. This was proved by Knutson, Miller, and Shimozono (2006) using a stabilization argument. We seek to give an elementary explanation for this formula using new combinatorial objects of Lam-Lee-Shimozono (2018) called bumpless pipe dreams.

**TITLE: **Quiver Polynomials and Bumpless Pipe Dreams (Pt 2)

**PRESENTERS: **Alexander Vidinas

**MENTORS:** Anna Weigandt

**ABSTRACT:** A quiver is a directed graph; a quiver representation is an assignment of vector spaces to vertices and linear transformations to directed edges. We will explain how isomorphism classes of representations of quivers can be encoded as orbits in the representation space of the quiver. These orbit closures are called quiver loci and give rise to quiver polynomials. Buch and Fulton (1999) conjectured a formula for type A equioriented quiver polynomials as a positive sum of products of specialized Schur functions. This was proved by Knutson, Miller, and Shimozono (2006) using a stabilization argument. We seek to give an elementary explanation for this formula using new combinatorial objects of Lam-Lee-Shimozono (2018) called bumpless pipe dreams.

## June 23, 2020

**TITLE: **Backpropogation with Biological Neurons

**PRESENTER: **James Hazelden

**MENTOR: **Daniel Forger

**ABSTRACT: ** Traditional artificial neural networks have been of insurmountable importance in recent developments in machine learning; however, recent research has suggested that neural networks that take into account the concept of time (spiking neural networks) may be more accurate for certain machine learning tasks. However, learning with such networks is still a point of much research. Traditional backpropogation used for artificial neural networks is quite hard to adapt in this context. In our research, we derive equations for backpropogation for the Hodgkin Huxley model of biological spiking neurons.

**TITLE**: Medium-Scale Ricci Curvature for Hyperbolic Groups

**PRESENTER**: Andrew Keisling

**MENTOR**: Thang Nguyen

**ABSTRACT**: Many notions of Ricci curvature have recently been developed that generalize classical curvature definitions from Riemannian geometry. In this talk, we will discuss a definition of curvature for finitely generated groups and compute some explicit examples. We will discuss our main results relating this curvature to other notions of curvature, particularly that of delta-hyperbolicity as defined by Gromov. As we will show, these two types of curvature agree in some expected ways, but they can also disagree in surprising ways. For example, delta-hyperbolic groups can be shown to have negative curvature under sufficient conditions, but there also exist negatively-curved groups that are not delta-hyperbolic.

**TITLE**: The Relativistic Vlasov-Maxwell System

**PRESENTER**: Advika Rajapakse

**MENTOR**: Neel Patel

**ABSTRACT**: When modeling the motion of collisionless plasma, especially when the particles are traveling at high speeds, the relativistic Vlasov-Maxwell system of equations is particularly useful. The global existence and smoothness of the 3-dimensional Vlasov-Maxwell system, a longstanding problem in analysis, has not yet been proved without further preliminary assumptions. In this project, we attempt to study how this system of equations governs the velocities of particles in lower-dimensional cases before possibly working on the 3-dimensional problem.

**TITLE**: Coding theory and Grassmannian variety

**PRESENTER**: Pengrun Huang

**MENTOR**: Eric Canton

**ABSTRACT**: In coding theory, one question is to look for codes that has great error correcting properties. In this talk, we will first introduce the concept of linear codes, generator matrix, hamming weight, weight distribution and minimum weight in coding theory. We will also explain why a linear code with length n and rank k is a linear subspace and how to embed a subspace into a point using Plucker embedding. Then, we will introduce isometric codes, and equivalence class defined upon isometry. Finally, we will discuss our result on relationship between isometric code and hamming weight of the Grassmannian variety.

## June 18, 2020

**TITLE: **Hilbert Functions and Square-Free Monomial Ideals

**PRESENTERS: **Marianne DeBrito & Meixuan Sun

**MENTORS: **Janet Page & Timothy Ryan

** ABSTRACT: **In commutative algebra, a common question is to ask how the number of generators of ideal changes as you change the ideal. In this talk, we will first introduce the class of ideals for which we want to study this question, which are known as square-free monomial ideals. We will then introduce symbolic powers of an ideal and ask how the number of generators changes as you increase the power. The answer to this question is phrased as a Hilbert quasi-polynomial which we will introduce next. Finally, we will discuss our current results in computing these polynomials.