2021 Summer Undergraduate Research in Mathematics Virtual Symposium
This event will be an opportunity for anyone interested to hear about the research work that was done by all the participating students in Columbia Math summer research programs (CSUREMM and REU).
When
08/13/2021, 10:00 AM - 2:45 PM ET
Where
Join Zoom Meeting https://columbiauniversity.zoom.us/j/92016498172?pwd=SEIyamwrRU96bmlYMU1ibjREbW55UT09
Meeting ID: 920 1649 8172 Passcode: 653784
Schedule
| Time | Event |
|---|---|
| 10:00-10:15 | Opening Remarks |
| 10:15-11:30 | Ignite Talks |
| 11:30-12:30 | Lunch Break |
| 12:30- 2:30 | Poster Presentations |
| 2:30- 2:45 | Closing Remarks & Group Photo |
Title & Abstracts
Math REU:
Team: Elena Gribelyuk, Ruoxi Li, Iris Rosenblum-Sellers, Henry Zhang
Project Lead: Joshua Pfeffer
Graduate TA: Zoe Himwich
Thick points in a random square subdivision model generated by Bernoulli trials
ABSTRACT: Random planar maps are fundamental models of random discrete surfaces with applications in combinatorics, probability, physics, and computer science. We consider a sequence of random planar maps embedded in the unit square defined in terms of Bernoulli trials with fixed parameter $p \in (0,1)$. This is a simplified version of a sequence of random planar maps that has been used to approximate a continuum model of random geometry known as Liouville quantum gravity. We analyze the set of $\alpha$-thick points, which are points in the square at which the sequence of random planar maps is asymptotically “$\alpha$-concentrated”. A Euclidean-typical point is $p$-thick, so the set of points with thickness $\alpha$ has measure zero for each $\alpha \neq p$. Our main result computes the a.s. Hausdorff dimension of the set of $\alpha$-thick points, which gives a finer description of the size of the set that distinguishes between different values of $\alpha$.
Team: David Chen, Wangdong Jia, Bryce Joseph Monier, Shiyang Shen
Project Lead: Carsten Chong
Graduate TA: Christian Serio
Constructing an estimator of FBM under noise
ABSTRACT: Fractional Brownian Motion (FBM) is an important generalization of Brownian Motion parameterized by the Hurst index. The Hurst index is a real-valued number H ∈ (0,1) that, informally, characterizes continuous processes that are “rougher” (when H < 1/2) or “smoother” (when H > 1/2) than standard Brownian motion, which is exactly defined by fBM with H = 1/2. In this presentation, we show how one may construct an estimator for an unknown Hurst index H given a single realized path of an fBM whose values are only observed with measurement noise. The main tool to develop this estimator is a theorem considering what quadratic variation-type functionals of these processes would converge to. The proof of this theorem is the main result of this REU project and adapts previously seen techniques such as pre-averaging, truncation and discretization to our current setting. Our results also allow us to derive estimators for H in the presence of stochastic volatility.
Team: Nicholas Gaston Molina, Siddharth Mehr Mane, Erdem Baha Topbas, Zhenfeng Tu
Project Lead: Konstantin Matestski
Graduate TA: Hindy Drillick
Title: TBA
ABSTRACT: TBA
CSUREMM:
Team: Anthony Ozerov, Steven DiSilvio, Yu (Anna) Luo
Graduate Mentor: Alejandra Quintos Lima
Traders in a Strange Land: Agent-based discrete-event market simulation of the Figgie card game
ABSTRACT: Figgie is a card game that approximates open-outcry commodities trading. We design strategies for Figgie and study their performance and the resulting market behavior. To do this, we develop a flexible agent-based discrete-event market simulation in which agents operating under our strategies can play Figgie. Our simulation builds upon previous work by simulating latencies between agents and the market in a novel and efficient way. The fundamentalist strategy we develop takes advantage of Figgie’s unique notion of asset value, which allows different traders to have different but equally valid estimates of value. We find that the fundamentalist strategy is, on average, the profit-maximizing one in all combinations of strategies tested. We develop a strategy which estimates value by observing orders sent by other agents, and find that it limits the success of fundamentalists. We also find that the chartist strategies implemented, including one from the literature, fail by going into feedback loops in the small Figgie market. Positive autocorrelation in the returns in at least one strategy combination indicates that a more robust chartist strategy could be successful in Figgie. We further develop a bootstrap method for statistically comparing strategies in a zero-sum game. Our results demonstrate the wide-ranging applicability of agent-based discrete-event simulations in studying markets.
Accounting for Disruptions in the Structural Connectome in a Network Diffusion Model for Tau Spread
Team: Maria Stuebner, Silvia Toderas, Yiqi Wang
Graduate Mentor: Alejandra Quintos Lima
ABSTRACT: Recent neuropathological studies of Alzheimer’s disease suggest that aggregation of the tau protein spreads across connected neurons in an activity-dependent manner, contributing to cognitive decline. While a linear diffusion model based on quantitative pathology is widely accepted and used to predict the progression of tau pathology, all first applications of this model to tau spread fail to take into consideration the factor of neuronal death, which would cast a significant impact on several parameters influencing the spread. To allow for more accurate predictions, we are building onto the existing framework by incorporating neuronal loss. This modifies the overall connectome of the brain and in turn changes the spread dynamics.
A Generalized Multi-Agent Framework To Simulate Deviant Mining Strategies on Bitcoin-like Blockchains
Team: Ketan Jog, Andrew Magid, Oliver Li
Graduate Mentor: Joe Suk
ABSTRACT: A big challenge in evaluating the profitability of deviant mining strategies on bitcoin-like blockchain systems is the lack of a generalized framework that sufficiently describes and simulates each strategy. We target this lacuna in blockchain mining literature. Our work is twofold. After a brief description of the honest protocol, we first present a taxonomy of existing deviant mining strategies that is succinctly parameterized. We define a set of variables that can describe every mining strategy we have come across in the literature: Selfish Mining, Smart Mining, Smarter Mining, Lead-Stubborn Mining, Intermittent Selfish Mining, and MDP-based formulations. Although individual studies have been done to show the effectiveness of these selfish mining techniques, there has not been a consolidatory work that compares the relative effectiveness of each strategy. In past work, the effectiveness of deviant mining has been established using varying assumptions and frameworks, making comparison between them extremely difficult. The second part of our work aims to gener alize Agent-Blockchain interactions with a single simulation-model which models agent dynamics independently from the blockchain state. This extensible system simulates the applicability of said strategies on blockchain systems with proof-of-work consensus protocols. We analyze the profitability of each mining strategy, and discuss how this advantage changes upon adjustment of certain parameters in the blockchain protocol.
Team: Jackman Liu, Rain Wei, Ribhav Talwar
Graduate Mentor: Joe Suk
Stock Price Prediction Using CNN-LSTM-Based Reddit Sentiment Analysis
ABSTRACT: In this paper, we conduct sentiment analyses over Reddit data to predict meme stock prices, specifically the GameStop Corp. (GME) and AMC Entertainment Holdings, Inc. (AMC) stock. We used wall- streetbets subreddit data: daily discussion thread submission titles relating to the GME and AMC stock to compute the average bullish and bearish scores for each stock for each day from 2019 to July 2021. These scores were then adjusted, normalized using the Fourier transform, and directed into the LSTM and CNN-LSTM neural net work prediction models for price prediction. Both models could pre dict the overall price trends of meme stocks - GME and AMC but were less reliable around volatile price variations. Moreover, the CNN- LSTM model was generally more accurate than the LSTM model.
Team: Jonathan Socoy, Diana Gregoire, Ahkeel Timothy
Graduate Mentor: Maithreya Sitaraman
Assessing Atlantic Hurricane Damage Using Satellite Imagery and Pixel Analysis
ABSTRACT: Tropical storms plague coastal communities around the world, threatening millions of lives and causing billions of dollars in damage to infrastructure. With this upsetting fact in mind, our research focuses on the Caribbean is- lands and US coastal regions to evaluate and quantify damage through satel- lite imagery and pixel analysis. We developed computational models to evalu- ate percent changes in greenery, homes and beaches for pre- and post-hurricane events throughout coastal regions while observing associated impacts, in par- ticular from wind, surge and varying elevation. From these observations, we developed linear regression models and ultimately found that results from our models aligned with common trends observed in tropical storm damages on coastal regions. With significant improvements to our current models, these pixel analysis models on satellite imagery have the potential to most effectively and swiftly analyse damage on coastal regions most impacted by tropical storms through a more objective approach than current economic models assessing damage on coastal environments. Not only do these models have potential to quantify damage on coastal regions, they are also great sources of information to the communities in these coastal regions. And overall these communities may be better informed and make the best decisions to establish protective measures against these intense, rotating vortices of air.
Team: Andrew Jin, Thiago Otto, Dennis Cruz
Graduate Mentor: Maithreya Sitaraman
Detecting Exoplanets Using Light Curve Data from the K2 Mission: A Supervised Learning Algorithm
ABSTRACT: One method to detect an exoplanet is to examine the light curve of a star and find a transit. We present a machine-learning algorithm using this method that trains and validates on the data from the K2 mission. Our method finds clusters of boundary points near transits, then uses classification on the size of clusters to determine whether a light curve corresponds to an exoplanet. We achieved a maximum of 65.2% accuracy on training data, whose parameters were 58.6% accurate on testing data. This is significantly lower than previous machine-learning methods developed to find exoplanets from light curves. Nevertheless, we suggest improvements to this method could improve its accuracy and hope that future work will enhance its capabilities to provide additional information about planets detected.
Independent Project
Student: Anda Tenie
Faculty Supervisor: Francesco Lin
Geometry and Topology of isospectral hyperbolic 3-manifolds
ABSTRACT: The project is concerned with constructing examples to better understand the geometry and topology of isospectral hyperbolic 3-manifolds. Two manifolds are said to be isospectral if the eigenvalues of their Laplace operators (on forms) are the same. Intuitively, this corresponds to “drums” of different shapes whose frequencies are heard the same. We implemented Sunada’s general construction which gives a pair of isospectral manifolds as covering spaces of the same base. As the set of all possible volumes of hyperbolic 3-manifolds is well ordered, we ask what the smallest volume of a Sunada pair can be. Previous work of Linowitz and Voight using number theoretic methods found a volume 51.02 example. Searching the Snappy OrientableClosedCensus we were able to construct an example of a Sunada pair of volume 25.51. In a different direction, we ask how much of the topology a pair of isospectral hyperbolic 3-manifolds has to share. It is known that their first rational cohomology has to be the same. It is then natural to ask if their rational cohomology rings are isomorphic. Using the work of Suciu and Wang we implemented an algorithm that computes the cup product map on the first cohomology. We present an example where the maps have kernels of different ranks and so the rational cohomology rings are not isomorphic.
Student: Alan Du
Faculty Supervisor: Kyle Hayden
Distinguishing Surfaces Using Khovanov Homology
ABSTRACT: Previous work established a Khovanov Homology invariant of smooth, compact, orientable, properly-embedded surfaces in the 4-ball. We perform a detailed and complete analysis of maps on Khovanov Homology arising from disks bounding the knot 6_1. In particular, we prove that these maps distinguish the two slice disks of 6_1. Our proof highlights different approaches that can be potentially generalized to other applications.