Hurricanes are among the most destructive natural disasters in the world, causing immense economic damage and loss of life. Although modern forecasting methods have improved substantially, predictions beyond a few days remain challenging because of the atmosphere’s chaotic nature. In this project, I investigated whether atmospheric temperatures, polar motion data, and sunspot numbers could be combined with historical hurricane records in a multivariable machine learning framework. By incorporating these unconventional predictor variables into a time-series forecasting approach, I explored new ways to improve long-range hurricane prediction and better understand the environmental factors associated with hurricane occurrence.

Flood detection from satellite imagery is often complicated by noise, cloud cover, and changing environmental conditions. In this project, I explored whether tools from topological data analysis could improve the ability of machine learning systems to recognize flooding events. Using persistent homology and satellite observations from the SEN12-FLOOD dataset, I developed models that capture large-scale geometric features of flooded regions rather than relying solely on local image patterns. This work demonstrates how mathematical ideas from topology can provide new approaches for environmental monitoring and disaster response.

Medical screening tests save lives, but their results are often misunderstood by patients and even healthcare professionals. Because many diseases are rare, positive test results can frequently be false positives despite highly accurate screening methods. In this project, I investigated how different presentations of conditional probability and medical risk affect understanding and anxiety about screening outcomes. By studying how people interpret these probabilities, I explored ways to improve communication between healthcare providers and patients and promote more informed medical decisions.

In the summer of 2021, I got an email that would change my life. I’d been attending the UCLA math circle for almost a decade at that point, and I had a good relationship with the head of the program, Prof. Oleg Gleizer. But I was surprised when he sent me an email saying, “We may have a research opportunity at USC this Summer.” Prof. Salman Avestimehr of USC (University of Southern California) was doing research into distributed machine learning techniques, and he had reached out to Prof. Gleizer looking for mathematically advanced high school students for an eight-week apprenticeship … At the time I knew very little about the inner workings of artificial intelligence, and nothing about any of the subjects Prof. Avestimehr was researching. But I’d been interested in AI for a long while and loved seeing all the new technologies being created. This was when GPT-3 had just been released, and ChatGPT was more than a year away, so the available software was very basic by today’s standards: one AI could write classical music, another could replace the background of a video, another could generate realistic faces.

During the first visit to my friend’s house since the pandemic had started, the first thing I did was wash my hands. The first thing I noticed was the eye-catching shape of the water projecting from the faucet. While the sinks I had used for the past year emitted frothy, turbulent jets, the water in this sink fell over a flat edge and created a laminar cascade of water that appeared to take the shape of mutually orthogonal chain links. I decided I had to discover what was going on. When I got home, I scoured the internet and found some qualitative explanations, but not much more. Maybe it was because I had just taken differential geometry at school and a course on planetary-scale ocean dynamics with another educational program, or that I had worked on fluid dynamics simulations for an autonomous underwater robotics project, but I really wanted to devise a mathematical description of this phenomenon. I love understanding not just the intuition behind why things happen, but the equations as well … I present a theoretical model of gravity-driven laminar fluid chains. A stream of fluid falls under this categorization based on the distinctive shape of its mutually orthogonal sheets of fluid bounded by curved jets, which will be referred to throughout this discussion as “chain links.” If the fluid falls through specific orifice geometries under the influence of gravity, the chain will be oriented downwards. The jets that bound the sheets collide over and over again, pulled towards each other by surface tension and cascading down through the system. This paper will present both a qualitative explanation of this phenomenon and simulation of it based on a mathematical model. The figure below shows an example of one such chain link. For the purposes of this paper, the flow within the sheet and jets will both be treated as laminar. The analysis will consider the case with no motion of the fluid in directions perpendicular to the direction of flow, nor eddies, vortices, or turbulence of any other kind in the stream …

Human brains are way too complicated with billions of neurons and hundreds and even thousands of trillion connections that are still not completely understood. Because of this, studying a smaller “brain” permits one to better understand how the brain and the neural network influences the behavior of a creature. The C. elegans’ connectome is the ideal network to research because of its simplicity (only consisting of 302 neurons and the fact that it has been completely mapped out. In this project, I strive to find the most important connections within different C. elegan sub-neural networks (chemical forward, gap junction forward, chemical backward, and gap-junction backward neural networks) using the symmetrized neural sub-networks. I randomly break the symmetry and use stochastic binary simulations to approximate its dynamics. My study focuses mainly on the locomotion neural circuits of C. elegans. These neurons are categorized as forward or backward, controlling the forward and backward motion of the C. elegans, respectively. The forward and backward neural networks are further broken down into the gap-junction and chemical circuits where the gap junction circuit connections carry information that travels to and from the neurons connected (bi-directional) while the chemical synapse connections carry information that travels only in one direction (uni-directional). The locomotion circuit consists of two main functional classes of neurons called command interneurons and motor neurons. Command interneurons function as information processors as they receive input from sensory neurons (not studied) and pass on information and decisions to motor neurons or other interneurons.

One evening in the summer of 2020, well after the severity and endurance of the COVID-19 pandemic had become evident, I was having a chat with my father at the dinner table. Both STEM people, our talks often leaned towards the topic of science, especially in the realm of current events. This time, it was the matter of COVID testing that made its appearance. Testing, so essential to managing an outbreak, yet so scarce when it was needed. That night, I learned about a mostly unemployed method in the pandemic called “pooled testing,” where, rather than testing polymerase chain reaction (PCR) samples individually, multiple are combined and tested together. In theory, if a pooled sample were to test negative, it would indicate that all individual samples of that pool are also negative, meaning many tests can be saved. So why not squeeze as many samples as possible (without jeopardizing accuracy) into each pool? The issue here is that with so many samples per pool, more pools are likely to test positive, and all individuals of positive pools must be retested to identify those with the disease. This posed an interesting problem. If some balance between too many and too few individuals per pool could be found, then the amount of tests and thus resources/money saved could be greatly boosted … Over the course of my project, which spanned from late fall of 2020 to summer of 2021 (with large pauses in between), I received advice from a mentor on the process of reviewing literature, producing novel contributions in the field, and writing a paper. However, my research and derivations were performed almost exclusively from the desk in my room. But even without an extravagant lab or experiments to run, this intersection of science and mathematics and public health was enough to fully entertain me.

The main problem of our project was investigating whether or not there was an efficient 2D analog to the segment tree. Here, instead of updating and querying arbitrary ranges of a list of numbers, we want to update and query arbitrary submatrices of a matrix of numbers. When updating a submatrix, we add all numbers in the submatrix with an arbitrarily chosen constant value; when querying a submatrix, we find the minimum of all numbers in the submatrix. We wanted to see whether there was a data structure that could efficiently perform each of these operations being repeatedly done one after the other, where each update can affect future queries. The answer we found to this question is a conditional no if a specific long-standing conjecture is assumed. The conjecture is the “All-Pairs Shortest Path Conjecture”, which states that the “All-Pairs Shortest Path” problem cannot be solved in “truly subcubic” time. Since at least the 1970s, it has been known that the “All-Pairs Shortest Path” problem has equal difficulty to computing a certain operation on matrices, called “min-plus matrix multiplication”, which is similar to the standard matrix multiplication in linear algebra but where the summation is replaced with the minimum() function and the multiplication is replaced with addition. Also since the 1970s, this conjecture has remained open, and not too much progress has been made towards refuting it. The key to achieving our research result was realizing that if there was an efficient 2D segment tree, then it could perform min-plus matrix multiplication in truly subcubic time, which would refute the conjecture. Therefore, if the conjecture is true, which many believe to be the case, then an efficient 2D segment tree is impossible.

My research was motivated by my profound interest in how adolescents use social media, and TikTok in particular, which stemmed from my own personal experience with the platform and my previous scientific research in this area. I had already begun to conduct research on TikTok during my junior year, so I knew that there was very little scientific literature about this youth-dominated platform. My junior year research project was focused on the socioemotional well-being of adolescent TikTok users, and was conducted to satisfy my personal concern about how my usage might be impacting my own moods. After I completed the research in 2019, the app was even more popular, so there was more to learn about how it was being used and what impact it was having. In early 2020, I began to search for mentors that could guide me. When my school went remote due to the pandemic, my research mentor and I were already exploring potential dimensions for a study and concluded independently that we had to capture this moment while it was happening. I began data collection almost immediately, without a clear research question, because we had no idea about the scope or duration of the pandemic. I was eager to collect as much data as possible about how people were posting about Coronavirus on TikTok to see where the data would lead me and what insights it could offer. The results of the final research contribute to our burgeoning knowledge of trending content by offering a snapshot of the social media landscape during a crisis, showcasing the connection social media provides, and exposing its commercial facets.

My first real encounters with physics were in Year 9 (the United Kingdom equivalent of 8th grade). It was here that my passion for the subject was sparked, thanks to my teacher at the time: Mr. Andrew Brittain. His immense enthusiasm rubbed off on me. Two of his lessons, in particular, are imprinted in my memory. One was devoted entirely to particle physics and in the second, he discussed quantum tunnelling. Both these topics were well beyond our curriculum, but he thought that they would be of interest and serve as a valuable reminder that there was more to physics than studying changes of state or rolling a ball down a ramp. I remember being instantly captivated by the mystical world of quarks, and intrigued that it was, in fact, possible (though extremely unlikely) to run through a wall and appear on the other side! As a result of my journey into calculus, I was able to participate in the Senior Physics Challenge 2019, where I completed over 600 challenging questions in an eight-month period. This led to me being one of 41 students invited to a four-day residential at Cambridge University at the start of July. At the famous Cavendish Laboratory, Professor Mark Warner delivered a series of enthralling lectures in which he gave us a rigorous mathematical introduction to quantum mechanics. I was finally able to make sense of ideas I had run into previously like eigenvalues, eigenstates, the Schrodinger wave equation, infinite and finite potential wells, and forbidden regions. I also learnt about how the structure of atoms, and thus of the universe, is governed by the complex interplay of electrical attraction and kinetic energy of localisation. The residential prompted me to write this paper, returning to where it all began in Year 9 (when I got my first look at quantum physics) but with a more informed perspective and an in-depth calculus-based approach which lent clarity to this inherently unintuitive field. I truly relished writing this paper and hope you can draw some inspiration from it!

Climate change impacts all people living on the Earth. The El Ni�o Southern Oscillation (ENSO) is a system which influences the climate around the globe. For this reason, it would be helpful to create a procedure for predicting ENSO each year, allowing the population to understand and prepare for a potential climate in their area, months in advance. This study developed a procedure to create predictions of ENSO every year. This procedure is simple, using basic statistics and computer science to create forecasts more accurate than those currently existing. Additionally, the study helped specify the relationship between the pressure systems surrounding the Pacific and ENSO, assisting in creating stronger predictions and allowing us to better understand the phenomenon.

As I entered high school I became interested in learning about different areas of mathematics but I was also interested in the research process. It was one thing to read about mathematics, but what was inventing it like? Eventually, this path led me to the MIT PRIMES program where I was paired up with a graduate student to work on a project in combinatorics. The problem I worked on was a conjecture by Prof. Richard Stanley about q-binomial coefficients, and generalization of binomial coefficients … My mentor had me read through some related papers concerning the topic so that I got a good sense of the area and which directions people were interested in exploring.

Last year, the researcher outlined the mathematical basis of a new quantum secure direct communication (QSDC) protocol. QSDC protocols are methods of information transfer which gain security from the use of quantum mechanical effects. Due to the measurement principle, quantum communication reveals eavesdroppers with a probability arbitrarily near unity. In a world where traditional encryption is increasingly threatened by quantum computers and Shor’s Algorithm, QSDC protocols provide impregnable security to banking transfers, diplomatic wires, and general communications. In contrast to existing QSDC protocols, the researcher’s protocol does not require the use of entanglement, which can be technically difficult to create and maintain without succumbing to decoherence and collapse. Further, the researcher’s protocol can be implemented with simple optical elements, transmits information directly, and retains quantum security advantages … Recently, IBM allowed the researcher to run trials on a real quantum computer, with a success rate above 90%. Given the promising results of the project, the protocol may soon be used to protect businesses, governments, and private citizens from certain types of monitoring, espionage, and cybercrime.

During an evening quantum computing seminar at Portland State University, Professor Marek Perkowski discussed the relatively new area of quantum machine learning. As the lecture continued, I gradually realized that I had come across a field with an ideal intersection between my passions for mathematics and machine learning. Following this seminar, I started reading published works in the field which introduced quantum analogs of algorithms ranging from support vector machines to k-nearest neighbors … My proposed quantum machine learning algorithm is a novel approach which converts the learning samples into an incompletely specified Boolean function for classification problems. An incompletely specified Boolean function is a function with multiple don’t know values (values which are not covered by the training data) along with values 0 or 1 corresponding to negative and positive decisions, respectively. A complex system can have multiple incompletely specified Boolean functions to handle multiple inputs and outputs. A new Kronecker Reed-Muller (KRO) spectral transform, to convert the Boolean function into a KRO form, is used to minimize such functions (i.e. with as many zero spectral coefficients as possible). Spectral coefficients are defined as the constants corresponding to each product term in a KRO form. There are a total of 3-to-the-n KRO forms (informally proven in Section 2.2) and the proposed algorithm is able to identify the exact minimum expression out of all such forms which cover each don’t know for a function of n input variables …

My advice to high school students looking to work on a research project that combines science and mathematics is to be invested in learning a lot of new things and keep an open mind as high schoolers, there’s a lot of mathematics we are unfamiliar with, and sometimes it’s easy to get overwhelmed. However, I would say that the beauty of combined science and math research is that you learn so much about mathematical theory that can be applied to the science a double-pronged approach, if you will! It’s a fantastic opportunity, and with an open mind and willingness to learn it becomes that much better and more rewarding … This research describes how a concentrically embedded bubble alters the surface wave of a suspended viscous drop. The analysis considers small amplitudes of the interfacial pulsation so that the nonlinear convective terms can be neglected in the flow equation. The consequent linearized system is represented by a matrix formulation which predicts the natural frequencies and decay constants for different modes of oscillation. The involved matrices have a block diagonal structure separating the deformational and rotational groups. The presented work includes the mathematical derivations for both groups proving that the former manifests both oscillating and decaying features while the latter only exhibits monotonic decrease. The results reveal the variation in decay constants with bubble-drop size-ratio for different rotational modes …

When I was in sixth grade, I first became aware of my family history of heart disease. My dad battling chronic hypertension, close relatives passing away from heart failure, and me knowing that I could be next in line … During my freshman year of high school, my curiosity to investigate better heart disease therapies drove me to take honors biology. I was especially intrigued by our discussion of the incredible healing potential of pluripotent stem cells. Eager to learn more, I decided to take a summer class in biotechnology at the University of Pennsylvania, through the Summer Academy of Applied Science and Technology … As I began to code my algorithm, I found myself exploring the exciting interface between biology and vector calculus. I realized that, by thinking of a tissue contracting as a point being displaced in the cartesian coordinate system, I could make the problem of identifying contractile force magnitude and direction much simpler. Thinking about displacement immediately called to mind vectors, which I had just learned about in my junior year multivariable calculus class. As I reviewed the literature for previous attempts at modeling tissue contractions using vectors, I encountered a paper from UCSF that employed vector fields to represent tissue displacement.

It is actually impossible to explain my experience in math research without beginning with my experience in math contests. As a relatively accomplished contestant over my high school years, including participating the United States of America Junior Mathematical Olympiad (USAJMO) and twice in United States of America Mathematical Olympiad (USAMO), I fell in love with the mathematics and the often slick and beautiful solutions in these contests. However math contest can be incredibly deceiving as in most serious mathematics the necessary background knowledge can be quite cumbersome for high school students. Research in analytic number theory, my mentor once joked requires a PhD to understand. But in modern mathematics there is at least one notable exception, combinatorics.

I researched the efficiency of modern algorithms that test whether large integers are prime or not. As it turns out, this question is fundamental to modern cryptography: many modern encryption algorithms used for internet security purposes require a steady supply of large prime numbers. Although many different primality tests are used in cryptography, I focused on the strong Lucas pseudoprime test, which relies on concepts from algebraic number theory. To begin working on my project, I did over a month of background reading on algebraic number theory.

Since my freshman year of high school, when I became increasingly bored with school math, I would look up things on my own which interested me. More than that, I would play around with them. Continued fractions? Sounds interesting - now let’s try and see if I can derive a closed form if I vary the parameters this way. Sums involving the harmonic number? Let’s see if I can generalize them with another parameter. Of course, I didn’t find anything truly interesting for a very long time - I would find out that what I’d done had been done two hundred years ago, in a much simpler way. But eventually, I hit gold - and it turned into this very project. I knew absolutely nothing about number theory when I started - I was working with polynomial roots at first, and only later did I realize I was staring at functions I’d seen in number theory.

With the advent of the smartphone, cameras have suddenly become very convenient. Like many others, I enjoy taking pictures on my rather old, outdated phone. I became interested in how to take better-quality photos despite my device’s limitations, and a Google search pointed me toward high dynamic range (HDR) imaging. The process involves taking multiple photos instead of just one to produce a final image that shows a real-world scene more completely. I started researching the limits of what HDR imaging currently has to offer, and many of the methods had yet to overcome obstacles, most of which often originated common problems (e.g. camera shake). I thought it was intuitive for there to be a way to automatically correct a picture when taken, especially when the problems precluded HDR imaging from working properly, and that’s where my investigation for a solution began.

Number theory is a field that has, for me, always held a special kind of magic. There is something about reading an elegant proof that sticks with me, gives me a certain feeling of gratification like the sensation experienced at the conclusion of a mystery novel. I have always believed that the objective of any mystery is to figure it all out before the grand reveal, to make the leaps of intuition before Sherlock Holmes. It is exactly that desire to investigate that compels me to study numbers. In number theory, when proving a theorem, you start with a problem, uncover clues, try out possibilities and Eureka! you’ve constructed a solution; case closed!

About a year ago, I fascinated over a computer simulation called Conway’s Game of Life. This simulation consisted of a two-dimensional grid. Its cells could be filled or empty, which corresponded to alive and dead, respectively. This grid followed four simple rules: any live cells with fewer than two live neighbors dies, any live cell with two or three live neighbors lives to the next generation, any live cell with more than three neighbors dies, and any dead cell with exactly three neighbors becomes a live cell. Filling in the grid, the player could create patterns with different properties. What I found fascinating was not the game itself, but the properties the game exhibited.

Bioinformatics is a field that draws from mathematics, computer science, and engineering to develop biological understanding [26]. Bioinformatics uses many techniques and analyses to identify the biological mechanisms that underlie biological data. Bioinformatic analysis begins with data such as sequences of DNA, structural information about a protein, or measures of gene expression. Much of this data is available online in publicly accessible repositories. Using these repositories, researchers can apply various machine learning techniques to high-quality data without incurring the cost of generating the data themselves.

Constructing and n-dividing different polar curves through the use of straight- edge and compass constructions are proven using Field Theory. Some of the solved systems include arc length divisions for circles, hypocycloids, and lemniscates. However, little has been done in terms of n-dividing area of pre-drawn polar curves. Using theory allows for a closed determination of the possible n-divisions, since it gives a closed form for possible lengths and angles. Constructibility can be applied to the long-standing ancient Greek “unsolvable” problems, roots of unity in complex analysis, and computer science through binary digits.

That summer, I applied to and was accepted to the Wolfram Science Summer School (WSSS) WSSS2012 was hosted at Curry College in Milton, Massachusetts. At WSSS2012, I met Stephen Wolfram, members of the Wolfram Science team, and numerous computer science enthusiasts from around the world, all with unique and interesting backgrounds. It was after talking to Dr. Wolfram for the first time that I decided to study long-distance cellular automata, or LDCA, a field of cellular automata that had not been extensively documented before. I began by created a nomenclature for LDCA, and started to study their basic characteristics … Cellular automata (CA) have been utilized for decades as discrete models of physical, mathematical, chemical, and biological systems. The most common form of CA, the elementary cellular automaton (ECA), has been studied intensively in the past due to its simple form and versatility. However, ECA are constrained to evolve according to a neighborhood of adjacent cells, which limits their sampling radius and the environments in which that they can be used. The purpose of my study was to explore the behavior of one-dimensional CA in configurations other than that of ECA. Namely, long-distance cellular automata (LDCA), a construct that had been described in the past but never studied …

A year ago I investigated a mathematical problem relating to Latin squares. Most people, whether knowing it or not, have actually seen a Latin square at some point in their lives and many newspapers actually include partial Latin squares on a daily basis in the form of a sudoku puzzle. A Latin square is a grid of cells with numbers in each cell such that no number is repeated in any row or column, so any completed sudoku puzzle is really a 9x9 Latin square. Although Latin squares have been around for a while, providing entertainment in the form of puzzles to people ranging from Benjamin Franklin to high school students like me, there are actually quite a few open mathematical problems surrounding Latin squares. Latin squares have been used not only as puzzles, but also as tools to aid in eliminating bias in experimental design, and they are mathematically very interesting and have connections to areas like group theory and graph theory …

Combinatorics is a field of mathematics that has always fascinated me. Specifically, graph theory, a branch of combinatorics, has always piqued my interest. In general, graph theory deals with the study of mathematical structures, modeled by vertices with edges connecting them. While these graphs can be very simple, they can also get exceedingly complicated in structure; indeed, there are very interesting properties we can say about these graphs. The field is both enormously complex as well as incredibly enlightening … In the summer of ninth grade, I had my first experience with graph theory at a summer math camp called PROMYS. There, I researched the invariant measures of graphs under arbitrary permutations of vertices. An invariant measure is a certain quality of a graph that is preserved by any permutation of the set of vertices. In that project, the question at hand was the following: given a graph G, what methods can be used to determine if the graph has an invariant measure? Furthermore, what constructions of this invariant measure are possible? This topic is of importance in several issues relating to network connectivity. By examining the invariant measures on graphs, one can relate the network connectivity of graphs under, say, arbitrary permutations (or any other measure) and show possible relatedness between structures …

I’ve always been interested in mathematics. It is the pinnacle of human logic and is unquestionably correct, leading to wonderful models of predicting weather and making transistors. I found math to be a beautiful art form with a personality; some equations are humble, some are lawless, and some are mysterious, teasing for further inquiry … I have conducted research in representation theory, the backbone of many mathematical ideas in algebra, topology, and particle physics. A major part of this field is the interplay between symmetries and the algebraic objects which control them. In the 1980’s, Charles Dunkl introduced certain operations involving both derivatives (rates of change) and certain reflections naturally associated to the symmetry of ordinary Euclidean space. These Dunkl operators have proven useful in both physics and mathematics, where they are used to study quantum many-body problems, conformal field theory, Lie theory, and harmonic analysis. In my project, I studied new Dunkl-type operators better adapted to a type of noncommutative space, which is a space in which the multiplication of quantities does not satisfy the familiar relation ab = ba …

It all started with social media. Like many Facebook fans of my age, a significant part of my life was spent on social media. As we take knowledge from the infinite pool of cyberspace, cyberspace, in return, instilled appalling social habits, and my social interactions simply became competitions of who can glue eyes to their screen the most. Consequently, for me (and my 819 friends), my speech patterns rescinded to a level akin to OMG LOL I have to get to class. I lived in social media, knowing it inside and out … In sophomore year, I had an opportunity to put my social media expertise to some use as a paid summer intern at a Silicon Valley startup that automatically tracks public opinions and sentiments from social media. Their system uses natural language technology to do sentiment analysis of consumer opinions about a brand or topic … My initial job was to incorporate social media jargon into the system, especially the emotional expressions from Urban Dictionary. I was also assigned to test entries from Facebook fan pages, sorting positive sentiment from negative. I soon immersed myself into my work routine but noticed that the system always disregarded smiley faces (emoticons) as these are things beyond words, extra-linguistic symbols. As visible representations of emotion, isn’t that a missed opportunity to help gather sentiment? A happy face like :) usually denotes a positive tone of sentiment while a sad face :( a negative tone. Intuitively, it should help the system for the purpose of sentiment analysis … This research presents a novel study of how emoticons can help sentiment analysis precision. Data analysis shows that emoticons alone cannot determine sentiments towards a brand and they can only be used together with other evidence. Further study has discovered a use of emoticons as counter evidence to block glaring errors in sentiment analysis …

The title of my paper was “The Sky’s the Limit- An Investigation of Cloud Cover on Major League Baseball Performance.” My research project was inspired by a genuine passion for the game of baseball and my desire to learn more about its subtle nuances. I often wondered how much weather variables such as sun, clouds and shadows affected the outcome of a game or individual player performance. My curiosity prompted me to do some preliminary research to identify whether these questions were previously investigated. This served as the impetus for my project and interestingly the results only led me to formulate more questions … The focus of my project was to investigate whether different percentages of cloud cover during a baseball game favored batters or pitchers. Additionally, what effect cloud cover percentages had on fielders during a Major League Baseball game? The data collected for this investigation was obtained from the years 2007 to 2010. Baseball data gathered during day conditions were also compared to baseball data gathered during night conditions to see if any significant differences existed. Baseball data was collected from www.baseball-reference.com and weather data was gathered from the National Climatic Data Center. Seven variables were compared across three categories of cloud cover, 0-29%, 30-79% and 80-100% cloud cover during day games and night games, which served as the control for my study. The overall results suggest that clearer conditions tend to favor the pitcher in each day game while cloudier day games tend to favor the batter.

To predict the response of estuarine ecosystems to anthropogenic and natural changes, process-based physical computer models serve as an important tool for simulation of estuarine salinity. Among the school of data-driven parametric models as alternative tools for process-based physical models to simulate environmental variables, artificial neural networks (ANNs) have become an increasingly popular modeling technique over the past two decades (Maren et al., 1990; Schalkoff, 1997; Dawson and Wilby, 2001; Maier and Dandy, 2001; Dawson et al., 2005; Pao, 2008). ANNs is a programming logic model using multivariable calculus and an algorithmic learning process to simulate various functions related with information processing, including pattern recognition, forecasting, and data compression. The logic of ANNs aims to imitate the workings of individual neurons in the human brain, making it able to dynamically model non-linear functions with very high accuracy. In this way, a modeler using ANNs has no need to explore the intermediate processes that occur in the relationship between an input variable and the final output. Instead, the ANN implicitly takes them into account during its learning process. Transport of salt in estuaries is influenced by multiple factors such as freshwater inflows and tide, and their relationship with salinity is highly complex and non-linear, making it ideal cases for the application of ANNs. The objective of this study is to develop ANNs to predict estuarine salinity using the Loxahatchee River as a case study. The Loxahatchee River is selected because of concerns about saltwater intrusion into the river (SFWMD, 2002; 2006; Kaplan et al., 2010; Liu et al., 2011). The hypothesis is that salinity in the Loxahatchee River can be effectively simulated with ANNs, through properly training and testing, using freshwater inflow, rainfall, and tide as inputs.

Oftentimes, there are “bad guys” such as diseases, wildfires, or thieves that the “good guys” such as the CDC, firemen, or police wish to control or capture. However, the good guys only have a limited quantity of assets such as money, people, and time, so it is important for them to use the least amount of resources. One well-known way of analyzing such problems is known as “Cops and Robbers on a Graph”. I analyzed a different version of this model to find out how to minimize the cost (called the Workday Number) to catch the bad guys. I discovered how to compute a way to catch the bad guys in two days while still minimizing the cost … How did I come up with my research topic? During the proof-based power round of the national American Regions Mathematics League Competition (ARML), there was one problem which introduced and asked questions about the Workday Number. In answering it, I realized that I could combine the idea of flow networks from computer science with monovariants from my math experience to give bounds on the Workday number. Unknown to me at the time, the panel of judges, all mathematicians, had deliberated for over thirty minutes over the correctness of my solution. Although it appeared correct, and they could not find any holes in it, it simply did not match any of the official proofs that they had.

I am fascinated by problems that require a blend of computational topology, geometry, and number theory. I have also been studying fractals which interesting geometrical objects that have been used in diverse applications such as cryptography, seismology, network optimization, and even weather forecasting. However, despite the wide range of applications and interest in fractals, the general theory of these objects is still in its infancy. My work on this research project has developed some theorems and conjectures in the field of combinatorics and has begun to shed some light on some areas of fractals, one-cell automata and dynamical systems … This need for optimization has become increasingly more important in today’s society from the perspective of both resource management as well as leveraging new opportunities. In terms of resource allocation, combinatoric optimization is being used to improve the efficiency of scheduling transportation (the traveling salesperson problem) to allocating scarce resources (such as militaryequipment or food distribution), through improving internet network traffic throughput, latency, and infrastructure costs. But the field has broader impact than just efficient resource allocation as it can more help in more efficiently processing large amounts of data. Increasingly, we are producing more information that we can efficiently sort through and understand, whether it is the 100k plus tweets per minute of the Presidential debates, the information gathered about global warming, or the data mining of consumer information.

I began to appreciate such simplicity, and to redefine my understanding of mathematics. I came to see it as being much more than just its constituent symbols and equations, but a beautiful language capable of describing the logical foundations of all the natural sciences. Over time, that same beauty began to appear everywhere I looked … Aortic stenosis (AS) is a lethal disease that can lead to severe cardiac complications if left untreated. A new type of non-invasive treatment for AS, transcatheter aortic-valve replacement (TAVR), exhibits comparable success rates in comparison with conventional surgical aortic valve replacement. Nevertheless, it also demonstrates significantly greater rates of paravalvular regurgitation, a serious complication associated with increased rates of later mortality. In this study, we achieve three main objectives. First, we design a computer program for automatic 2-dimensional measurement of the aortic annulus that is statistically non-inferior to radiologists’ manual measurements. Secondly, we use these measurements in addition to the Agatston calcium score to identify significant predictor variables of paravalvular regurgitation. At a significance level of 0.05, the predictor variables were identified to be aortic valve calcification and prosthesis mis-sizing. Lastly, we use these predictor variables to construct a multivariate Bayesian model that predicts the incidence of moderate post-TAVR paravalvular aortic regurgitation with 70% accuracy, highlighting its potential for clinical use in recommending patients to the appropriate AS treatment. In light of the fact that 50% of medically treated AS patients die within two years of onset of symptoms and as many as 30% of these patients cannot undergo surgery, TAVR is a life-saving procedure that has the potential to positively impact many patients’ lives. Since TAVR cannot be conducted safely without prior assessment of risk, the proposed risk-stratification model reflects a significant advancement in AS patient care.

I am especially interested in Colony Collapse Disorder (CCD) as an environmental issue as well as an economic and policy issue. Within the United States honeybees contribute to the success of one-third of U.S. agriculture; furthermore these insects are responsible for countless jobs and many billions of dollars in revenue. They are the unsung heroes of harvests of numerous fruits, berries, and nuts, and therefore crucial to the long-term viability of our global economy. I have taken a special liking to these insects and, through my research; I hoped to find creative ways to enable the species to survive the current challenges to their existence. During the summer of 2010, I had a unique opportunity to conduct original research at Michigan State University in the entomology lab of Dr. Zachary Y. Huang through the High School Honors Science Program. My work focused on the impact of time on the duration of honeybees’ memory in relay learning. My field research required working in close proximity to thousands of bees to investigate degradation of memory as a possible cause of CCD . . . My advice to students who are undertaking a project combining mathematics and science would be to never give up on your initial goals. Although you may have to tweak your methodology, you should never give up on answering your initial questions. My research encountered several stumbling blocks related to replicating conditions across trials, which I was able to overcome through perseverance and enthusiasm. The honeybees were not returning to the hives, and after careful observation, I realized it was due to the placement of the hives.

In order to build my own guitar I clearly needed to research how I would go about the task. Thus began, unbeknownst to me, my first true application of the scientific method that would eventually inspire me to become an Intel participant. I researched multiple designs in order to create something that felt both original and practical and then I made detailed lists of the necessary supplies. I collected materials from around my house, online, and occasionally even the garbage can. I scrounged everywhere for the parts, all the while keeping within a tight budget. Sometimes I even had to create parts rather than purchase them. With the help of my own dad, I learned to create a functioning circuit, wire the pick-ups, solder the wires to the pots and capacitors, and hook up all the electronics to the input . . . I spent countless hours in my garage toiling in the heat. Seven in the morning and I’d be there, surrounded by a fog of paint fumes, diligently constructing amidst my clutter of scattered materials. But that was just it. It was my clutter, my mess, which would eventually transform into my guitar . . . Thus, in order to conduct my research project I needed to master statistics. Encouraged by the research I had conducted on guitar models and designs, and the knowledge that I had asked my dad to show me how to create a functioning circuit, I sought help from the mathematics department in my school. I scoured the library and pestered my math teachers until I felt I had a firm understanding of the basics of statistics and modeling. I discovered a completely unfamiliar branch of mathematics and I learned it with gusto.

In 1997, The New York Times quoted a Bronx High School of Science administrator regarding the then- surprising increase in behavioral science honorees in the prestigious Westinghouse (now Intel) Science Talent Search competition: ‘It [behavioral science] does provide another outlet for some students whose strength may not be in empirical science and math,’ said the chairwoman of the school’s biology department (A Fine Hour For Squishy Sciences, NYT 2/16/97). Oh really? I thought my diverse fifteen hundred subject sample population and multiple analysis of covariance-based statistical analyses were pretty darned scientific. Silly me. I was also impressed with the way my friend at a local school developed and piloted her own instruments in two languages, using factor analysis and Cronbach’s alpha. A professor in California has asked permission to use the instruments in her own future research among Latino/a adolescents. It’s been fifteen years since the Times article, which otherwise offered a very positive review of all the wonderful work being done by young social scientists. Hopefully, we’re all past this squishy science / hard science nonsense. At heart, I’m a physicist. Nothing I’ve read in the last three years has excited me more than last week’s ‘Higgs boson’ discovery, but at 16 or 17, I lacked both the expertise and the opportunity to talk my way into an internship in Geneva. Social science represented an important area where I could both apply and develop my skills to an important project one that could actually make a tangible difference to girls not much younger than myself . . . Has the movement to reform middle grade education had unexpected social consequences for preteen girls struggling with self-esteem, body and weight issues? This study examined body-consciousness and sociocultural appearance attitudes among 1537 girls in seven different towns with three different grade groupings: Middle School (K-5/6-8), Modified Middle School (K-4/5-8); and Junior High (K-6/7-8). Do grade groupings within school districts affect the age at which girls become body-conscious? Do girls in Middle School districts report worse body image than those in Junior High districts?

Autism is a mental disorder that impairs the mental and social development of children on their way to adulthood. Not everyone with autism has the same severity of symptoms and therefore researchers refer to the variance of the disorder as autism spectrum disorders (ASDs). In recent years, there has been an increase in children diagnosed with autism (Groom, 2009). Reasons for such a peak in diagnoses range from a vaccine link to simply just more accurate methods of testing (Downs, 2009). No matter the cause, children with ASDs need assistance in progressing as individuals throughout life. . . The topic of autism is very personal to me. Due to the fact that my brother has autism, I have always been intrigued by the progress he has made with behavioral intervention. I want to help others with ASDs communicate and express their feelings just like any regular person has the luxury of doing. By increasing positive behavior in children with autism, they would gain the ability to socialize with normal peers and enjoy the same experiences a normal functioning child goes through. Numerous types of interventions have been implemented to aid kids with autism. These interventions span various settings and conditions, which creates a sense of spontaneity that these kids would otherwise lack. Decreasing bad behavior in kids with autism during school hours allows teachers to maximize the children’s potential . . . There is a great demand for successful interventions in the realm of behavioral intervention for children with autism. Much attention has been paid to behavioral interventions such as applied behavioral analysis (ABA) and the Lovaas method, but such a strict, rigorous method can be very hard on the parents. The treatment in this experiment is known as a social story technique, and it can be a lot less time-intensive and therefore a very useful tool for parents if it is effective.

How many times does a child hear his or her parent say, turn your phone off before going to bed or don’t sleep with your phone on next to you or stop texting at night because you won’t get enough sleep? I know I’ve heard those words countless times. But, I’ve always wondered if using my phone, or any other technological device, could actually hinder me from getting the best quality sleep I can get. This is what led me to develop my present study for the Intel Science Talent Search competition. I was curious as to whether or not there was truth to what my parents had been saying to me for all these years . . . One word of advice for those who are interested in undertaking a project combining science and mathematics would be to choose a topic of interest. What kept me going and helped me to focus on my task was the fact that I was eager to find out if my hypotheses were true. I was genuinely interested in my topic. I think that having a firm interest in one’s research is the best way to ensure success.

It is difficult to say when my passion for mathematics was first kindled. I have liked doing number problems and logic puzzles for as long as I can recall: one of my earliest memories is of using toothpicks to guide a brave mouse across shark-infested waters to steal the king’s cheese (a challenge found in The Puzzle Book, which I owned when I was younger). My interest was further strengthened by participation in math competitions and a math club organized by Professor Ron Ji at IUPUI. However, soon after entering high school, I began to feel that I would like to go beyond solving contest problems and engage in the creative process of mathematical discovery . . . Solutions to the Yang-Baxter equation, an important equation in mathematics and physics, lead to matrix representations of a collection of all braids known as the braid group. Such representations have applications in fields such as knot theory, statistical mechanics, and, most recently, quantum information science. In particular, representations with a special property called unitarity are desired because they generate braiding quantum gates. These quantum analogs of classical gates are actively studied in the ongoing quest to build a topological quantum computer that could be exponentially more powerful than our computers today. A generalized form of the Yang-Baxter equation was proposed a few years ago by Eric Rowell et al. By solving the generalized Yang-Baxter equation, we found new unitary braid group representations. Our representations give rise to braiding quantum gates and thus have the potential to aid in the construction of useful quantum computers.

I have long been fascinated by math, and more recently by biology. When my high school presented the opportunity to participate in research at a local university two years ago, I looked for a project that could help me see how the mathematics I learned in the classroom could be applied to help us better understand questions in biology . . . My advisor found Dr. David Eisenberg’s lab at the Molecular Biology Institute at UCLA, a lab studying, among other things, amyloid fibers and Alzheimer’s disease. I was introduced to Dr. James Stroud, who had developed a method applying Bayes’ theorem to data of amyloid fiber formation. With James’ help, I began working on writing code to create realistic simulations to test and refine the method. By testing the method and improving it where possible, we created a method that can be used to analyze real data . . When I began working in the lab, I knew next to nothing about Bayes’ theorem or amyloid fibers. Diving into the project meant learning things from an area of mathematics completely foreign to me, and simultaneously attempting to apply it to the real world. With a lot of help from my mentor, I came to understand and even contribute to the project.

Despite the changes in custom and language that occurred simultaneously in my life as I moved from South Korea to America, math was the one thing that made me feel secure and confident. Math provided me with a sense of direction through the confusion and cultural difference I had to overcome. Therefore, I began to devote my time to mathematics, taking advantage of all the research opportunities that were available in the field . . . I had not initially held an interest in astrophysics, but this project helped me develop a new inquisitiveness. This project introduced me to a realm of science that captured my attention from the start. It is a type of science that cannot be physically observed but has tremendous effects on life on Earth: nuclear physics.

I wanted to work on something related to game theory. During my sophomore and junior years, I had bounced back and forth between various math concepts, but I always came back to game theory because it can describe interpersonal interactions in mathematical terms, an idea that was very intriguing to me. However, I looked for something beyond game theory’s most common applications, namely in economics, social psychology, and evolutionary biology. While searching for this new application of game theory, I noticed that cells, especially cancer cells, can behave strategically. The development of a malignant tumor requires the emergence of more aggressive subclones of cells. I imagined that during the development of malignancy, there must be some form of competition4 and cooperation5 among the tumor cells. Each individual cell can be a player with a strategy determined by its phenotype. With these thoughts in mind, I began researching the possibility of applying game theory to cancer.

I love working on conjectures. Just as in the various Mathematical Olympi- ads in which I have participated, the conditions are already set; the challenge consists in cleverly using the hypotheses of the problem to produce the conjectured conclusion. Number theory, the study of properties of the ordinary counting numbers 1; 2; 3; : : :, is particularly rich in this type of problems, which range from puzzles for the general audience to the challenges on the Interna- tional Mathematical Olympiad to famous conjectures, such as Fermat’s Last Theorem and the Twin Prime Conjecture, which commonly remain unsolved for hundreds of years. My instincts told me that Conrad’s problem would not be one of these enduring conjectures, and eagerly I set to work.

The summer after my junior year, I went to the Research Science Institute (RSI) program at MIT. I had a blast there, and I strongly encourage any eligible students reading this to apply. I had two mentors: a graduate student named Dustin Clausen and a professor named Pavel Etingof. My mentors contacted me before the program to tell me about a potential project on representation theory in complex rank, following a paper of Deligne that laid the groundwork and beginning work on a program that Etingof himself had proposed in a talk at the Newton Institute. There were a few obstacles. First, Deligne writes in French. It’s a good thing that I take the language in school, but I’m not terribly fluent. Fortunately, mathematicians tend not to use diffcult words; most of the technical math jargon consists of cognates anyway. Recognizing “categorie” as “category” does not require translator-level skills. A more serious diffculty was that Deligne’s paper is hard. Academic math papers in general have a tendency to focus on correctness over understandability (the word “trivial” is used very differently by research mathematicians and other people, for instance). Deligne’s paper also heavily uses the language of category theory, a branch of mathematics whose dryness has earned it the nickname “abstract nonsense” among mathematicians…

Mathematics has always been important to me. When I was little, I liked to do mathematical puzzles out of a book I had bought from a catalog. I would think and ponder about these puzzles and discuss them with my family and friends. The answers were not straightforward, and I found the solutions interesting to read. Doing the puzzles, I discovered that mathematics is more than just a set of drills to memorize. It is a subject full of interesting, clever ideas . . . Gradually, I became more interested in proof-based mathematics, so I pursued it further in a research internship at CSULB. My mentor, Dr. Wen-Qing Xu, had published several articles in the field of error-correction coding theory. Since that field of mathematics does not require the use of calculus and other collegiate mathematical preparations, but it does involve serious mathematical arguments, so I decided to pursue research in this area. I studied the separation of symbols on two-dimensional arrays, and came up with formulas for the maximum separating distance in various cases. I wrote proofs of my results, and spent many, many days discussing them with my mentor and revising them over and over again. It was intense, grueling work, but in the end it paid off…

I hit numerous impasses and sometimes spent hours at my local university’s library thinking and looking for theorems and tools. For example, when I started to parameterize my space curve, I ran into the problem that one of Wang et al.’s theorem that was crucial in parameterization failed for my intersection space curve projection. I was stuck, but I knew I could adapt it some way. I read ahead. I tried multiple ways of attacking this problem. Many times I failed. I didn’t give up. I just went back to my notes, and looked for new ideas that I had written down. Weeks later, while I was reading the parameterization section in Wang et al.’s paper, I suddenly realized that if I lowered the degree of the space curve projection, I could successfully adapt the failed theorem . . . Eventually I was able to derive the parameterization for a special case of the intersection curve.

My advice to other high school students who wish to pursue math and science is to never feel intimidated. Problem solving techniques in engineering and science involve application of mathematics spanning various facets, such as statistics to calculus to abstract algebra and topology. When it becomes necessary to learn and apply them, do not feel intimidated by their complexity; it may be hard to understand some of these concepts at first, but it is important to be persistent. A sudden epiphany is all it takes for it to click, and it can occur at any time, as long as you have been doing all you can to expand your boundaries and learn all there is to learn about the fundamental concepts…..

Math can be an intimidating field. To work on some problems, one must know decades or centuries of background before one can even understand the question. However, what tends to get lost in all of that is that math can be fun, even for the relatively uninitiated. There are problems in mathematics that are discrete (essentially, self-contained) and with some combination of background research, mathematical thought, and appropriate mentoring, they are easily within reach of the high school student. Generally, random walks on graphs are approximated by computing the expected hitting time, or probable number of random moves required to go from one vertex to another. Although random walks are useful in mathematics and computer science, probabilistic systems do not offer sufficient precision for some applications. There are, however, several emerging methods of deterministically simulating random walks which can be used to more efficiently compute hitting times [4, 6]. One such deterministic simulation uses a process known as chip-firing.

For nearly an entire week, I spent many hours a day in MIT’s Hayden Library, immersing myself in introductory books on number theory and abstract algebra. It was a near-overwhelming amount of new notations, concepts, and ideas to absorb in such a short period of time, but I enjoyed the challenge and enthusiastically threw myself into mastering the mathematics necessary to solve my problem. In that short period of time, I learned the basics of group theory, radix representations, and residue classes, just to name a few…..

I was no longer at a junction where I could ask my parents for the answers, nor indeed any being living or dead…

I have been interested in mathematics since kindergarten. I always loved searching for and finding patterns in sequences of numbers. I used to read whatever information I could in order to learn more about the wonders of math. In fifth grade I was part of my school’s program for gifted math students. There the teacher introduced me to a wonderful sequence of numbers called the Fibonacci sequence, named after the 13th century mathematician Leonardo da Pisa Fibonacci. She gave me a list of the first 80 or so numbers in the sequence. My first impression was an amazement at how large these numbers seemed to get, and so quickly. She showed me how the sequence was formed: each term was the sum of the two previous terms beginning with 1, 1, 2, 3, 5, 8, …. She showed me how the sequence showed up in the turns of a pinecone and a sunflower

In the summer of 2002, my family and I traveled to St. Petersburg, Russia to visit my grandparents. While there, my father told me about an intriguing theorem in geometry that he himself learned in 1972 from Dr. Zalman A. Skopets (1917-1984) who was married to my grandmother’s best friend. Dr. Skopets was a prominent geometer and headed the Department of Geometry at the Yaroslavl’ State Pedagogical University for many years. In the summers, my father would visit Dr. Skopets who would give him challenging problems in geometry. One such problem appealed to him so much that he remembered it even on that summer day in St. Petersburg some 30 years later. Here is that remarkable problem:

Prior to my Intel research project, I had tried to get a research project with several local universities. However, all of my attempts collapsed for a variety of reasons. To be honest, I had given up on trying to find research opportunities when I received a letter of acceptance from the Research Science Institute (RSI). Sponsored by the Center for Excellence for Education (CEE), RSI accepts about 50 high school students every year, typically students who have finished their junior year. At RSI, these 50 US students plus about 25 international students conduct research with faculty at Harvard, MIT, and other Boston-area institutions with topics ranging from carbon nanotubes to dark matter to Z-DNA to coding for peer-to-peer networks to graph theory.

Intel Science Talent Search project was a four year labor of love that tested my patience more than any undertaking that I had ever attempted in my life. Once, after a particularly frustrating day in the lab, I told my mentor how upset I was about spending the previous three hours wiring a cryogenic thermometer only to have it break as it was inserted into the liquid helium Dewar. ìSounds like youíre doing research,î he replied. This incident was emblematic of my entire research project. For almost two years, I researched low temperature superconductors before realizing that I would not be able to conduct a valid experiment, and I was forced to change my topic in the middle of my junior year. I spent a hectic final nine months collecting data and writing the paper. Looking back, I realize that my project taught me two very important lessons: one, the power of perseverance and determination and; two, how not to conduct a research project.