2024
Students
A list of student and mentor groups. Click here to see their posters.
Aakash Gurung: "Möbius Transformation and Continued Fraction" (Mentor: Ruipeng Xu)
Adityo Mamun : "The Dictionary Between Algebra and Geometry" (Mentor: Nathaniel Kingsbury)
Alec Smith (Mentor: Emma Dinowitz)
Alex Vtorov (Mentor: Ryan Utke)
Alsu Flare: "Decoding Messages 22 Light Hours Away: The Algebra of Error Correction" (Mentor: Nathaniel Kingsbury)
Anna Borenstein (Mentor: Michael Pallante)
Colin Pratasevich: "An Introduction to Brownian Motion and Stochastic Differential Equations" (Mentor: Kurt Butler)
Edgar Cuapio-Diaz: "Bayesian Optimization" (Mentor: Dan Waxman)
Feiyi Ma & Huzaifa Azfal: "Ehrenfest Model - A Model of Markov Chain" (Mentor: Joshua Meisel)
Giovanna Ricevuto: "Gödel's Incompleteness Theorem" (Mentor: Michael Pallante)
Ishraq Mahid: "Stochastic Differential Equations" (Mentor: Kurt Butler)
Jamie Rivera Aparicio: "Planar Vector Fields" (Mentor: Ryan Utke)
Jim McCurley: "Shortest Paths and Variational Calculus" (Mentor: Santiago Cordero Misteli)
Jonathan Jaimangal: "Hamiltonian Monte Carlo" (Mentor: Dan Waxman)
Justin Salamon: "Topics in Topology" (Mentor: Ryan Utke)
Kripamoye Biswas: "Curves in Projective Geometry" (Mentor: Valeriy Sergeev)
Lauren Harrington: "Poincaré Recurrence" (Mentor: Ruipeng Xu)
Micah Gentry: "Infinite Sets" (Mentor: Michael Pallante)
Nicole Froitzheim & Samyah Ahmed: "Can You See the Values of a Quadratic Form?" (Mentor: Ajmain Yamin)
Qing Chen (Mentor: Lucy Hyde)
Randy Lu: "Optimal Transport" (Mentor: Daniel Grange)
Reilly Fortune: "Axiomatizing Arithmetic" (Mentor: Elijah Gadsby)
tahda queer (Mentor: Kurt Butler)
Vladislav Vostrikov: "Braid Groups, Word Problem, and Cryptography" (Mentor: Lucy Hyde)
Yitao Li: "Geometric Group Theory" (Mentor: Weiyan Lin)
Mentors
A list of mentors and their proposed projects.
Interests:
Machine learning, Dynamics, Statistics, Geometry
Projects:
Stochastic Differential Equations
Machine Learning with Kernels
Generative Models and Unsupervised Machine Learning
Project Descriptions
Stochastic Differential Equations
A stochastic differential equation (SDE) is like a differential equation but you add 'random noise' into the equation. Because the noise is random, every time you solve a SDE, you get a different randomly-determined solution. The mathematics that we use to solve SDEs are super interesting, and SDEs are applied in physics, machine learning, and quantitative finance to do a lot of very cool things. The goal of this project is to get a sense of what SDEs are and how we can solve them. Based on the interests of the students, we may look at one or both of these textbooks, which I feel have really good exposition on the topic.
References: Applied Stochastic Differential Equations, by Särkkä and Solin; Stochastic Calculus for Finance II: Continuous-time models, by Shreve
Prerequisites: Calculus and introductory linear algebra are required. Some familiarity with probability is greatly preferred. A course in differential equations would also be useful.
Machine Learning with Kernels
In machine learning, kernels are functions that measure the similarity between two vectors. Support vector machines, or "kernel machines," have proven themselves to be a very robust, principled and theoretically grounded way to do machine learning, and their performance is often comparable to that of artificial neural networks. The theory of kernels is basically just geometry, and this leads us to methods that are as visually motivated as they are theoretically grounded. In this project I want to discuss what kernels are, and what are some of the key ideas in how we use kernels to solve machine learning problems.
References: "Learning with kernels" by Smola and Scholkopf
Prerequisites: Calculus and introductory linear algebra are required. A basic course in set theory and proofs would be helpful for me to explain some of the details in this reading.
Generative Models and Unsupervised Machine Learning
Generative models are machine learning models that learn how to imitate or recreate data that is similar in pattern to the data used in training. For example, a generative model that is trained on a collection of dog photos will allow us to generate new photos of dogs that were not seen in the training set. Aside from being interesting in applications, generative models have a rich theory, borrowing concepts from topology, analysis and statistics. In this project, we will spend some time discussing the theory of machine learning and statistical inference, and building a bridge with the geometric ideas that are at play in the background.
References: "Geometry of Deep Learning: A Signal Processing Perspective" by Jong Chul Ye
Prerequisites: Calculus, linear algebra and probability are required. Any experience with statistical inference, machine learning or statistics is a plus.
Interests:
I'm interested in anything that is related to analysis and differential geometry. Specially in geometric analysis, minimal surfaces, PDE, calculus of variations and geometric measure theory. These fields have great applications and interactions with other fields such as general relativity and other real world phenomena modelled by PDEs. One of the most important things that we study in this fields is curvature and this is a measure that allows for very precise quantitative descriptions of shapes.
Projects:
Minimal surfaces
Calculus of Variations
(Geometric) Measure Theory
Project Descriptions
Minimal surfaces
Finding surfaces that behave nicely can tell us a lot of information about the ambient space. Minimal surfaces can be seen in nature as soap films, approximating membranes or interfaces between fluids and, more abstractly, they also model the apparent horizon of black holes. They can serve as powerful tools for proving theorems in many fields and are an active topic of research. Depending on the student's level of mathematical background, we can either focus on classical facts or more recent developments. Students may also choose to focus on more analytical aspects of minimal surfaces such as those related to PDE or more geometrical ones.
References: A course in minimal surfaces by T Colding and W Minicozzi, Geometric Relativity by Dan Lee
Prerequisites: Some knowledge and or intuition in topology, manifolds, analysis and differential equations would be ideal. At the very minimum good working knowledge of Calculus, but the student would ideally also have experience with reading and writing proofs.
Calculus of Variations
In Calculus one learns about how to find minimum and maximum values of scalar-valued functions using derivatives. While this procedure can be readily understood in finite-dimensional settings, sometimes we are interested in finding extrema in settings with an infinite number of degrees of freedom; this requires one to develop more theory. We'll define the general setting, show how to obtain Euler-Lagrange equations and apply it to finding shortest curves, quickest profiles for slides, and other classical and not so classical problems.
References: Variational Calculus and Optimal control by JL Troutman for example, but any other Calculus of variations book should do the trick.
Prerequisites: Only multivariable calculus and some linear algebra. And some high interest in proofs!
(Geometric) Measure Theory
Depending on the students interests we can start studying some basic measure theory or start further ahead. At the very least, one goal is to define what a Hausdorff measure is and to compute the Hausdorff dimensions of common fractals, as well some develop some properties of Lipschitz functions. For more advanced projects we could instead talk about sets of finite perimeter, rectifiable sets, or currents.
References: Real analysis modern techniques and their applications by Folland, Geometry of sets and measures in Euclidean Spaces by P Mattila, and Sets of finite perimeter and geometric variational problems: an introduction to geometric measure theory by F Maggi.
Prerequisites: Calculus, some geometric intuition and some familiarity with proofs.
Emma Dinowitz
Interests:
Dynamical systems, symbolic dynamics, Hausdorff dimension, hyperbolic dynamics
Project:
Hausdorff Dimension
Project Description
Hausdorff Dimension
We will read Kenneth Falconers book "Fractal Geometry: Mathematical Foundations and Applications", understand different notions of fractal dimension, their intuitive meaning and how they relate to each other. We will compute dimensions of various sets from the basic definitions and then prove theorems enabling us to compute many more examples of Hausdorff and other dimension quantities.
References: Kenneth Falconer "Fractal Geometry: Mathematical Foundations and Applications"
Prerequisites: Real analysis, measure theory, basic topology
Elijah Gadsby
Interests:
I study logic. Some of my key interests are modal logic and formal arithmetic, but I am happy to supervise a project related to any aspect of logic.
Projects:
Axiomatizing Arithmetic
Modal Logic
Project Descriptions
Axiomatizing Arithmetic
Can the process of mathematical research be automated? Using a formal system such as first-order logic, we can encode some basic axioms about addition and multiplication and then create a program that enumerates true statements about arithmetic. But would this capture everything? Could we do this in such a way that every true statement is eventually proven? Can the mathematician be removed entirely from the process? In this project, we would explore Peano arithmetic, the most well-known set of axioms for the natural numbers. Depending on prior background, we could examine deductive systems for first-order logic, positive results regarding what can be proven inside of Peano Arithmetic (which is a lot more than you would expect), or Gödel’s (in)famous first incompleteness theorem which says that fortunately for mathematicians, the answer to the questions above is no.
References: Elliot Mendelson, “Introduction to Mathematical Logic”, Chapter 3
Prerequisites: At least one proof-based mathematics course.
Modal Logic
Given a sentence, we can ask whether it is true or false, but that is not all there is to say. A true statement could be necessary, known, believed, provable in a particular axiomatic system, etc. Modal logic provides simple, yet powerful tools that enable us to analyze various notions that qualify the truth of a statement. It also has important applications in computer science. In this project, we would cover some of the key ideas and techniques of modal logic and then explore some applications to an area of interest to the student.
References: Melvin Fitting & Richard Mendelsohn, “First-Order Modal Logic.”
Prerequisites: None
Daniel Grange
Interests:
Optimal transport
Project:
Introduction to Optimal Transport
Project Description
Introduction to Optimal Transport
Optimal Transport was first formulated by French Mathematician Gaspard Monge in 1781. From the modest ambitions of moving piles of dirt efficiently, the role of optimal transport in applied mathematics has dramatically grown, with applications being found in image processing and fluid dynamics, to even understanding how the cells in our body differentiate! The book chosen is meant for applied mathematicians and enables focused theoretical discussions as well as cross-disciplinary projects.
References: F. Santambrogio, "Optimal Transport for Applied Mathematicians"
Prerequisites: Calculus and linear algebra. Familiarity with real analysis and differential equations is an optional bonus, but all levels in between are welcome.
Lucy Hyde
Interests:
My current main focus is in logic, primarily computability theory, but I have broad side interests across pure mathematics and computer science, particularly in group theory, category theory, and algorithms.
Project:
An Introduction to Nonabelian Infinite Groups
Project Description
An Introduction to Nonabelian Infinite Groups
Finite groups are a well trodden territory in introductory group theory courses, and usually when forays are made into infinite groups, only the abelian infinite groups are touched on. However, the world of nonabelian infinite groups is incredibly rich with groups connected to braids, knots, and many geometric objects. Over the course of the semester we will work through the main reference learning about infinite groups through the lens of geometry, making detours as necessary to explore groups not covered there.
References: John Meier, "Groups, Graphs and Trees: An Introduction to the Geometry of Infinite Groups", and possibly other sources to cover particular topics
Prerequisites: Basic familiarity with group theory
Nathaniel Kingsbury
Interests:
I'm mostly interested in number theory, but end up picking up all sorts of interesting math in service thereof. As of late I've been teaching myself algebraic geometry.
Projects:
Number Theory
Algebraic Geometry
Project Descriptions
Number Theory
Did you know that an odd prime is the sum of two square numbers exactly when it leaves a remainder of 1 when divided by 4? Or that the number of primes less than some big number N is roughly N/ln(N)? Have you ever wondered why that “sum the digits” trick detects divisibility by 3? Or been fascinated by the twin primes, Goldbach, or Collatz conjectures? These are all examples of questions and results from number theory -- the study of properties and patterns of the natural numbers.
Number theory uses all sorts of techniques to study these problems, from modular arithmetic (accessible to a middle schooler) to the fanciest modern tools. In this broad and flexible project, we would select a subfield to study based on your background and interests. We could discuss topics like modular arithmetic and divisibility and treat them as a training ground for reading and writing proofs -- that is, for thinking and communicating as a mathematician. Or use calculus to understand the distribution of primes. If you are comfortable with proofs and computer programming, we could study number-theoretic cryptography or modern factorization and primality testing algorithms. If you have experience with analysis, abstract algebra, or complex variables, we could study more advanced topics, such as the p-adic numbers and exotic distances on the rationals; number theory in rings other than the integers (e.g. ℤ[√2]); or the prime number theorem. Or, if you have ideas not listed here, I’d be happy to hear them!
References: Hardy and Wright, “An Introduction to the Theory of Numbers;” Bressoud, “Factorization and Primality Testing;” Shahriari, “Approximately Calculus;” Gouvêa , “p-adic Numbers;” Stewart and Tall, “Algebraic Number Theory;” Jameson, “The Prime Number Theorem”
Prerequisites: Whether you just love math, have taken two semesters each of algebra and real and complex analysis, or are somewhere in between, I have projects at your level.
Algebraic Geometry
Algebraic Geometry is the geometry of polynomials: we study spaces that arise as the solutions to polynomial equations and properties preserved by polynomial or rational maps between them. We can prove some amazing things about these spaces: in the plane, a curve of degree n and a curve of degree m will intersect in exactly m*n points, provided we count them correctly. Every cubic surface contains exactly 27 lines. “Holes” over the complex numbers influence the number of rational points.
Algebraic geometry is vast and typically only studied in grad school. This project will give you a first glimpse of the subject, tracing one of several possible paths based on your interest and background. We could follow the route I first did, playing with examples to develop an intuition for the fundamental idea of “projective space.” We could focus on working over the complex numbers and see what complex analysis tells us about curves. We could study the computational side of algebraic geometry, perhaps implementing algorithms in Python. Or, we could concentrate on commutative algebra, which underpins both algebraic geometry and algebraic number theory.
References: Cox, Little, and O'Shea, "Ideals, Varieties, and Algorithms;" Kendig, “Elementary Algebraic Geometry;” Reid, “Undergraduate Algebraic Geometry;” Fulton, “Algebraic Curves”
Prerequisites: Complete comfort reading and writing proofs, and strong background in linear algebra. Courses in abstract algebra or complex analysis are helpful, not required.
Weiyan Lin
Interests:
Geometric Group Theory
Projects:
Group and Group Actions
Introduction to Geometric Group Theory
Project Descriptions
Group and Group Actions
The beginning of the modern group theory comes from the study of the solutions of the polynomials of degree higher than 4. Many mathematicians, including Cachy, Galois, and Abel, develop the concept of the modern group theory. Later on, other mathematicians, like Klein, Lie, and Cachy, use this concept to study symmetries. This group of symmetries may seem that this object is very abstract and hard to study. But we can better understand what this group “look” like by analyzing its action. The goal for the semester is: first we will begin by some history of group theory and some basic examples of abstract groups, such as cyclic groups, permutation groups, alternating groups, dihedral groups, etc.; then, we are going to learn about the group actions, with geometric examples (more pictures). If time permits, we will finish with the one “big” theorem (Cachy’s Theorem or Sylow’s Theorem).
References: Davis S. Dummit and Richard M. Foote, “Abstract algebra”, Chapter 1,4; Oleg Bogopolski, “Introduction to Group Theory”, Chapter 1.
Prerequisites: No prerequisites, but at least one proof-based course is recommended.
Introduction to Geometric Group Theory
Group theory, by definition, can be thought as a part of abstract algebra and groups could be studied as algebraic objects, with “symbols” and “words”. However, there is another perspective for groups, which is to endow them with a metric and view them as geometric space. This new perspective allows us to naturally draw connections with other areas of mathematics, such as topology, geometry, dynamic systems, etc. More specifically, we are interested in the “large-scale” geometry, and two spaces share the same “large-scale” geometry are called quasi-isometric. One key source of such quasi-isometry comes from a “geometric” group action by the Milnor-Svac Theorem. Throughout the semester, we will cover Calay graph of groups, quasi-isometry, examples of quasi-isometry, group actions, and the Milnor-Svac Theorem. If time permits, we will cover Gromov hyperbolic groups and some examples of these.
Reference: Cornelia Druţu and Michael Kapovich, “Geometric Group Theory”, Chapter 8,11.
Prerequisites: At least one proof-based course. Some prior knowledge of basic group theory is recommended.
Jessica Liu
Interests:
Dynamical Systems and Ergodic Theory
Project:
An Introduction to Mathematical Thinking Via Number Theory
Project Description
An Introduction to Mathematical Thinking Via Number Theory
Number theory, the study of the patterns which emerge within the arithmetic of the counting numbers (1,2,3,...) has long been of fascination to mathematicians. Gauss famously said: "Mathematics is the queen of the sciences - and number theory is the queen of mathematics". Today, number theory forms the foundation of many of the cryptographic algorithms which are used to send secure messages over the internet. In this project we will study some beautiful and accessible results in number theory. Along the way, we will introduce techniques of mathematical proof as well as data visualization. If there is interest, we may implement some cryptography algorithms in python notebooks.
This project will appeal to those who consider themselves visual or creative thinkers, and I welcome participants who do not identify as a "math person".
References: An Illustrated Theory of Numbers by Martin H. Weissman.
Prerequisites: Precalculus is required. Calculus is recommended.
Joshua Meisel
Interests:
Discrete Probability
Project:
Discrete Probability and Stochastic Processes
Project Description
Discrete Probability and Stochastic Processes
The main goal would be to learn about random processes in discrete time. The most famous is the random walk, also called the drunkard's walk. In it, a particle moves left or right 1 unit every second randomly. The walk can be in multiple dimensions: for instance in 3d, you can move left, right, forwards, backwards, up or down. One astonishing result that would be great to cover is that in dimensions 1 and 2, particles almost surely return home, while in dimension 3 and above they may leave (a bird that leaves its nest won’t return home, but a chicken that can’t fly always does). Another random process for which not much is known is the bullet process: bullets are fired every second from the origin along the x-axis at random speeds. When a fast bullet catches a slow one, they disappear. It is unknown if the first bullet can survive forever, but this doesn’t stop us from thinking about the problem and performing computer simulations!
References: Discrete Probability (Undergraduate Texts in Mathematics) by Hugh Gordon, The bullet problem with discrete speeds (paper by Brittany Dygert, Christoph Kinzel, Jennifer Zhu, Matthew Junge, Annie Raymond, Erik Slivken)
Prerequisites: Some experience with probability (sample spaces, conditioning, Bayes’ rule) would be helpful, or some experience with proof-based math.
Michael Pallante
Interests:
Algebra, Dynamical Systems
Project:
Philosophy of Mathematics
Project Description
Philosophy of Mathematics
Pythagoras, Hypatia, René Descartes, Gottfried Leibniz, Bernard Bolzano, Willard Quine, Bertrand Russell, Alfred North Whitehead, Hilary Putnam, … the list goes on. These great minds made significant contributions to both the fields of philosophy and mathematics. From the preface to our text: “Philosophical conundrums pervade mathematics, from fundamental questions of mathematical ontology — What is a number? What is infinity? — to questions about the relations among truth, proof, and meaning. What is the role of figures in geometric argument? Do mathematical objects exist that we cannot construct? Can every mathematical question be solved in principle by computation? Is every truth of mathematics true for a reason? Can every mathematical truth be proved?” The author is himself both a mathematician and a philosopher, and also a former CUNY professor whose classroom I had the privilege of sitting in. I did my undergraduate degree in philosophy at Hunter College before going on to study mathematics.
Reference: Lectures on the Philosophy of Mathematics, by Joel David Hamkins
Prerequisites: Some experience with proofs.
Valeriy Sergeev
Interests:
Commutative Algebra, Algebraic Geometry, Model Theory
Projects:
A Concrete Introduction to Algebraic Curves
Computational Algebraic Geometry and Commutative Algebra
An Introduction to Algebraic Geometry
Project Descriptions
A Concrete Introduction to Algebraic Curves
Algebraic geometry is a branch of mathematics that unifies algebra, geometry, topology, and analysis. Usually it requires a prohibitive amount of mathematical machinery. However, some ideas from algebraic geometry don't require much more than high school algebra to understand. The goal of this project is to study curves of degree at most 3 using relatively simple tools.
Reference: Robert Bix, "Conics and Cubics: A Concrete Introduction to Algebraic Curves"
Prerequisites: Some Calculus, familiarity with proofs would be useful.
Computational Algebraic Geometry and Commutative Algebra
Algebraic geometry studies systems of polynomial equations. Computational algebraic geometry has emerged at the intersection of algebraic geometry and computer algebra, with the rise of computers. The goal of this project is to introduce basic ideas of algebraic geometry using computational techniques.
Reference: David A. Cox , John Little , Donal O’Shea, "Ideals, Varieties, and Algorithms
Prerequisites: A proof based linear algebra course, familiarity with ring theory is useful, but not essential.
An Introduction to Algebraic Geometry
The goal of this project is to introduce basic ideas of classical algebraic geometry. The focus will be on understanding singularities of curves and how to resolve them.
Reference: Igor R.Shafarevich, "Basic Algebraic Geometry I"
Prerequisites: Abstract algebra, a little bit of point set topology.
Connor Stewart
Interests:
Algebraic Geometry, Number Theory
Project:
Intro to Complex Dynamics
Project Description
Intro to Complex Dynamics
Complex dynamics studies the orbits of points in the complex plane under iterates of holomorphic maps. We will introduce the Julia and Fatou sets of a map, as well as the famous Mandelbrot set, and study their properties. If time permits, we will read a short expository paper showing that the exponential map is chaotic.
References: Notes by Lasse Rempe
Prerequisites: Complex analysis and a proof-based real analysis course; some topology is useful
Ryan Utke
Interests:
Algebraic and Geometric Topology, Dynamical Systems
Project:
Topics in Topology
Project Description
Topics in Topology
Topology is a branch of mathematics that deals with shapes in a 'looser' sense than geometry. For instance, even though a basketball and football are different shapes, they have the 'same topology' in the sense that the surface is one unbroken piece that wraps around, with no edges or holes. Compare this with the shape of a donut or bagel, which is clearly different. Topology is a rich subject, which can be studied from many points of view: using either continuous or discrete mathematics, using techniques from algebra, analysis, combinatorics, and geometry proper. Depending on student interests, specific topics might include: the study of knots; the concept of a manifold; hyperbolic geometry and the theory of Riemann surfaces; the fundamental group and covering spaces; different models of homology; ....
References: Prasolov, Intuitive Topology; Singer & Thorpe, Lecture Notes on Elementary Topology and Geometry; Milnor, Topology from a Differentiable Viewpoint; Bredon, Topology and Geometry
Prerequisites: None, topics can be tailored to the students level and interests.
Interests:
I work primarily in Bayesian machine learning and causality -- my corresponding mathematical interests are in probability, real analysis, and mathematical statistics.
Projects:
Measure Theory and Filtering
Gaussian Processes and Bayesian Optimization
Approximate Inference in Machine Learning
Project Descriptions
Measure Theory and Filtering
Noise ruins everything — it takes a signal that we’re interested in, and corrupts it to make analysis much more difficult. Filtering is the general task of determining a probability distribution over some “hidden” process X by observing a related signal Y. Not only is filtering one of the most important practical problems in the world of data, it’s also mathematically interesting, and a natural place to motivate and explore measure theory. If the student has not yet seen any measure theory/measure-theoretic probability, we’ll first spend considerable time with that formalism. We can jump straight to filtering if the student has seen measure theory and/or measure-theoretic probability.
Reference: Measure Theory and Filtering, Aggoun & Elliott.
Prerequisites: Real analysis. Experience with measure theory and/or measure-theoretic probability is a plus.
Gaussian Processes and Bayesian Optimization
Description: Gaussian processes are an infinite-dimensional generalization of the familiar, finite-dimensional Gaussian distribution. Interesting mathematical objects in their own right, the first part of this project would consist of an introduction to Gaussian processes and how they’re used in statistics machine learning. The second part of this project will look at an interesting application of Gaussian processes to optimization, often called “Bayesian optimization.” Depending on the student’s interests, this project can take a more applied flavor — coding Bayesian optimization and applying it to real-world problems — or a more theoretical flavor, taking a look at the corresponding “regret analysis.”
References: Bayesian Optimization, Garnett.
Prerequisites: Multivariate calculus, linear algebra, and introduction to probability or statistics.
Approximate Inference in Machine Learning
As machine learning models get larger, the cost of prediction does too. This is particularly challenging in probabilistic machine learning, where problems often involve solving an integral that’s not analytically tractable. Two particularly popular approaches to solve this are Monte Carlo methods and variational inference. Monte Carlo methods work by making random guesses and correcting them to approximate the integral in question. Variational inference works instead by approximating the integrands with functions we know how to integrate, and finding the solution “closest” to the intractable integral. In this project, we will take a broad view of this active area, and implement and compare a few methods that the student finds interesting.
References: Many are possible based on the student’s interests. We can start with Probabilistic Machine Learning: Advanced Topics by Murphy and branch out from there.
Prerequisites: Multivariate calculus, linear algebra, and introduction to probability or statistics.
Ruipeng Xu
Interests:
Dynamical Systems
Projects:
Symbolic Dynamic
The Ergodic Theorem and its Application
Vector Spaces and Linear Transformation
Project Descriptions
Symbolic Dynamics
First objective: Get familiar with basic definitions relating to shift spaces and their properties. Then we will move into some basic graph theory and learn about their association with shift spaces. We will then talk about Sofic Shifts and Entropy. Second Objective (If the student is interested): we will read a paper relating to topological pressure on compact shift spaces beyond finite type.
Reference: We will most likely read off the following reference. Lind, B. Marcus, L. Douglas, and M. Brian. An Introduction to Symbolic Dynamics and Coding. An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, 1995.
Prerequisites: Undergrad Analysis and linear algebra.
The Ergodic Theorem and its Applications
We will start by covering some basics of recurrence and ergodicity, we will learn about ergodic transformations and important theorems such as Poincare Recurrence Theorem. The ultimate goal is to at least get to the Birkhoff Ergodic Theorem and then look at its application such as characterization of ergodicity (Depending on how comfortable the student is with measure theory. We can start by covering the basics of Lebesgue measure and integration if it's needed.)
Reference: "Invitation to Ergodic Theory" by C.E. Silva
Prerequisites: Linear Algebra, Undergrad Analysis.
Vector Spaces and Linear Transformation
This project is oriented toward beginner students who have strong interests in pure math. We will begin by learning the basics of abstract vector spaces and linear transformations using our reference. Then we will learn about inner product spaces and their operators. We will learn many interesting and FUN facts about linear algebra! Which are typically not covered in an undergraduate LA course at CUNY. After reading through the book, we can look at some applications of linear algebra in other fields of mathematics.
Reference: "Linear Algebra Done Right" by S. Axler.
Prerequisites: Precalculus (a well-equipped College Algebra Course would also be sufficient).
Ajmain Yamin
Interests:
Algebra, Algebraic Geometry, Complex Analysis, Number Theory
Project:
Topology of Numbers
Project Description
Topology of Numbers
We will explore ideas in number theory through a topological viewpoint. If you like visual thinking and arithmetic, this is the project for you! Topics include Farey fractions, Farey diagram, Ford circles, continued fractions, quadratic forms, lattices, Conway's topograph, trees, modular group, hyperbolic geometry, mobius transformations, rational approximation, Pell's equation, quadratic irrationalities, Apollonian circle packings, Schmidt arrangements, etc. There are many topics in this circle of ideas which are all related to each other, and can be visualized.
Note: If you like programming, we can incorporate that into this project as well! Various coding projects can be done to visually understand certain ideas in number theory.
References: The Sensual Quadratic Form by John Conway, Topology of Numbers by Allen Hatcher, various papers of Katherine Stange
Prerequisites: Linear Algebra, knowledge of complex numbers. Helpful, but not necessary: Group Theory, Elementary Number Theory, Abstract Algebra, familiarity with computer programming
Organizers
Kurt Butler, Nathaniel Kingsbury, Vincent Martinez, Eunice Ng, Ruipeng Xu, Ajmain Yamin