Applications to become a CUNY DRP student for the Spring 2025 term are closed.
Make sure to read the FAQ before applying.
In the application, you'll be asked to rank your top three mentors/projects. You will also need to answer the following short-answer questions:
What math courses have you taken? What is your spring course load?
Have you engaged with mathematics outside of coursework? (It's okay if not!)
Is there a project that you would like to propose outside of the current list of projects?
Why are you interested in the directed reading program?
What do you like about math? Do you have a favorite idea or topic?
(*) = requires calculus or less
(**) = requires linear algebra (possibly proof-based) or a proofs course or less
Low-dimensional topology, hyperbolic geometry
The Game of Cops and Robbers on Graphs*
Geometry in Groups*
Discrete Morse Theory*
The Game of Cops and Robbers on Graphs
The game of Cops and Robbers and its associated graph parameter, the cop number, have been the subject of study in graph theory for many years and have recently seen an increased interest due to a variety of new open problems. The game is played by placing n cops and one robber at different vertices of a given graph, and alternately letting cops and the robber move by one step. The cops win if one of them is able to land on the same vertex as the robber in finitely many steps, and the robber wins if he is able to evade the cops forever. The cop number of the graph is the minimum number of cops required to guarantee a cop-win. In this project we will learn about the techniques used to compute the cop number of a graph, on different variations of the game. We will play around with different versions of the game and try find algorithms for cop-win situations.
Reference: "An Invitation to Pursuit-Evasion Games and Graph Theory" by Anthony Bonato
Prerequisites: None
Geometry in Groups
A group is an algebraic object that in some sense captures the symmetries of a shape. The algebraic properties of the group can be studied by considering the geometry of the shape. On the other hand, the group itself can be thought of as a geometric space. Geometric group theory is a field of mathematics that uses techniques from geometry to study algebraic properties. In this project we will learn about groups and group actions, some hyperbolic geometry and the large-scale geometry of spaces.
References:
"From Groups to Geometry and Back" by Vaughn Climenhaga
"Office Hours with a Geometric Group Theorist" by Matt Clay, Dan Margalit
Prerequisites: None
Discrete Morse Theory
Morse theory studies the shape of a space using real-valued functions on the space. Discrete Morse theory combines ideas from Morse theory and topology with combinatorics and has applications in computer science, topological data analysis, data compression and noise removal. In this project we will learn how simple shapes can be built using real-valued functions on them, and how the shape of a dataset can be used to analyze and study the sample.
Reference: "Discrete Morse Theory" by Nicholas Scoville
Prerequisites: Point-set topology (optional)
My research interests broadly lie in the area of low dimensional topology and geometric group theory.
Introduction to Hyperbolic Geometry
Basics of Knot Theory
Introduction to Hyperbolic geometry
Have you ever wondered why trees branch out in such intricate patterns, or why coral reefs exhibit folded surfaces? Have you spent time contemplating the mesmerizing designs in M.C. Escher's paintings? The answer to these fascinating phenomena lies in hyperbolic geometry—a geometry that emerges from the negation of Euclid's fifth postulate.
In this project, we will delve into various aspects of hyperbolic geometry, including: different models of Hyperbolic Geometry, measuring distances on these models, polygons in hyperbolic space, and many more such topics. By examining different models and their properties, we will gain a deeper understanding of how this fascinating geometry informs not only mathematics but also the very structures we see in nature.
References:
"Hyperbolic geometry" by James W Anderson
"Geometry of surfaces" by John Stillwell
"Notes on hyperbolic geometry" by Caroline Series
Prerequisites: Calculus sequence, point set topology, linear algebra.
Basics of knot theory
From the intricate structure of DNA to the simple shoelace, knots are an omnipresent feature in both nature and everyday life. In this project, we will delve into the fascinating realm of knot theory, exploring its mathematical underpinnings and practical applications.We will examine various knot invariants that help classify knots and also learn about prime knots, which cannot be represented as the knot sum of two non-trivial knots. We will also investigate combinatorial approaches to knot theory. We will also take a look at a few magic tricks that utilize knot concepts, showcasing the surprising connections between mathematics and entertainment. This project will provide a comprehensive overview of knot theory, emphasizing both its mathematical significance and its presence in the world around us.
References:
"Intuitive topology" by V. V. Prasalov
"Interactive introduction to knot theory" by Allison K. Henrich and Inga Johnson
"Knot theory" by Colin Adam
Prerequisites: Calculus sequence, point set topology, linear algebra.
I'm interested in anything that is related to analysis and differential geometry, specifically 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, which is a measure that allows for very precise quantitative descriptions of shapes.
Minimal surfaces
Semi-Riemannian Geometry
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.
Semi-Riemannian Geometry
Riemannian geometry studies the shape of a huge range of objects by means of varying the metric from point to point. This metric is a definite positive quadratic forms and it is what allows us to measure angles and lengths in not necessarily symmetric shapes. Semi-Riemannian geometry generalizes this concept further by allowing the quadratic form to have other signatures (which means that now vectors can have zero or negative length(!).
The most studied case is the one of Lorentzian geometry which is the base for modern general relativity (where time is associated with a negative direction and space with positive ones).
Reference: "Semi-Riemannian Geometry" by Barrett O'Neill
Prerequisites: Some Differential Geometry notions, linear algebra and calculus, but we can review stuff as needed.
I'm a logician. My main interests are formal arithmetic, proof theory, and modal logic, but I'm happy to supervise a project related to any aspect of logic.
Axiomatizing Arithmetic**
Algorithms and Undecidability**
Axiomatizing Arithmetic
Are all true mathematical statements provable? In a formal proof system, we precisely specify a language, axioms and inference rules in such a way that a computer could verify that our proofs are correct. Can we construct such a system for arithmetic so that all (and only) true statements about the natural numbers are provable. Gödel’s (in)famous incompleteness theorem says that the answer to this question is no. No matter how we try, any such system is necessarily incomplete.
In this project, we would examine explore first-order logic, proof systems, and Peano Arithmetic (PA), the standard axioms for the natural numbers. We would some positive results about what can be proven in PA (which is a lot more than you would expect), and (depending on prior background) prove Gödel’s incompleteness theorem.
Reference: "Logic and Structure" by Dirk van Dalen
Prerequisites: At least one proof based course
Algorithms and Undecidability
We all have an intuitive idea of an algorithm, a procedure that can be carried out in a finite number of steps according to specific rules or instructions. Such effective procedures are encountered frequently in mathematics, from the Euclidean algorithm for computing quotients and remainders, to algorithms for determining if a graph has a Hamiltonian cycle. It is possible to give a precise definition of an algorithm or effective procedure so that it becomes a genuine mathematical concept. With this in hand, it turns that some problems, such as the halting problem, are not algorithmically decidable. Beyond just decidability in principle, there are questions of complexity and efficiency. What sort of procedures can we feasibly perform given constraints on computation time and memory space. This project would cover the basics of computability and complexity theory.
Reference: "Computability Theory" by Herbert Enderton
Prerequisites: At least one proof based course
Hyperbolic geometry and geometry of Lie groups, Teichmüller theory and complex analysis, dynamical systems and ergodic theory (differentiable dynamics, homogeneous dynamics, dynamics on moduli spaces and dynamics of group actions, relationships to other fields), low-dimensional topology (2 & 3 dim manifolds, related to Thurston’s Geometrization), geometric group theory (large scale geometry of metric spaces, mapping class groups, random walks on groups)
Hyperbolic Geometry and Many Friends: Topology, Dynamics, Complex Analysis, Group Theory…
Hyperbolic Geometry and Many Friends: Topology, Dynamics, Complex Analysis, Group Theory…
Hyperbolic geometry is a special non-Euclidean geometry. Not like flat Euclidean space, hyperbolic space is negatively curved and doesn't satisfy the Parallel Postulate. By some studies (e.g. by Poincaré, Klein…), it turns out hyperbolic geometry is a fantastic “toy model” to play with. However, it was later realized that hyperbolic geometry is a gateway to many other important and interesting mathematics.
At first, we will learn some basic geometric properties, including how to visualize some models of hyperbolic spaces, e.g. their geodesics and isometries. Also, we will introduce the definitions of Fuchsian groups (2-dim) and Kleinian groups (3-dim) (vaguely say, they are fundamental groups of hyperbolic manifolds), and we focus on some basic properties of hyperbolic 2 & 3 manifolds. Then we study some big pictures of this non-Euclidean world, e.g. 2-dim case is flexible: due to Riemann, there can be tons of hyperbolic structures on a closed surface (we can also visualize this phenomenon by using pair of pants to glue a surface); however higher dimensions (n>=3) are rigid: due to Mostow, there is at most one hyperbolic structure on a closed n-manifold.
Later, if we still have time, based on your interests, there are four topics you can try. Pick one of them, get a feeling and some big pictures about it:
Low-dimensional topology: heading to mapping class groups of surfaces, Thurston’s Geometrization of 3-manifolds and the Poincaré Conjecture.
Dynamics: dynamical properties of the geodesic flow and horocyclic flow, how the chaos of the previous one applies to the Mostow Rigidity and the rigidity of the second one inspired the dynamical proof of the Oppenheim Conjecture in analytic number theory.
Complex Analysis: quasiconformal maps, Teichmüller theory of hyperbolic Riemann surfaces, measurable Riemann mapping theorem and quasi-Fuchsian 3-manifolds.
Geometric Group Theory: the coarse geometry of hyperbolic spaces and finitely generated groups, Gromov hyperbolicity and beyond like the coarse geometry of mapping class groups and Teichmüller spaces.
References:
Lectures on Hyperbolic Geometry (Benedetti & Petronio)
The Geometry and Topology of Three-manifolds (W. P. Thurston)
A Primer on Mapping Class Groups (Farb & Margalit)
An Introduction to Geometric Topology (B. Martelli)
Ergodic Theory with a View Towards Number Theory (Einsiedler & Ward)
Ratner’s Theorems on Unipotent Flows (D. W. Morris)
Teichmüller Theory and Applications to Geometry, Topology, and Dynamics, Volume 1: Teichmüller Theory (J. H. Hubbard)
Geometric Group Theory (Drutu & Kapovich)
Coarse Geometry and Teichmüller Theory (Note by Alex Wright)
Prerequisites: Linear algebra, basic point set topology (especially the covering spaces and deck transformations), basic complex analysis (understand Mobius transformations will be enough), basic knowledge of differentiable manifold and Riemannian geometry will be helpful but not necessary, basic measure theory if you would like to choose topic 2 or 3.
Geometric Analysis, PDEs, Mathematical Physics
Matrix Groups and Applications**
Curves and Curvature: An Introduction to Differential Geometry**
Matrix Groups and Applications
Matrix groups bring together algebra and geometry to tackle problems in both pure and applied mathematics. At a fundamental level, matrix groups consist of collections of matrices that satisfy certain properties, such as the ability to be multiplied and inverted while maintaining group structure. Studying these groups can reveal symmetries and transformations that appear across physics, computer science, and engineering. For example, rotations and reflections in space are encoded by special types of matrix groups, making them essential in many fields, from computer graphics to quantum mechanics. Matrix groups also provide a concrete way to understand abstract algebraic concepts like group theory and Lie algebras, so they serve as an excellent bridge between pure math and practical applications. Depending on your background and interests, we could focus on topics like the 3D rotation group, quaternions, or Pauli matrices (which model particles with “spin”, and are an example of a Lie algebra). There are many exciting directions to travel!
References:
"Matrix Groups for Undergraduates", Kristopher Tapp
"Matrix Groups: An Introduction to Lie Group Theory", Andrew Baker
Prerequisites: Linear algebra (required); Abstract Algebra, Ordinary Differential Equations, Complex Variables (useful but not required)
Curves and Curvature: An Introduction to Differential Geometry
Differential geometry uses calculus to explore curves, surfaces, and other smooth shapes in spaces beyond ordinary Euclidean geometry. The central concept is “curvature.” The right notions of curvature allow us to understand how a sphere (or a distorted sphere—like a potato), a donut shape (called a torus), or even the shape of the universe behaves. Using tools from ordinary differential equations, you can investigate geodesics, which are the "straightest paths" on curved surfaces. These ideas are fundamental in physics, especially in Einstein's theory of general relativity, where the curvature of spacetime is central to understanding gravity. Beyond physics, differential geometry has applications in many areas like computer graphics and biology.
The most natural place to start is with the differential geometry of curves and surfaces—smooth shapes in 3D space. This leads to Gauss’s “Theorema Egregium” (the “Remarkable Theorem”), which says that curvature is actually intrinsic to a surface—we don’t have to think about shapes as being “embedded” in an ambient space. Depending on your interests, we could look at more abstract notions of curvature by understanding a bit about manifolds, metrics, and the curvature tensor. We could also dabble in some General Relativity!
References:
"Elementary Differential Geometry", Andrew Pressley
"Differential Geometry of Curves and Surfaces", do Carmo
Prerequisites: Linear Algebra, Calculus III
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 also been learning algebraic geometry and dynamical systems.
Exploring the Number Jungle**
Ramsey Theory on the Integers**
Exploring the Number Jungle
Real numbers come in all sorts of kinds: rational numbers, irrational numbers, algebraic numbers, and transcendental numbers. All numbers can be approximated as well as you want by rational numbers (just truncate the decimal expansion), but it turns out we can quantify how complex a rational approximation is --- for example, 314/100 = 157/50 is a more complicated approximation for π than 22/7, in that it requires you to use much larger numbers in the numerator and denominator for about the same level of accuracy. This quantification reveals very surprising facts: if an irrational number can be approximated really well by rational numbers that aren’t too complicated, then it cannot be a root of a polynomial with rational coefficients (that is, it cannot be algebraic)! In this project we will explore these rational approximations (also known as “Diophantine approximations”) and how they relate to properties like irrationality, algebraicity, and transcendence of different numbers. We will learn why the Golden Ratio is the hardest number to approximate with rational numbers, and why this causes it to appear in nature. We may also learn why certain polynomial equations in two variables (defining so-called “elliptic curves”) only have finitely many integer solutions.
Reference: “Exploring the Number Jungle: A Journey into Diophantine Analysis” by Edward Burger
Prerequisites: Calculus, basic Linear Algebra, and comfort with reading and writing proofs (such as from having taken one proof-based mathematics course)
Ramsey Theory on the Integers
In a room of 6 people, there will either be a collection of 3 people, all of whom have met each other before, or a collection of 3 people, none of whom have met each other before. In an elementary school class of at least 18 people, there will always be 4 kids who are all friends with each other, or 4 kids none of whom are friends with each other. In a large enough collection of people or objects, complete disorder is impossible. These facts are studied in a branch of combinatorics known as Ramsey Theory. When applied to the integers, Ramsey Theory allows us to show that in a large enough set of integers, you can always find an arithmetic progression (that is, an evenly spaced sequence of numbers like 3, 7, 11, 15) of whatever length you like, and to show that certain other equations must have solutions in some set of integers, so long as the set is large enough. This project will explore Ramsey Theory and its application to sets of integers, introducing you to important ideas in Arithmetic Combinatorics (and Combinatorics more generally) and to Erdős-style problems and results.
Reference: “Ramsey Theory on the Integers” by Bruce Landman and Aaron Robertson
Prerequisites: Comfort with reading and writing proofs, such as from having taken one proof-based course in mathematics. Exposure to discrete mathematics or combinatorics is helpful, but not required.
Algebraic Number Theory, Representation Theory, Commutative Algebra.
Representation Theory of Finite Groups
Algebraic Theory of Numbers
Rational Points on Elliptic Curves
Representation Theory of Finite Groups
Representation theory is the study of group actions on vector spaces. It turns out, there is a rich and beautiful theory that comes about, that has startling implications on the structure of finite groups. For example, it allows us to give a natural proof of Burnside’s theorem, which says that all groups with exactly two prime factors are solvable. Representation theory also has a vast array of applications in a diverse set of areas of mathematics. For example, we will see it plays an important role in the discrete Fourier transform, which has applications to signal and image processing, combinatorics, and number theory. Depending on student interest, we can also explore applications in graph theory, card shuffling, and even quantum chemistry.
References:
"Representation Theory of Finite Groups" by Benjamin Steinberg
"Representations and Characters of Groups" by Gordon James and Martin Liebeck
Prerequisites: Linear algebra, group theory, ring theory.
Algebraic Theory of Numbers
An appeal of number theory comes from the fact that many questions can be stated to a high school student, but whose solution may require an abstract algebraic or analytic theory to properly handle. For example, Pell’s equation x^2 - dy^2 = 1 (d a square free integer) is easily stated, but a satisfying and elegant solution involves the study of units of subrings of quadratic field extensions of Q. In this project, we will develop fundamental algebraic machinery enabling a more intuitive and systematic approach to answering all sorts of these number theoretic questions.
References:
"Algebraic Theory of Numbers" by Pierre Samuel
"Algebraic Number Theory and Fermat’s Last Theorem" by Ian Stewart and Ian Tall
Prerequisites: Linear algebra and a first course in abstract algebra. Galois theory is recommended, but not required.
Rational Points on Elliptic Curves
An elliptic curve is the set of zeros of a cubic polynomial in two variables. When the polynomial has rational coefficients, a major interest is to describe integer and rational solutions. It turns out, there is a simple geometric operation one can assign to these solutions, which turn them into a group. The study of these groups lead to a powerful theory. For example, it can be used to show that the elliptic curve y^2 = x^3 + 3 has infinitely many rational solutions, which is by no means obvious. This is our principle objective in this project. We can also explore the many applications elliptic curves have to cryptography.
Reference: "Rational Points on Elliptic Curves" by Joseph H. Silverman and John T. Tate
Prerequisites: Calculus I & II, group theory. Some linear algebra, real & complex analysis, and basic number theory may be useful, but is not required.
My main interests are in discrete dynamical systems, ergodic theory, and fractal geometry. Adjacent to these topics are metric number theory, information theory, computability theory, conformal geometry, probability theory, data science and machine learning.
Discovering Discrete Dynamical Systems*
The Mathematics of Data**
Invitation to Ergodic Theory
(Note: Marco is a postdoc mentor.)
Discovering Discrete Dynamical Systems
This will be an inquiry-based introduction to dynamical systems based on the book with the same title, by Johnson, Madden, and Șahin. Topics include chaos, fractals, symbolic dynamics, among others. This book has short expositions, many exercises as well as projects that encourage open exploration. This book is appropriate for beginning and intermediate undergraduate students.
Reference: "Discovering Discrete Dynamical Systems" by Aimee S. A. Johnson, Kathleen M. Madden, and Ayșe A. Șahin.
Prerequisites: Calculus. Familiarity with proofs and real analysis is helpful but can be learned concurrently.
The Mathematics of Data
This reading project focuses on various mathematical methods that form the foundations of machine learning and data science, such as numerical linear algebra, stochastic optimization, and probability methods. This project is suitable for advanced undergraduates or graduate students.
Reference: "The Mathematics of Data" by Mahoney, Duchi, and Gilbert.
Prerequisites: Multivariable Calculus, Linear Algebra, and familiarity with proofs (preferably real analysis).
Invitation to Ergodic Theory
Ergodic theory is basically a statistical approach to the study of chaotic dynamical systems. In this project we will learn concepts such as ergodicity, recurrence, and mixing of dynamical systems. Further topics may be incorporated as well, depending on the student’s interest and background, such as entropy and thermodynamic formalism, or connections with other areas such as fractal geometry or metric number theory. Along the way basic knowledge of measure theory and metric topology can be developed if the student doesn’t already possess this background. This reading project is suitable for advanced undergraduates or beginning Master's students with background knowledge of undergraduate analysis (a.k.a. advanced calculus).
Reference:
"Invitation to Ergodic Theory" by Cesar Silva
"Foundations of Ergodic Theory" by Krerley Oliveira and Marcelo Viana
Prerequisites: Undergraduate real analysis
I'm primarily interested in Algebraic Geometry/commutative algebra, with a strong secondary interest in Topology.
Topology: A Categorical Approach
Category Theory in Context
(Note: Adityo is an undergraduate mentor.)
Topology: A Categorical Approach
Over the last century many of the basic tools of topology have come to permeate mathematics, making it an essential part of one's toolkit in studying number theory, analysis, algebraic geometry, etc. To that end, "point-set" topology describes the basic set-theoretic definitions of many of these concepts, and is useful as a foundation. Nonetheless, many fields start to require more sophisticated tools and ideas (i.e., topological groups, homology groups, abelian varieties, etc.) and there tends to be a significant conceptual leap going from basic point-set topology to the more abstract/algebraic topology needed in these areas.
Topology: A Categorical Approach attempts to bridge this gap by taking familiar point-set constructions and reintroducing them via easily generalizable categorical methods. In doing so, it also serves as a relatively gentle way to gain some familiarity with category theory (see Project 2), and eventually builds up to some important algebro-topological constructions like the fundamental group.
References:
"Introduction to Topology: Third Edition" by Bert Mendelson (for basic point-set topology)
"Topology" by James Munkres (for basic point-set topology)
"Topology: A Categorical Approach" by Tai-Danae Bradley, Tyler Bryson, and John Terilla
www.math3ma.com (Tai-Danae Bradley's blog -- look for "category theory" and "topology" in the categories section)
Prerequisites: Comfort with proofs. Some prior familiarity with point-set topology is recommended. Some exposure to abstract algebra (i.e., rings, groups) could also be helpful for some examples, but isn't necessary.
Category Theory in Context
Since its advent in the 1940s, "category theory" has become increasingly useful as a language with which to describe mathematical ideas-- in particular, letting one easily transfer concepts from one context to another. For instance, when we talk about a "product" in mathematics, we could mean a product of: sets, vector spaces, rings, groups, topological spaces, etc. Despite this sheer breadth of definitions however, category theory tells us that these are all just examples of one type of product: the "categorical product" of two objects. If one wishes to learn, say, algebraic geometry, algebraic number theory, or even functional programming, categorical concepts are essential!
Unfortunately, however, the same generality that makes category theory so useful is also what lends it a reputation for being unnecessarily abstract and opaque. Riehl's book attempts to demystify the language of category theory and provides many examples for a reader to sink their teeth into.
References:
"A Book of Abstract Algebra" by Charles Pinter (for abstract algebra)
"Category Theory in Context" by Emily Riehl
"Basic Category Theory" by Tom Leinster
Prerequisites: Comfort with proofs and proof-based linear algebra. Some abstract algebra (i.e., groups or rings) is strongly recommended. A bit of point-set topology could be helpful as well.
I am a PhD mathematics student at the CUNY Graduate Center advised by Andrew Obus and Dennis Sullivan. My interests span a menagerie of mathematical ideas, e.g. algebraic and arithmetic geometry, and more specifically, the problem of resolution of singularities, regular models of curves, and the topology of algebraic varieties. On the applied side of mathematics, I find quantum computation fascinating, and dabble in machine learning & artificial intelligence, applied linear algebra & tensor networks, and the satisfiability problem.
Introduction to Commutative Algebra with a View Toward Algebraic (& Arithmetic) Geometry*
Topics in (Arithmetic &) Algebraic Geometry*
Quantum Computation*
Introduction to Commutative Algebra with a View Toward Algebraic (& Arithmetic) Geometry
The title of the projects nods to (and is the title of) David Eisenbud's book leading us up the spire of commutative algebra prerequisite to appreciate algebraic (& arithmetic) geometry. We will maintain an eye toward geometric considerations underlying the algebra throughout. A student carrying little to no prerequisite knowledge of (commutative) algebra may gain an appreciation of algebraic varieties, while a student far along in their study of (commutative) algebra may hope to gain an appreciation of the language of schemes à la Grothendieck by semester's end.
References:
"Commutative Algebra with a View Toward Algebraic Geometry" by David Eisenbud.
"An Invitation to Algebraic Geometry" by Karen Smith et al. (The first chapter of) "Algebraic Geometry" by Robin Hartshorne.
Prerequisites: Mathematical maturity and eagerness to learn should suffice — nearly everything can be developed as needed. However, completion of a course in abstract algebra, e.g. so the definition of a (commutative) ring is understood, is preferred.
Topics in (Arithmetic &) Algebraic Geometry
We may study any of:
The problem of resolution of singularities,
Regular models of curves, or
The topology of algebraic varieties.
References: Contingent on topic choice.
Prerequisites: Contingent on topic choice.
Quantum Computation
This project aims to reaffirm the classical story of computation, and then augment it to the phantasmagorical world of quantum computation. Sufficiently advanced students might progress to implementing code for a famous quantum algorithm, e.g. Shor's algorithm, on a quantum computer in, e.g. a Jupyter Notebook.
References:
"A Course in Quantum Computing (for the Community College) Volume I" by Michael Loceff.
"Quantum Computing: An Applied Approach" by Jack D. Hidary
Prerequisites: Mathematical maturity and eagerness to learn should suffice — nearly everything can be developed as needed. However, completion of a course in linear algebra, e.g. so the definition of a basis of a vector space is understood, is preferred.
Commutative Algebra, Algebraic Geometry, Model Theory
A Concrete Introduction to Algebraic Curves**
Computational Algebraic Geometry and Commutative Algebra**
An Introduction to Algebraic Geometry
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.
References: Robert Bix, "Conics and Cubics: A Concrete Introduction to Algebraic Curves"
Prerequisites: Some Calculus and linear algebra, 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.
References: 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. Potential topics to focus on are: singularities of curves and how to resolve them; divisors on curves and the Riemann-Roch Theorem; Bezout's Theorem and intersection Theory.
References: Igor R.Shafarevich, "Basic Algebraic Geometry I", William Fullton "Algebraic Curves: An Introduction to Algebraic Geometry", Miles Reid "Undergraduate Algebraic Geometry"
Prerequisites: Abstract algebra, a little bit of point set topology.
Algebraic geometry; number theory
Introduction to Projective Geometry*
Matrix Groups**
Introduction to Projective Geometry
Generically, two lines in Euclidean space meet at a unique point. This statement fails for parallel lines in Euclidean space but remains true if we consider the lines to live in a larger projective space, in which the lines “intersect at infinity”. This project will introduce projective spaces, projective transformations and invariants, with applications to classical geometry.
References: Lecture notes on projective geometry by Thomas Baird; Introduction to Projective Geometry by C.R. Wylie, Jr
Prerequisites: Two versions of this project are possible, one with no prerequisites and one for a student with strong background in linear algebra.
Matrix Groups
This project will be a study of the classical real and complex matrix groups and the geometric contexts in which they arise. The project can be tailored to students’ backgrounds, ranging from a first introduction to these groups, to topics in algebraic group theory or Lie theory.
References: Varies based on background
Prerequisites: Linear algebra. Abstract algebra and topology are helpful but not necessary.
Topology and Dynamics
Introduction to Dynamics*
Introduction to Dynamics
Dynamical systems are systems which change over time. Some classic examples are a pendulum swinging, the planets orbiting the sun, and populations of rabbits and foxes. Some questions might be, how much time does the pendulum take between swings (and with friction, how quickly does it stop swinging), does an orbiting planet ever come back to the exact same position, and will the foxes hunt the rabbits to extinction?
Our project would be about learning some of the tools used to study dynamical systems (ideas like 'stability', 'periodicity', 'bifurcation theory', and 'chaos theory'), and exploring some basic models appearing in biology, chemistry, or physics from this viewpoint. Depending on interest, we might also talk about dynamical systems from a purely mathematical perspective, both as a field of study in itself and how dynamical tools are used by other areas such as geometry, number theory, or probability.
References:
"Differential Equations, Dynamical Systems, and an Introduction to Chaos" by Hirsch, Smale, Devaney
"Nonlinear Dynamics and Chaos" by Strogatz
Prerequisites: Ideally the student would have taken multivariable calculus. This is not strictly required.
Bayesian statistics; machine learning; online learning
Gaussian Processes and Bayesian Optimization
Gaussian Processes and Bayesian Optimization
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.”
Reference: "Bayesian Optimization", Garnett.
Prerequisites: Multivariate calculus, linear algebra, and introduction to probability or statistics.
High Dimensional Probability and Mathematical Data Science
High-Dimensional Probability and Mathematical Data Science
(Note: This project was added on 12/17. If you already submitted an application and would like to change your mentor preferences, please use the link that was emailed to you after your submission to edit your response.)
High-Dimensional Probability and Mathematical Data Science
The topic is a highly relevant and exciting field today. It is not easy to define what "mathematical data science" is. In general, this involves mathematical foundations of high dimensional spaces and probability theory plays a big role in it!
If you and your friends have an interest in probability theory and linear algebra, and are considering pursuing theoretical data science research in the future, please do not hesitate to reach out to me!
References: (All freely available online and we can discuss if you want to read something else coherent to this topic.)
"High-Dimensional Probability" by Roman Vershynin
"Foundations of Data Science" by Avrim Blum, John Hopcroft, and Ravindran Kannan
Prerequisites: Proof-based linear algebra and probability theory, undergraduate real analysis. Of course, the more you know the better!
Dynamical System and PDE
Topology From the Differentiable Viewpoint
Topology From the Differentiable Viewpoint
We will study notions like smooth manifolds, smooth maps and tangent spaces and derivations...etc. Facts like the theorem of Sard and Brown...etc. As the title suggested, this project aim to study topology using tools from Smooth Manifold Theory, depending on a student's background, we may end at different topology fact. For instances, if a student with no background in measure theory we may end the semester proving the Fundamental Theorem of Algebra. Else, we can dive into Brouwer fixed point theorem using Sard and Brown. If time allows, we can also dive into deeper results.
References:
"Intro. to Smooth Manifolds" by John Lee
"Topology from the Differentiable Viewpoint" by John W. Milnor
Prerequisites: Minimum requirements are Linear Algebra, Multivariable Calculus and Point Set Topology
Number Theory, everything
An Invitation to Representation Theory**
An Invitation to Representation Theory
The subject of representation theory is one of the most connected in mathematics, with applications to group theory, geometry, number theory and combinatorics, as well as physics and chemistry. It can however be daunting for beginners and inaccessible to undergraduates. In this project, we will read R. Michael Howe's "An Invitation to Representation Theory", which provides a gateway to representation theory via the ubiquitous symmetric group and its natural action on polynomials.
Reference: "An Invitation to Representation Theory" by R. Michael Howe
Prerequisites: A solid grounding in (proof-based, abstract) linear algebra.
Randomized Matrix Computations, Random Matrix Theory, Iterative Methods, Numerical Linear Algebra, Topology of Neural Networks
Low-Rank Approximations and Linear Transformations: Bridging Theory and Application**
Low-Rank Approximations and Linear Transformations: Bridging Theory and Application
Low-rank approximations are essential tools in both mathematics and computer science which enable the reduction of high-dimensional matrices to more manageable lower-dimensional forms. This process not only simplifies complex structures but also preserves key geometric properties, offering valuable insights into linear algebra, optimization, and manifold theory. Understanding these interactions between geometric and algebraic frameworks is crucial for advancing in these fields. For instance, in genomics, low-rank approximations help analyze vast gene expression datasets, allowing researchers to uncover significant patterns associated with diseases. Similarly, in computational fluid dynamics, techniques like Proper Orthogonal Decomposition enable the efficient simulation of fluid flows, facilitating deeper insights into complex systems.This project seeks to approach the study of matrices in a more abstract way than is typically presented at the undergraduate level. Students will explore foundational decomposition methods, spectral properties, and sensitivity analysis to gain insights into how linear transformations affect various datasets. By delving into these abstract concepts, participants will not only strengthen their theoretical understanding but also enhance their practical skills in matrix computations.A key objective of this project is to build an intuitive grasp of matrix transformations, preparing students for advanced courses in courses that then deal with theoretical properties of spaces that are preserved under continuous transformations and deformation. This deeper theoretical foundation is vital for success in applied fields where matrix computations are critical, including computational sciences, signal processing, machine learning, algorithm development, image compression, and more.
Reference: "Numerical Linear Algebra" by Trefethan and Bau
Prerequisites: Linear Algebra, Knowledge of Python, Calculus