1. 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

2. 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: 

     1. "From Groups to Geometry and Back" by Vaughn Climenhaga

     2. "Office Hours with a Geometric Group Theorist" by Matt Clay, Dan Margalit

   • Prerequisites: None

3. 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)

1. 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: 

1. "A course in minimal surfaces" by T Colding and W Minicozzi

2. "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.

2. 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.

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:

1. Low-dimensional topology: heading to mapping class groups of surfaces, Thurston’s Geometrization of 3-manifolds and the Poincaré Conjecture.

2.  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.

3. Complex Analysis: quasiconformal maps, Teichmüller theory of hyperbolic Riemann surfaces, measurable Riemann mapping theorem and quasi-Fuchsian 3-manifolds.

4. 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.

1. • References: 

1. Lectures on Hyperbolic Geometry (Benedetti & Petronio)

2. The Geometry and Topology of Three-manifolds (W. P. Thurston)

3. A Primer on Mapping Class Groups (Farb & Margalit)

4. An Introduction to Geometric Topology (B. Martelli)

5. Ergodic Theory with a View Towards Number Theory (Einsiedler & Ward)

6. Ratner’s Theorems on Unipotent Flows (D. W. Morris)

7. Teichmüller Theory and Applications to Geometry, Topology, and Dynamics, Volume 1: Teichmüller Theory (J. H. Hubbard)

8. Geometric Group Theory (Drutu & Kapovich)

9. Coarse Geometry and Teichmüller Theory (Note by Alex Wright)

1. • 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.

1. 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)

2. 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.

1. 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.

2. 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).

3. 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: 

     1. "Invitation to Ergodic Theory" by Cesar Silva

     2. "Foundations of Ergodic Theory" by Krerley Oliveira and Marcelo Viana

   • Prerequisites: Undergraduate real analysis

1. 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: 

     1. "Commutative Algebra with a View Toward Algebraic Geometry" by David Eisenbud. 

     2. "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.

2. Topics in (Arithmetic &) Algebraic Geometry

   • We may study any of: 

     1. The problem of resolution of singularities, 

     2. Regular models of curves, or 

     3. The topology of algebraic varieties.

   • References: Contingent on topic choice.

   • Prerequisites:  Contingent on topic choice.

3. 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:

     1. "A Course in Quantum Computing (for the Community College) Volume I" by Michael Loceff.  

     2. "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. 

1. 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.

2. 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.

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.

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: 

1. "Intro. to Smooth Manifolds" by John Lee

2. "Topology from the Differentiable Viewpoint" by John W. Milnor

• Prerequisites: Minimum requirements are Linear Algebra, Multivariable Calculus and Point Set Topology

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