1. An investigation into the theory of minimal surfaces*

   • We will spend some time investigating the historical significance of some of the founding questions of the field and try to understand the varied formulations of the topic in some depth. Depending on the students' background ranging from knowledge of multivariable calculus to more sophisticated knowledge about Riemannian geometry and analysis and anywhere in between, we can consult the varied and expansive literature on the topic and formulate a curriculum that would give the student a proper taste into the field as a whole! 

1. • Reference: A reference that I plan on using is Robert Osserman's "A Survey of Minimal Surfaces." This reference is self contained, readily available online, and contains just enough of the differential geometry required to talk about the theory without being too cumbersome. Overall, I think that this reading project would be a motivated introduction to some differential geometry with the goal in mind of understanding minimal surfaces. 

1. "A Survey of Minimal Surfaces" by Robert Osserman

2. "A Survey on Classical Minimal Surface Theory" by Joaquín Pérez and William Hamilton Meeks

3. "A Course in Minimal Surfaces" by Tobias Colding and William Minicozzi

1. • Prerequisites: I want to make this as accessible as possible and hope to cover any required background, but here is a written out background.

     • Students should know multivariable calculus, including gradients, divergences, surface integrals, and Jacobians.

     • They should understand basic linear algebra, including eigenvalues, matrices, and bilinear forms.

     • They should be comfortable with real analysis concepts like limits, continuity, differentiability in ℝⁿ, and uniform convergence.

1. 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. (Formal theorem provers such as Lean or Rcoq are examples of this.) 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.

1. • Reference: "Logic and Structure" by Van Dalen

   • Prerequisites: At least one proof-based mathematics course

2. Constructive Mathematics**

   • It’s easy to see that there are irrational numbers a and b such that a^b is rational. If sqrt(2)^sqrt(2) is rational then we are done. Otherwise, take a = sqrt(2)^sqrt(2) and b = sqrt(2). However, you may feel cheated by this proof in that I have not actually given you an example. If you feel this way then you are in good company. Constructive mathematics is a family of approaches to mathematics requiring that proofs of existence to involve the explicit construction of a witness. The result is that from a constructive proof that for each x there is y satisfying some condition, one can extract an algorithm that takes input x and computes a witnessing y. Constructive reasoning thus preserves computational content and constructive proofs are equivalent to programs in a way that can be made precise. 

The cost of this approach is that certain classical principles are no longer valid. The chief culprit is the principle of the excluded middle stating that for any proposition “A”, “A” or “not A” holds. (Spot where this principle was used in the above proof.) A ground up reformulation of mathematics is thus required. Once this has been done new mathematical worlds become available where every function on the natural numbers is computable and every function on the reals is continuous. This project would involve an introduction to ideas and principles of constructive logic and mathematics. The precise content will be tailored to the background and interests of the student. 

2. • Reference: Depends on the background and interests of the student  

   • Prerequisites: At least one proof-based mathematics course. Any familiarity with logic, compatibility theory or typed programming languages would be useful but is not essential. 

1. Analytic Methods in Enumeration

• Analytic combinatorics aims at predicting precisely the properties of large structured combinatorial configurations, through an approach based extensively on analytic methods. This project explores how generating functions serve as a bridge between combinatorial structures and their asymptotic properties. We will study the symbolic method for translating combinatorial constructions into generating functions and apply singularity analysis to derive asymptotic formulas for counting problems such as Catalan numbers and set partitions.

• References: 

1. "Analytic Combinatorics" by Philippe Flajolet and Robert Sedgwick

2. "Principles and Techniques in Combinatorics" by Chen Chuan-Chong and Koh Khee-Meng

• Prerequisites: Some knowledge in enumerative combinatorics, a proof based course, Calc 1 and 2 (for understanding series)

2. Computational Abstract Algebra

• Computer algebra systems (CAS) provide impressive mathematical capabilities that have revolutionized the way mathematics is done and taught. They can perform without error gargantuan calculations that would be infeasible to attempt by hand. Accordingly, CAS have become commonplace computational and modeling tools in the scientific and engineering workplace. They have also been incorporated into the mathematics curricula at many colleges and universities starting with calculus and moving to more advanced topics such as linear algebra and differential equations. Yet despite their widespread application and importance, few users are familiar with the inner workings of CAS. Through the study of algorithms for polynomial arithmetic, GCD computation, factorization over finite fields, and Gröbner bases, the project aims to bridge the gap between theoretical algebra and symbolic computation.

• References:

  1. "Computer Algebra, Concepts and Techniques" by Edmund Lamagna

  2. "Algorithms for Computer Algebra" by Keith Geddes, Stephen Czapor, and George Labahn

  3. "An Elementary Introduction to the Wolfram Language" by Stephen Wolfram (free online)

• Prerequisites: Some knowledge in abstract algebra

1. "Recategorizing" core topics in mathematics**

• We will read chapters 1-4 in "Category Theory in Context" by Emily Riehl. This introductory text in category theory is approachable and rich in examples across core topics from topology, algebra and others. We will divert to "Topology" by Terilla, et al, and "Algebra Chapter 0" by Aluffi for a deeper look at these examples. Recontextualizing in category theory will make typically difficult proofs and concepts readily available, e.g. the Stone-Cech compactification, or universality of the tensor product. (And category theory doesn't always deliver on this promise! We'll discuss and debate.) Time permitting we'll continue to chapters 5 and 6, or read further in a specific application.

• References: (All available as free PDFs.)

  1. "Category Theory in Context" by Emily Riehl

  2. "Topology: A Categorical Approach" by John Terilla, Tai-Danae Bradley, and Tyler Bryson

  3. "Algebra Chapter 0" by Paolo Aluffi

• Prerequisites: An introductory proof writing course is a great preparation (esp. basic set theory and logic). I intend to focus on algebra and topology, familiarity in those is good but not mandatory. Any junior or senior math undergrad will be totally prepared.

1. Introduction to Fourier Analysis

   • Fourier analysis is a useful subject with broad applications in a variety of fields, including partial differential equations, image & signal processing, physics, electrical engineering and more. In this project, we'll read through selected chapters on Stein and Shakarchi's book on Fourier Analysis.

   • Reference: "Fourier Analysis: An Introduction" by Elias M. Stein, Rami Shakarchi

   • Prerequisites: Undergraduate real analysis

2. Introduction to Statistical Learning 

• Statistical learning encompasses a variety of tools used to draw insights from complex data. In this project, we will cover a subset of topics from An Introduction to Statistical Learning with Applications in Python (ISLP), tailored to the student’s interests. Some topics include classification, linear regression, unsupervised learning, and survival analysis. Our goal will be to introduce fundamental concepts in several areas and then explore one topic in great detail.

1. • Reference: "An Introduction to Statistical Learning with Applications in Python" by Gareth James, Daniela Witten, Trevor Hastie, Robert Tibshirani

   • Prerequisites: An introductory course in probability & statistics. Some familiarity with Python programming and linear algebra are a plus.

1. A View from the Top: Analysis, Combinatorics, and Number Theory*

• We will chart a scenic path through some beautiful math, starting by learning some fundamental bounds (the Cauchy-Schwarz inequality and its cousins), before learning all the wonderful things we can use it to estimate throughout math! Some questions we will consider:

  • • • Given a bunch of points in 3-dimensional space, we can “project” them onto each of the 3 coordinate planes, finding the “shadows” they cast onto each plane. What’s the least number of such shadows? 

      • Given a set of integers, A, is it possible that neither the set of all sums of pairs of elements of A nor the set of all products of pairs of elements of A is much larger than A?

      • If we have a graph with e edges and n vertices, how many edges must cross over one another?

      • If two positive integers are chosen at random, how likely are they to have no common factor?

We will see how some of the same fundamental techniques can be used on all these problems and more, and in the process glimpse the “unity of mathematics:” seemingly distinct lines of inquiry converge, rhyme, and support one another. 

• Reference: “A View from the Top,” Alex Iosevich, Student Mathematical Library Vol. 39 

• Prerequisites: At minimum, all you need for this project is a solid grasp of high school math and a desire to learn. Some experience with proofs will allow us to move more quickly, and knowledge of multivariable calculus will open up a few subsections. If you have studied more advanced math, there’s still plenty for you to learn from this project, for both research and the Putnam competition.

2. p-adic Analysis Compared with Real

   • We will learn about the p-adic numbers: an infinite family of new number systems, one for each prime number p. In the weird and wacky world of the p-adic numbers, an infinite series will converge if and only if the nth term tends to 0, positive powers of p are small, while the fraction 1/p is larger than any integer. We will study the p-adics as you may have studied the real numbers in real analysis, defining them as a metric space constructed by completing the rational numbers with respect to a p-adic absolute value. We will discuss the “topology” of the p-adics by describing their open sets, and see that they “look like” Cantor sets. We may talk about special classes of functions on the p-adics. Along the way, we’ll review and clarify ideas from real analysis.

Now, you might wonder about the significance of the p-adics --- they’re a fun world to explore, but are they just a mere curiosity? In fact, the p-adics have a many applications in number theory; as a simple example, applying Newton’s method in a p-adic context will let us prove results about modular arithmetic. Time permitting, we may also discuss an application of the 2-adic numbers to combinatorial geometry.

2. • Reference: “p-adic Analysis Compared with Real,” Svetlana Katok, Student Mathematical Library Vol. 37

   • Prerequisites: One semester of mathematical analysis/real analysis/advanced calculus/equivalent. Exposure to modular arithmetic, elementary number theory, and/or abstract algebra is helpful, but not necessary.

3. Student’s Choice: Number Theory or Harmonic Analysis

• • If there is some other area of number theory or harmonic analysis that interests you, I would be happy to develop a project in that area with you. If you sign up for this project, please sketch your interests in your application.

1. Computational Algebraic Geometry

   • Algebraic geometry is the study of solutions to polynomials equations in several variables. It is a hot topic in part due to its vast array of real world applications such as computer vision, robotics, genetics, particle physics, statistics, cryptography, and automatic theorem proving to name a few. It is also extremely pervasive across virtually all areas of pure math. In this project, we will break into this subject, from a computational perspective. We will both develop the theoretical and technical foundations, while working with many concrete examples, including implementing algorithms using computer algebra systems. And we can explore some applications such as those listed above. 

   • Reference: “Ideals, Varieties, and Algorithms" by David A. Cox, John Little, Donal O’Shea

   • Prerequisites: Linear algebra, group theory

2. 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.

   • Reference: “Algebraic Number Theory and Fermat’s Last Theorem" by Ian Stewart and Ian Tall, Number Fields by Daniel A. Marcus

   • Prerequisites: Linear algebra and a first course in abstract algebra. Galois theory is recommended, but not required.

3. Rational Points on Elliptic Curves

   • An elliptic curve is the set of solutions 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 equation y^2 = x^3 + 3 has infinitely many rational solutions, which is by no means obvious. Elliptic curves are central to number theory, having applications to cryptography, diophantine approximation, and played a major role in the proof of Fermat’s Last Theorem. We will explore all this and more in this project. 

   • 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 elementary number theory may be useful, but is not required. 

1. 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).

• References: 

1. "Invitation to Ergodic Theory" by Cesar Silva

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

• Prerequisites: Undergraduate real analysis

2. Mathematical Finance*

•  The main focus of this reading course is to learn the mathematical development of options pricing. This entails covering topics such as probability theory, random processes such as random walks and Brownian motion. Applications will include derivation of the Black-Scholes equation and its solution, as well as develop an understanding of the financial concepts involved (e.g. call options, volatility, hedging, etc.) If the student is interested we could also incorporate Monte Carlo methods in Python to simulate options pricing with and without hedging. 

• References: 

1. "The concepts and practice of mathematical finance" by Mark S. Joshi

2. "Dynamic Asset pricing theory" by Darrell Duffie

3. "Stochastic Calculus for Finance" by Steven E. Shreve

• Prerequisites: Calculus. Familiarity with probability theory and advanced calculus (i.e. analysis) would be beneficial. Computer programming if the student is interested in the Monte Carlo simulations.

3. Mathematics for Machine Learning**

• This reading project focuses on various mathematical methods that form the foundations of machine learning and data science, such as linear algebra and probability theory. We will learn how to apply this mathematical theory to Machine Learning problems such as parameter estimation, linear regression, dimensionality reduction, or others depending on the student's interests.

• References: 

  1. "Mathematics for Machine Learning" by Marc P. Deisenroth, Aldo Faisal, and Cheng Soon Ong 

  2. Statistical Learning with Python -- Stanford Online

• Prerequisites: Multivariable calculus and linear algebra. Computer programming if the student is interested in implementing ML methods on datasets.

1. 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, and eventually builds up to some important algebro-topological constructions like the fundamental group.

1. • References:

     1. "Introduction to Topology: Third Edition" by Bert Mendelson (for basic point-set topology)

     2. "Topology" by James Munkres (for basic point-set topology)

     3. "Topology: A Categorical Approach" by Tai-Danae Bradley, Tyler Bryson, and John Terilla

   • 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.

2. 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.

2. • References:

     1. "A Book of Abstract Algebra" by Charles Pinter (for abstract algebra)

     2. "Category Theory in Context" by Emily Riehl

   • 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.

An Introduction to Symmetric Functions*

• Symmetric functions might look like ordinary polynomials at first, but they hide a surprising world of structure. These functions stay the same when you shuffle their variables, and that simple rule leads to beautiful formulas for partitions, identities, and connections to representation theory. In this project, we’ll explore the main families: elementary, homogeneous, monomial, power, and Schur functions. We will learn how one translates between them. We’ll also see how generating functions can organize and simplify complicated counting problems. Stanley’s Enumerative Combinatorics, Vol. 2 will be our main guide. Some passages can feel dense, but the central concepts are very doable, and with guidance we’ll keep the focus on concrete examples and key insights rather than technical detail. Many of the perquisites can be picked up quickly if not already familiar. 

• Reference: "Enumerative Combinatorics, Vol. 2" by Richard P. Stanley

• Prerequisites: The background needed varies wildly and be approached in most forms, from scratch to all necessary perquisites. To get the most of this project an understanding of algebra and linear algebra is recommended but absolutely not necessary.

1. Category Theory**

   • Category Theory is increasingly becoming the language of abstract mathematics. Except for those working in analysis or some niche fields, no modern mathematician can really do without it! Depending on the reader's level, we will cover anything from the basics of categories and universal constructions, to functors and representability, to some content on simplicial sets.

   • References: 

     1. "Category Theory: A Gentle Introduction" by Peter Smith

     2. "Category Theory" by Steve Awodei

     3. "Category Theory In Context" by Emily Riehl

   • Prerequisites: Discrete mathematics or a proofs course. A first pass with some abstract algebra would be helpful but is not required.

2. Algebra from a Categorical Viewpoint**

   • Study groups, rings, and modules from a categorical perspective. Category theory is a (highly visual!) language for mathematics that clarifies constructions like products and quotients and formalizes analogies between different areas of math. If you like drawing diagrams involving a bunch of arrows, you might be interested!

   • Reference: "Algebra Chapter 0" by Paolo Aluffi

   • Prerequisites: Linear algebra and discrete mathematics or a proofs course. A first pass with some abstract algebra (group, rings, fields, ...) would be helpful, but it isn't necessary. This text is perfectly fine for a first course.

3. Homological Algebra

   • The study of sequences of modules. A module is an algebraic object analogous to a vector space except with an underlying ring rather than a field. Proofs in this subject are very fun -- it often feels like solving a puzzle! Definitely a more abstract and advanced topic, but very rewarding!

   • Reference: "An Introduction to Homological Algebra" by Joseph Rotman

   • Prerequisites: A first pass with some algebra is required. A first pass with some algebraic topology is helpful for motivation, but not required.

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 algebraic 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 E. Smith, Lauri Kahanpää, Pekka Kekäläinen, and William Traves

  3. (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. Mapping Class Groups of Surfaces

• • One useful and fun way to study a mathematical object is to study its symmetries. You might learn about dihedral groups, the groups of symmetries of polygons, as early examples of groups in an undergraduate abstract algebra course. Working mathematicians are similarly interested in groups of symmetries of shapes, but these shapes are squishy ones that come from topology. In this project you’ll learn about a family of groups of symmetries of orientable surfaces called mapping class groups. You’ll get to draw lots of pictures of curves on donuts and understand how to solve (and make!) puzzles like the ones here: https://aharalab.sakura.ne.jp/teruaki.html

  • References: 

    1. "Office Hours with a Geometric Group Theorist" edited by Matt Clay and Dan Margalit

    2. "A Primer on Mapping Class Groups" by Benson Farb and Dan Margalit

  • Prerequisites: One undergraduate course in topology would suffice; an undergraduate course in abstract algebra would be very nice but not strictly required; someone who has also taken a differential geometry course or has had some educational encounters with hyperbolic geometry might get even more out of the project.

1. Infinity and Foundations: An Introduction to Set Theory**

   • How big is infinity? Are some infinities larger than others? This project explores the strange but beautiful world of set theory — the foundation of modern mathematics. We’ll discuss countable vs. uncountable sets, Cantor’s diagonal argument, the Axiom of Choice, and the Continuum Hypothesis. Along the way you’ll see how axioms shape mathematics and get a taste of how set theorists handle paradoxes.

   • Reference: "Set Theory: An Open Introduction" by Tim Button

   • Prerequisites:

     • Minimal: A proof-based math course (Intro to Proofs, Foundations, or similar).

     • Helpful: Some exposure to basic logic or real analysis.

2. Logic Beyond the Classical: Systems and Semantics**

   • Classical logic is just one point in a larger landscape. In this project you’ll explore alternative systems — paraconsistent, intuitionistic, relevant, or modal logics — that break or modify classical rules. You’ll pick one or two systems you find interesting and study both their proofs and semantics (such as Kripke frames or many-valued models). If you’ve ever wondered how contradictions can be tolerated or necessity formalized, this is your chance to see the math bexhind it.

   • Reference: "An Introduction to Non-Classical Logic" by Graham Priest (2nd edition)

   • Prerequisites:

     • Minimal: Comfort with symbolic reasoning and a proof-based math course.

     • Helpful: Prior exposure to elementary logic (truth tables, natural deduction).

3. What Is Mathematics, Really? Perspectives in Philosophy of Mathematics**

   • Is mathematics discovered or invented? Do numbers exist independently of us? In this project we’ll read and discuss classic and contemporary texts on major positions in the philosophy of mathematics — realism (Platonism), formalism, and constructivism. You’ll see how these views influence mathematical practice and debates over infinity, truth, and proof.

   • References: 

     • "Thinking About Mathematics: The Philosophy of Mathematics" by Stewart Shapiro

     • "Philosophy of Mathematics: Selected Readings", edited by Paul Benacerraf and Hilary Putnam

   • Prerequisites:

     • Minimal: Interest in conceptual questions and at least one proof-based math course.

     • Helpful: Prior exposure to logic or history of mathematics.

4. Infinity Through the Ages: A Historical Journey*

• Infinity once terrified mathematicians — now it’s a cornerstone of modern math. This project traces the history of the infinite from ancient Greek paradoxes to Cantor’s transfinite numbers. We’ll read primary and secondary sources on how ideas like limits, infinitesimals, and cardinality evolved, and how philosophical and religious debates shaped their acceptance. You can also focus on a specific figure or episode of your interest (e.g. Newton vs. Berkeley on calculus or Cantor vs. Kronecker on transfinite numbers).

• References: 

1. "Uses of Infinity" by Leo Zippin

2. "Journey Through Genius" by William Dunham

• Prerequisites:

  • • Minimal: Interest in history of mathematics; at least one college math course.

    • Helpful: Familiarity with basic calculus or logic.

1. 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: "Conics and Cubics: A Concrete Introduction to Algebraic Curves" by Robert Bix

   • Prerequisites: Some calculus and linear algebra, familiarity with proofs would be useful. 

2. 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: "Ideals, Varieties, and Algorithms" by David A. Cox , John Little, and Donal O’Shea

   • Prerequisites: A proof based linear algebra course, familiarity with ring theory is useful, but not essential. 

3. 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:

    1. "Basic Algebraic Geometry I" by Igor R.Shafarevich

    2. "Algebraic Curves: An Introduction to Algebraic Geometry" by William Fullton

    3. "Undergraduate Algebraic Geometry" by Miles Reid

  • Prerequisites: Abstract algebra, a little bit of point set topology.

1. Introduction to Projective Geometry**

   • In Euclidean geometry, two lines in the plane generically meet at a unique point; however, this statement fails for parallel lines, which never meet. This inconsistency is resolved if we consider a larger space, the projective plane, which includes extra "points at infinity" where parallel lines intersect. It turns out adding these points of infinity provides just the right framework to make many statements in classical geometry better behaved. This project will introduce projective spaces from the perspective of linear algebra, with applications to classical geometry. Along the way, we will touch on related topics including quadratic forms, the exterior algebra, and Grassmannians.

   • Reference: "Lecture notes on projective geometry" by Thomas Baird

   • Prerequisites: Linear algebra

2. Introduction to Algebraic Curves

   • Algebraic curves in their simplest form arise as the solution sets of polynomial equations in two variables. Examples appear all over mathematics, from the study of the arithmetic of elliptic curves in number theory to the characterization of compact Riemann surfaces as complex algebraic curves in complex analysis. In this project, we will take a computational, example-driven approach to learning the tools to work with algebraic curves, ideally culminating with resolution of curve singularities or the Riemann-Roch Theorem.

   • Reference: "Algebraic Curves" by William Fulton

   • Prerequisites: Abstract algebra. Basic knowledge of topology is helpful but not necessary.

3. Introduction to 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 groups or Lie theory.

• Reference:  Varies based on background

• Prerequisites: Linear algebra. Abstract algebra and topology are helpful but not necessary.

1. Statistical Modeling of Electrostatic and Charge-State Dynamics in Respiratory Complex I

• How do proteins control the movement of electrons and protons with such precision? This project invites students to explore that question through statistical bioinformatics. Using Complex I—a massive enzyme central to cellular respiration—as a model system, students will analyze how amino acid charge states fluctuate and how these fluctuations shape the enzyme’s electrostatic environment and function.

The project combines computational data analysis with statistical modeling to study residue-level charge distributions derived from molecular dynamics (MD) simulations and structural bioinformatics data. Students will apply methods such as correlation analysis, clustering, and principal component analysis (PCA) to detect patterns in electrostatic variability across subunits and species (Thermus thermophilus, E. coli, Cyanobacteria, and Mus musculus).

Participants will construct statistical models to identify residues that most influence charge regulation and proton-coupled electron transfer. The broader goal is to develop a data-driven understanding of how electrostatic dynamics support biological energy conversion. Students will gain practical skills in data handling, visualization, and statistical interpretation while connecting mathematical reasoning to molecular mechanisms.

This project will interest students drawn to computational biology, biophysics, or data science. It provides an opportunity to apply mathematical tools to biological questions and see how quantitative analysis reveals physical insights into protein behavior.

• References: 

1. Hirst, J. (2013). Mitochondrial Complex I. Annu. Rev. Biochem. 82, 551–575.

2. Kaila, V. R. I. (2018). Long-range proton-coupled electron transfer in biological energy conversion. J. R. Soc. Interface, 15(138), 20170916.

3. Gunner, M. R., et al. (2020). Computational electrostatics and bioenergetics of proteins. BBA - Bioenergetics, 1861(4), 148216.

4. "Numerical Recipes: The Art of Scientific Computing" by Saul Teukolsky, William H. Press, and William T. Vetterling

• Prerequisites:

  • Minimal: Basic knowledge of statistics (mean, variance, correlation), introductory programming in Python or R, and basic molecular biology (proteins and amino acids). Familiarity with Jupyter notebooks or data plotting tools is sufficient.

  • Helpful: Experience with bioinformatics tools, molecular dynamics, physical chemistry, or statistical modeling (e.g., regression, PCA, clustering) will allow deeper engagement.

1. The Mathematics of Waves*

   • The idea of a 'wave' can refer to many things: the waves of the ocean breaking on the beach, the sound coming from a violin string, the light coming from the sun. Our goal will be to give a mathematical discussion of these phenomena, from deriving and solving explicit equations to exploring aspects such as conservation laws and shockwaves. These ideas, and the techhniques we will develop, arise throughout mathematics, physics, and engineering.

   • Reference: "An Introduction to the Mathematical Theory of Waves" by Roger Knobel

   • Prerequisites:  Calculus III. Some knowledge of differential equations would be helpful but not strictly required.

2. Differential Geometry**

• • "There is a bear in the wild. The bear walks a mile south, then a mile west, then a mile north, and ends up back where they started. What color is the bears fur?" Calculus is an extraordinarily powerful tool. One of its main applications in modern mathematics is to understand angles and curvature on different kinds of shapes. For instance, if you walk in a straight line on a sphere, you will end up back where you started, while if you walk on a straight line on a plane, you will keep getting further away. Calculus can be used to describe how these shapes are different. These ideas form the basis of Einstein's theory of general relativity, where massive objects change the shape of space.

  • References: 

    1. "Differential Geometry of Curves and Surfaces" by Manfredo do Carmo

    2. "Elementary Differential Geometry" by Barrett O'Neill

    3. "Visual Differential Geometry and Forms" by Tristan Needham

  • Prerequisites:  Calculus III and Linear Algebra.

1. 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.

1. The Math of Nonlinear Activations: Geometry, Stability, and Approximation in Neural Networks (Beginner- Medium)**

   • Neural networks get their expressive power from nonlinear activation functions. This project takes a concise, math-first look at standard activations (ReLU, Leaky-ReLU, tanh, sigmoid) as maps on R^n , focusing on regularity (piecewise-linear vs. smooth), Lipschitz behavior, and how these properties shape representation. You’ll connect these facts to concrete experiments you can see and measure.

     • Geometry (space warping): Visualize how each activation transforms space in R^2 (grids, circles -> bent shapes).

     • Stability (Lipschitz & gradients): Empirically track gradient norms, prove the basic Lipschitz/derivative bounds, and explain geometric effects.

     • Approximation: fit small 1-hidden-layer nets to polynomials/sinusoids/etc. and a 2D circle, and compare activations.

   • Reference: "Neural Networks and Deep Learning" by Charu C. Aggarwal. (We will mostly be look at chapter 1.5).

   • Prerequisites:  Calculus I–II, Linear algebra, Very basic Python/NumPy. Background understanding of neural networks is not required. 

2. Gauge-Equivariant Convolutional Networks (Advanced)

• • Convolutional neural networks (CNNs) are everywhere from phones, self-driving cars, medical imaging, manufacturing. They excel on images because they are translation-equivariant: shift the input, features shift the same way. On curved surfaces (e.g., spheres) or under rotations/reflections, it’s nontrivial to “align” filters.

This project introduces a gauge view of CNNs:

• • • a gauge = local choice of axes (frame),

    • feature fields = vectors attached to each point,

    • equivariance = features transform predictably under coordinate changes.

Goals for this project: 

• • • Read and summarize a short, coordinate-independent intro to equivariance and gauges.

    • Prove simple kernel constraints for symmetry groups  (translations + rotations). 

    • Run a small experiment (rotated digits or circle signals): compare a standard CNN vs. a gauge-equivariant version for robustness and sample efficiency.

This project helps students interested in machine learning connect theory to practice and build strong applied models. 

• • Reference: "Equivariant and Coordinate Independent Convolutional Networks: A Gauge Field Theory of Neural Networks" (https://maurice-weiler.gitlab.io/cnn_book/EquivariantAndCoordinateIndependentCNNs.pdf)

  • Prerequisites:  Linear algebra, a course in abstract algebra or knowledge of group theory, Multivariable Calculus, and basics of python. Exposure to basics of neural networks (feed-forward networks and CNNs) is preferred.