Differential Equations and Dynamical Systems (Part I) (with Jaaziel Lopez de la Luz and Clara Park)
This directed reading explores the compelling world of dynamical systems through linear and nonlinear ordinary differential equations (ODEs). We’ll start by building a strong foundation in the essential theory—covering topics like existence and uniqueness, stability, bifurcations, and qualitative analysis. As we progress, we’ll dive deeper into nonlinear dynamics, exploring concepts like global existence, Poincaré maps, stable and unstable manifolds, and the Poincaré-Bendixson theorem.
Prerequisites: 3A, 3D, 13
A Combinatorial Introduction to Topology (with Sasha Kononova)
This is an intro-level course in combinatorial, or algebraic, topology. There will be a gentle introduction to ideas of compactness, connectedness, and point-set topology. We'll use combinatorial methods to prove Sperner's Lemma, the Brouwer Fixed Point Theorem, Poincare Index Theorem, Alexander's Lemma, and the Jordan Curve Theorem. We will also cover the topics of plane homology, surfaces and their classification, Betti numbers and Euler characteristic, map coloring, and orientability.
Prerequisites: 2A, 13
The Method of Forcing (with Brian Ransom)
Godel's first incompleteness theorem gives the startling result that reasonable axiomatic theories cannot prove every statement to be true or false. Such statements are said to be ``independent" of the axioms of the theory. This may seem to have concerning implications for the foundations of mathematics, but it also presents the opportunity for an exciting direction of research: how can we tell which statements are independent of set theory?
Set theory is a foundational theory for mathematics, so its axioms (the ZFC axioms) can be regarded as the standard assumptions made for (almost) all mathematics. In this sense, statements that are independent of the ZFC axioms explore properties that aren't fully described by the standard assumptions of mathematics.
In 1966, Paul Cohen won the Fields Medal for pioneering the method of forcing, which is an expansive tool to describe and explore new possible ``worlds" of set theory. The ZF(C) axioms are true in these worlds, and forcing can be used to control whether or not certain other statements are also true. To formally justify that a statement is independent of ZF(C), forcing can be used to produce both worlds in which the statement is true and worlds in which it is false.
The goal of this DRP is to develop intuition for what the method of forcing is and why it works. We will also learn how to use it to clearly see why certain statements like the Continuum Hypothesis are independent of the ZFC axioms.
Prerequisites: at least four upper division courses
Random Matrix Theory (with Yiyun He)
Random matrix is a common random structure which is getting more and more important with its application in the neural network and machine learning. It mainly studies the spectrum behavior of the matrix with random entries. In this reading program, we will cover the most classical and beautiful results in random matrix theory and explore the probability idea behind it.
Prerequisites: 130B (147 suggested)
Cardinal Arithmetic (with Julian Eshkol)
The natural numbers are equipped with an intuitive arithmetic that is compatible with their ability to describe the cardinality of finite sets. For example 3+2=5, and indeed, a set of 3 apples together with a set of 2 apples makes 5 apples total.
But infinite sets have cardinalities too, and surprisingly, their associated infinite cardinals can be endowed with arithmetic operations that generalize the ordinary operations on natural numbers. This makes precise ideas like "infinity plus infinity".
Cardinal arithmetic turns out to be closely related to other foundational topics in mathematics, such as the Axiom of Choice and the Continuum Hypothesis.
Prerequisites: 13 (140A suggested)
An Introduction to Algebraic Geometry (with Aurora Vogel)
The course will be a basic introduction to algebraic geometry. Algebraic geometry is a beautiful subject which demonstrates deep connections between algebra and geometry. The objects of study are varieties which locally look the like zero loci of polynomials. It turns out that that the algebraic properties associated to these polynomials and the geometry of the variety locally are highly connected. In order to understand these connections we will cover topics such as:
Affine, Projective, and Quasi-Projective varieties
Zariski topology, Hilbert's Nullstellensatz, and the spectrum of a ring,
Classical constructions such as the Hilbert function of a variety.
We will closely follow Chapters 1-5 of Smith et al.
Prerequisites: 120A, 120B, 121A
Introduction to Partial Differential Equations (with Anna Cusenza)
Intro to partial differential equations: wave, heat, and Laplace equations.
Properties of harmonic functions: the maximum principle, the mean value property, etc.
Prerequisites: 2E, 140A, (140B suggested)
Introduction to Homology (with Suhasiny Naik)
In this DRP, we will work through Allen Hatcher's Algebraic Topology, focusing on chapter 2: Homology, including simplicial, singular, and cellular homology. Topics include exact sequences, chain complexes, simplicial complexes, CW complexes, and Mayer-Vietoris sequences. Category theory will be introduced, and used to describe the formal feautures of these constructions. Based on time and student interest, cohomology and homotopy theory can be covered. The structure of the course will be centered around weekly student-led presentations and problem solving sessions.
Prerequisites: 13, 120A