Introduction to Representation Theory: Finite Groups (with Timothy Cho)
In representation theory, we examine how a group G acts on a vector space V (say over the complex numbers) as linear transformations. In this course, we will work with finite groups, which act on finite dimensional vector spaces. In this case, linear transformations correspond to square matrices, which we understand well. This view of finite groups is important both within math, and has applications outside of math (e.g., in chemistry). We begin with a crash course on tensor products of vector spaces, as well as basic notions in representation theory (irreducible representations and characters). Then, we will study character theory, and will use this to give a proof of Burnside's Theorem on solvable groups. Finally, if time permits, we will look at the connection between combinatorics (integer partitions) and the representation theory of the symmetric group.
Prerequisites: 120A, 121A (120B and/or 121B strongly preferred)
Introduction to Mathematical Control Theory (with Jaaziel de la Luz)
Control theory offers a rich landscape of structured problems in which Differential Equations and Dynamical Systems combine to answer questions in engineering. We will define controllability from a mathematical standpoint and use underlying ODE theory to analyze when a system can be controlled and optimized. Topics include: controllability, hysteresis, time-optimal control, Pontryagin Maximum Principle, Dynamic programming, and game theory. Some programming may be used to culminate with a final project. This DRP may continue into a second part during the spring.
Prerequisites: 3A, 3D, 13, 140AB, Programming in C++ or Python is encouraged.
Partial Differential Equations (with Anna Cusenza)
This is a continuing DRP from last spring. We will continue reading Evans PDE book, 2nd edition. We will start with the heat equation on page 44 and finish chapter 2. We will then continue to chapter 5.
Prerequisites: 140A, 13A, 2E. Evans Ch 2 up to the heat equation as well, for context.
An Intro to Algebraic Geometry (with Aurora Vogel)
Algebraic geometry can be thought of as the study of the geometry of spaces, called varieties, which are locally the zeroes of polynomials. For affine varieties, these look simply as the intersection and/or union of graphs of polynomials. It turns out, via theorems such as Hilbert Nullstellensatz, we have a sort of equivalence between algebra and geometry, and thus many of the geometric questions which we may ask about varieties we can answer using the tools of commutative algebra. This course will illustrate this while introducing the notion of varieties, functions on them, functions between them, as well as basic properties they possess such as irreducibility / dimension. In particular, we will cover the first portion of Shafarevich which includes the definition of affine varieties, regular and rational functions, quasi-projective varieties, maps and products between these varieties, and lastly dimension. If time permits we will cover the first sections of the "Local Properties" portion of the text.
Prerequisites: 120A,120B, and/or knowing the basics of commutative algebra
An Introduction to Category Theory (with Suhasiny Naik)
Category theory is a mathematical language that enables the transfer of ideas between different areas of math, as well as a fascinating abstract topic in its own right. In this reading, we will introduce categories and morphisms and learn how a categorical perspective can be useful in various contexts. Topics covered will include functors, natural transformations, equivalences of categories, limits and colimits, adjunctions, and the Yoneda lemma. Examples from various fields of math will be used throughout, and will be guided by students' particular interests.
Prerequisites: 120A