Lecture Notes

I have written various notes to help with my teaching. I still revise them from time to time, so in that sense they are not finished, but they are complete enough that I (and others) have used them to teach. 

Three Conjectures in Number Theory (version October 2024). 

These notes were used for a series of lectures at Università degli Studi della Campania "Luigi Vanvitelli" during October-November 2024. They cover Schanuel's conjecture, the (strong) exponential algebraic closedness conjecture, and the multiplicative Zilber—Pink conjecture (also known as the Conjecture on Intersection with Tori or CIT), as well as various results around these problems, such as Ax's theorem on the functional transcendence of the exponential function, results on exponential algebraic closedness, the multiplicative Manin—Mumford conjecture. The focus was kept entirely on complex exponentiation (instead of considering more general settings) so that this would simplify the presentation of the main ideas.

Introduction to Analysis (version October 2024)

Originally written for UC Berkeley's Math 104. They cover the basics of an analysis course over the reals, such as sequences, series, limits of functions, continuity, differentiability and Riemann integration. They also cover some more advanced topics like the construction of the reals, metric spaces, and improper integrals. 

Introduction to Complex Analysis (version October 2024)

Originally written for UC Berkeley's Math 185. They cover the basics of holomorphic functions, the Cauchy—Riemann equations, path integration, Cauchy's integral formula and its many consequences (maximum modulus principle, Schwarz's lemma, Rouché's theorem), the classification of isolated singularities, Cauchy's integral theorem and the Residue theorem. They also cover a few other topics, like Dirichlet series, the basics of Fourier analysis and infinite products. 

Introduction to Abstract Algebra (version February 2023)

Originally written for UC Berkeley's Math 113. They cover the basics of group theory will special emphasis on finite groups, like Lagrange's theorem and Sylow's theorem, group actions. Then we move briefly into rings (mostly commutative unital rings), and then into finite field extensions. We finish with a proof of the fundamental theorem of algebra.