# Lectures

• In this lecture we motivate the definition of a smooth manifold as an "ideal", yet generalized space to do calculus on. The majority of notions from topology are built up from the basics in the beginning of the lecture, and the remainder deals with the notions of topological and smooth manifolds. There is an addendum on the definition of continuity of functions.

• In this lecture we motivate and discuss the tangent and cotangent bundles of a manifold. A little knowledge of linear algebra (familiarity with vector spaces, matrices, and bases) is beneficial but not necessarily required. First the tangent bundle and derivatives of smooth functions is discussed, finishing off with vector fields. The rest of the lecture is devoted to discussion of the cotangent bundle, culminating with a complete calculation of the de Rham cohomology of the circle using the concepts developed throughout the lecture. Proofs are minimal as stress is placed on conceptual understanding.

• In this lecture we introduce the main notation and structure of the untyped lambda calculus, finishing with a brief discussion of combinatory calculus. Historical details are given throughout.

• A broad yet detailed introduction to the basic structure and function of the foreign exchange markets through real-world examples and well-known models in quantiative finance. Topics include the microstructure of foreign exchange, currency forwards, futures, swaps, and options. Essential components of quantiative finance such as risk-neutral pricing and the Black-Scholes model are introduced and applied in the context of foreign exchange and risk management.

• Motivating the study of combinatorics by introducing common examples in those courses with a somewhat alternative view.

• In this lecture, we will be discussing (definite) integration on the real numbers. We will not so much be interested in how one can apply the definite integral, but rather, how one should go about defining it in the first place. As is common in math, multiple definitions can capture the essence of the object at hand, and the definite integral is no different in that regard. We start with Riemann's theory up to the elegant and powerful theory of the gauge integral.

• In this first look at topology, we discuss how commutative diagrams provide an interesting setting to construct some of the most remarkable examples of topology we can put on a set. This first lecture on topology introduces categories and basic general topology, setting the stage for algebraic topology which will be taken up in a second lecture.

• Following up on the lecture on integration theory over R, this lecture explains how to generalize integration to other spaces. We use measure theory to construct and discuss the Lebesgue formulation of the integral. We take a short detour through Banach space theory before using integration to define probability. We derive important results in the theory of Lebesgue integration (such as the coveted dominated convergence theorem). The talk finishes with two important theorems of probability.

• This lecture discusses the geometrization of the natural ideas of analysis. Through differential forms, we explore what it means to do calculus without coordinates. Using this newfound language, we state and prove Gromov's non-squeezing theorem.