Teaching

Course description: This course introduces the students to academic mathematics. The big difference with high-school mathematics is its emphasis on proof. The student learns about logic and various forms of proof, such as the direct method, proof by contradiction and proof by complete induction. These concepts will be applied to various fields of mathematics, such as set theory and number theory. Along the way, the student becomes acquainted with the language and notations of mathematics.

The course highlights the main attraction mathematics has for its practitioners: the joy of solving a puzzle. Every proof contains a sparkle of ingenuity, and there is great intellectual satisfaction in discovering the essential step in a proof, or admiring the brilliance of someone who found it before you. A typical problem is for instance the question whether the square root of 2 is a fraction. The answer came as a great shock to the ancient Greeks and its proof is both simple and very clever.

Another feature of the course is an introduction to the mysteries and paradoxes of the concept ʽinfinityʼ. Are there more real numbers than integers? (Yes.) Is the set of fractions larger than the set of integers? (No.)

Finally, there is a big emphasis on writing proofs. A proof should be logical, clear and do precisely what it should: convince a reader of the truth of some mathematical statement. Writing good proofs is a difficult art, which requires practice and the highest intellectual precision.

Syllabus: syllabus_UCSIMAT14.pdf

UCSCIMAT11: Calculus and Linear Algebra

Course description: This introductory course focuses on basic concepts of calculus, starting with the functions of a single variable. First, we explore simple linear and nonlinear differential equations. Such equations are vital in explaining the dynamic behavior of many different systems in a wide variety of fields. This serves as a motivation to learn about calculus techniques, such as differentiation, integration, expansion in a small variable, and complex numbers.

Next, we learn to use powerful tools to study systems of many variables: linear vector spaces, linear operators (matrices) in such spaces, and key properties of matrices. We also extend techniques such as differentiation to functions of several variables and learn about their geometrical representation. The course concludes with various approaches to the optimization of functions of several variables.

The techniques we learn in this course have proven to be highly effective in a wealth of areas, as will be illustrated by examples in various fields. Some attention is paid to underlying mathematical foundations, but the focus is on understanding the methods and on learning to apply the techniques.

Syllabus: syllabus_UCSCIMAT11.pdf

Fall 2021

UCSCIMAT14: Foundations of Mathematics

Course description: This course introduces the students to academic mathematics. The big difference with high-school mathematics is its emphasis on proof. The student learns about logic and various forms of proof, such as the direct method, proof by contradiction and proof by complete induction. These concepts will be applied to various fields of mathematics, such as set theory and number theory. Along the way, the student becomes acquainted with the language and notations of mathematics.

The course highlights the main attraction mathematics has for its practitioners: the joy of solving a puzzle. Every proof contains a sparkle of ingenuity, and there is great intellectual satisfaction in discovering the essential step in a proof, or admiring the brilliance of someone who found it before you. A typical problem is for instance the question whether the square root of 2 is a fraction. The answer came as a great shock to the ancient Greeks and its proof is both simple and very clever.

Another feature of the course is an introduction to the mysteries and paradoxes of the concept ʽinfinityʼ. Are there more real numbers than integers? (Yes.) Is the set of fractions larger than the set of integers? (No.)

Finally, there is a big emphasis on writing proofs. A proof should be logical, clear and do precisely what it should: convince a reader of the truth of some mathematical statement. Writing good proofs is a difficult art, which requires practice and the highest intellectual precision.

Syllabus: syllabus_UCSIMAT14.pdf

Utrecht University

1st Period 2021

WISB102: Bewijzen in de Wiskunde (Proofs in Mathematics)

Course description: In the course "Proofs in Mathematics", the student is introduced to the abstract reasoning style in mathematics and practices different methods of proof. This subject is a compulsory subject for all mathematics students and provides essential prior knowledge for all other mathematics courses in the bachelor. The following topics are covered in this course: introduction to set theory; introduction to logic; proof methods; relations and functions; modular arithmetic; introduction to the analysis.

Syllabus: syllabus_WISB102.pdf

WISM102: Orientation on Mathematical Research

Course description: The aim of this first-year master course is to orient a student in contemporary mathematical research by doing projects in specialized research areas of their choice; possible specializations include algebraic geometry, number theory, differential geometry, algebraic topology, logic, scientific computing, applied analysis, probability, statistics and complex systems. The course has the following goals: students should be in contact with actual mathematical research; learn to work in groups; and practice written and oral presentation skills. This includes writing in latex and using literature and mathematical review systems.

Project supervised: How to construct an "exotic" group?

Project description: The study of growth of groups has a long history and goes back to Hilbert, Poincaré, and others. In 1968 it became apparent that all known classes of groups have either polynomial growth (e.g., abelian groups) or exponential growth (e.g., free groups). John Milnor was the first to ask whether groups of intermediate growth exist. This was answered in the positive by Grigorchuk in 1983, and since then there has been a lot of activity in the subject. The group constructed by Grigorchuk is a self-similar group acting on an in nite binary tree. Apparently, it is also the first solution to another classical problem of Burnside from 1902: Is there a finitely generated infinite torsion group? The goal of the project is to get familiar with self-similar groups and to study the paper by Grigorchuk and Pak about the properties of the Grigorchuk group. Depending on available time, we will also discuss other interesting examples of self-similar groups and open questions.

Syllabus: syllabus_WISM102.pdf

University of California, Los Angeles

Spring 2020

MATH 33B-2: Differential Equations

Course description: The purpose of MATH 33B is to provide an introduction to the basics of differential equations. We will cover first-order linear differential equations, second-order linear differential equations with constant coefficients, and linear systems of differential equations. We will also provide a review / summary of all necessary background from linear algebra.

Syllabus: syllabus_33B-2.pdf

Winter 2020

MATH 132H: Complex Analysis (Honors)

Course description: Math 132H provides a rigorous introduction to complex analysis. Unlike Math 132, this course has more emphasis on proofs and it is specifically designed for students who have strong commitment to pursue graduate studies in pure or applied mathematics. In particular, we will discuss Cauchy’s theorem, the residue theorem, the argument principle and their applications, as well as the Riemann mapping theorem. More specifically, we will cover Chapters 1, 2, Sections 3.1 - 3.6, and Sections 8.1 - 8.3 of the "Complex Analysis" textbook by Elias M. Stein and Rami Shakarchi.

Syllabus: syllabus_132H.pdf

MATH 33A-3: Linear Algebra and Applications

Course description: The purpose of Math 33A is to provide mathematicians, engineers, physical scientists, and economists with an introduction to the basic ideas of linear algebra in n-dimensional Euclidean spaces. More specifically, you will be introduced to systems of linear equations, matrix algebra, linear independence, subspaces, bases and dimension, orthogonality, least-squares methods, determinants, eigenvalues and eigenvectors, matrix diagonalization, and symmetric matrices.

Syllabus: syllabus_33A-3.pdf

Fall 2019

MATH 191: Introduction to Dynamical Systems

Course description: This is an introductory course on dynamical systems, which aims to give a glimpse at various aspects of topological dynamics, symbolic dynamics, complex dynamics, and some ergodic theory. With evident roots in physics, today dynamical systems have become very useful for both pure and applied mathematicians. For instance, Perron-Frobenius theorem is important in many areas, such as algorithms (e.g., Page Rank). We will discuss some fundamental examples in the field, including circle rotations, expanding maps, shifts and subshifts, and quadratic maps. The exact selection of topics is to be determined based on your background and interest, as well as the speed you progress through the new material.

Syllabus: syllabus_191.pdf

Spring 2019

MATH 33A-3: Linear Algebra and Applications

Course description: The purpose of Math 33A is to provide mathematicians, engineers, physical scientists, and economists with an introduction to the basic ideas of linear algebra in n-dimensional Euclidean spaces. More specifically, you will be introduced to systems of linear equations, matrix algebra, linear independence, subspaces, bases and dimension, orthogonality, least-squares methods, determinants, eigenvalues and eigenvectors, matrix diagonalization, and symmetric matrices.

Syllabus: syllabus_33A-3.pdf

Winter 2019

MATH 170B-1: Probability Theory

Course description: This course covers some more advanced topics in the fundamentals of probability theory. In particular, during our lectures we will have a closer look at transforms, conditioning, sums of random variables, and limit theorems. The second half of the course provides a fairly detailed introduction to Bernoulli, Poisson, and Markov random processes. The exact selection of topics is to be determined based on your initial knowledge of probability and the speed your progress through the new material. Click here for more course details.

Syllabus: syllabus_170B-1.pdf

MATH 170B-2: Probability Theory

Course description: This course covers some more advanced topics in the fundamentals of probability theory. In particular, during our lectures we will have a closer look at transforms, conditioning, sums of random variables, and limit theorems. The second half of the course provides a fairly detailed introduction to Bernoulli, Poisson, and Markov random processes. The exact selection of topics is to be determined based on your initial knowledge of probability and the speed your progress through the new material. Click here for more course details.

Syllabus: syllabus_170B-2.pdf

Fall 2018

MATH 115A-6: Linear Algebra

Course description: The course has two main goals. One is to rigorously establish the basics of linear algebra over a field. The other is to introduce students to the concept and practice of rigorous mathematical proof more broadly. The course material includes such topics as abstract vector spaces, linear transformations and matrices, determinants, eigenvector theory and diagonalization of matrices, inner product spaces. Click here for more course details.

Syllabus: syllabus_115A.pdf

Spring 2018

MATH 171-2: Stochastic Processes

Course description: The central topic of this course is stochastic processes and some of their applications, with focus on Markov chains, renewal processes, Poisson processes, and Brownian motion.We will also spend time on useful technical aspects underlying the theory, such as martingales.

Syllabus: syllabus_171.pdf

Winter 2018

MATH 117-1: Algebra for Applications

Course description: This course is an introduction to abstract algebra with a view towards applications. In particular, we will study division, the Euclidean algorithm, and factorization in the integers and polynomials rings over a field. We will also prove the theorems of Euler and Fermat, and the Chinese remainder theorem. Then we will look at applications, such as error correcting codes, fast polynomial multiplication, and the fast Fourier transform.

Syllabus: syllabus_117.pdf

MATH 171-2: Stochastic Processes

Course description: The central topic of this course is stochastic processes and some of their applications, with focus on Markov chains, renewal processes, Poisson processes, and Brownian motion.We will also spend time on useful technical aspects underlying the theory, such as martingales.

Syllabus: syllabus_171.pdf

Jacobs University Bremen

Fall 2016

10332: Discrete Mathematics

Course description: Discrete mathematics is a branch of mathematics that deals with discrete objects and naturally has many applications to computer science and cryptography. This course introduces the basics of the subject, in particular (enumerative) combinatorics and graph theory. The topics in enumerative combinatorics will include the binomial and multinomial coefficients, the pigeonhole principle, the inclusion-exclusion formula, generating functions, partitions, and Young diagrams. The topics in graph theory will include trees (spanning trees, enumeration of trees), cycles (Eulerian and Hamiltonian cycles), planar graphs (Kuratowski’s theorem), colorings, and matching (perfect matchings, Hall’s theorem). Additional topics may be chosen depending on interests of the instructor and students (for example, error-correcting codes and algebraic graph theory).