Courses
In this page, I will introduce the content of the postgraduate courses that I have finished during my master's.
2024 Spring
Master Project Mathematics (UvA) - 36EC
Supervisors: Dr. Ross Kang (UvA) and Prof. Dr. Daniel Dadush (CWI, Amsterdam and Utrecht University)
Coding Theory (Mastermath and DIAMANT) - 8EC
Lecturer: Prof. Dr. Alberto Ravagnani (Eindhoven University of Technology)
Course Content: In the context of digital communication, error-correcting codes are mathematical objects that correct errors in noisy and lossy channels. They find several applications in satellite communications, digital audio systems, wireless communications, error correction chips, deep space exploration probes, flash memories, networking, data storage, and in many other contexts.
This course offers an introduction to the mathematical theory of error correction (encoding and decoding information using mathematics). The main aims are:
to become familiar with the mathematical methods used to protect digital information from noise, damage, loss, and jamming attacks;
to become familiar with the mathematical structure of error-correcting objects;
to understand the connections between coding theory and other branches of mathematics (algebra, combinatorics, probability).
Tentative list of topics: communication channels, the Hamming distance, error-correcting codes, bounds, Reed-Solomon codes, Reed-Muller codes, decoding, duality theory of codes, locality, DNA data storage, rank-metric codes.
Queueing Theory (Mastermath and LNMB) - 6EC
Lecturers: Dr. Jacques Resing (Eindhoven University of Technology) and Dr. Rene Bekker (Vrije Universiteit Amsterdam)
Course Content: To provide insight in the theory of queueing models. The following subjects will be treated:
Fundamental queueing relations (Little's law, PASTA property)
Markovian queues (M/M/1 queue, M/M/c queue, M/E_r/1 queue)
M/G/1 queue and G/M/1 queue
Mean value technique
Priority queues
Variations of the M/G/1 queue
Insensitive queues (M/G/c/c queue and M/G/infinity queue)
2023 Fall
Algorithmic Game Theory (UvA - Master of Logic) - 6EC
Grade: 9.5 / 10.0
Lecturer: Prof. Dr. Guido Schaefer (CWI, Amsterdam and UvA)
Course Content: Algorithmic game theory (AGT) is a research area that lies in the intersection of theoretical computer science, discrete mathematics and economic theory. The area builds upon game-theoretic foundations to study situations of strategic decision making, with a particular focus on computational and algorithmic aspects. It combines methodologies and techniques from several disciplines such as complexity theory, algorithm design, discrete and continuous optimization, auction and mechanism design, online decision making and learning, etc.
The overall aim of the course is to get familiar with fundamental methodologies, techniques and results in AGT. We study the existence, computation and inefficiency of equilibria (ranging from pure Nash equilibria to learning outcomes) for various classes of games. We also consider the problem of designing coordination mechanisms to reduce the inefficiency. Further, we touch upon the computational limitations that arise in classic auction theory, and learn about techniques to develop simple and efficient algorithms for fundamental mechanism design problems. Throughout the course, we will build up a toolbox of models and techniques to study the impact of strategic decision making in various settings.
Graph Polynomials and Algorithms (UvA) - 6EC
Grade: 8.0 / 10.0
Lecturer: Dr. Guus Regts (UvA)
Course Content: Graph polynomials and partition functions play an important role in statistical physics and combinatorics. In this course we will introduce several of them such as the independence polynomial, the Tutte polynomial, the matching polynomial and several others. The main focus of the course will be on algorithmic techniques for (approximately) computing evaluations of these polynomials. We consider exact algorithms, based on determinants, for the dimer problem on planar graphs and for enumerating spanning trees. We consider three algorithmic approaches for approximate computations: one through rapid mixing of Markov Chains, one based on decay correlation decay (if time permits) and one more recent approach based on the locations of the complex roots of the polynomial.
Probabilistic and Extremal Combinatorics (Mastermath and DIAMANT/STAR) - 8EC
Grade: 8.0 / 10.0
Lecturers: Dr. Carla Groenland (TU Delft) and Dr. Ross Kang (UvA)
Course Content: This course covers selected topics and techniques in Ramsey theory, extremal set theory, random graphs, structural and extremal graph theory. Central to the course are applications of the "probabilistic method", a method that was championed by Erdös in the twentieth century, and by now has become standard in the modern mathematical repertoire. The principle is simple: to guarantee the existence of an object with some desired properties, it suffices to set up a probability space over the objects and show there's a chance the property holds. In spite of its simple and tautological nature, this has proven surprisingly powerful and versatile, with impact spanning mathematics, computer science, engineering, and networks, yielding many results that seemingly have nothing to do with probability. We focus on results in combinatorics, covering Ramsey's theorem, Turán's theorem, the Bollobás–Thomason threshold theorem, Dilworth's theorem, the Erdös–Ko–Rado theorem, and Thomassen's theorem. Although combinatorics has long been associated with a "problem-solving" style of mathematics, in this course we thread through underlying ideas and intuition, especially the link between combinatorics and algorithms, appearance of phase transitions, and relationship between local and global structure.
Stochastic Gradient Techniques in Optimization and Learning (Mastermath and LNMB) - 6EC
Grade: 9.0 / 10.0
Lecturers: Prof. Dr. Bernd Heidergott (Vrije Universiteit Amsterdam) and Dr. Ad Ridder (Vrije Universiteit Amsterdam)
Course Content: Adaptive algorithms are now utilized across varied applications in simulation/data-driven learning and optimization. These algorithms aim to estimate recursively an unknown time-invariant (or slowly varying) parameter vector, traditionally denoted by θ, from measurements whose statistics depend on it. A well-known example is the gradient descent method which is one of the main techniques in AI and machine learning for supervised and unsupervised learning. Such algorithms are subsumed under the name stochastic approximation (SA).
The focus of this course is on SA algorithms and their application to stochastic simulation/data-based optimization and learning algorithms. The ODE approach to iterative learning and optimization algorithms will be introduced. In this course, we will discuss the projection method and algorithms with biased gradient estimators. Next to discussing various SA algorithms, this course provides the theoretical insights needed for carrying out a proper statistical output analysis of SA-type algorithms. Optimal computer budget allocation in SA will be taught as well (batching updates is often not advisable). Applications will stem from a wide range of stochastic models.
The course is of interest to students in the area of Simulation Analytics, Computer Science, Operations Research, and Machine Learning. It offers a self-contained course on simulation-based techniques in optimization and (machine) learning. Participating students have majors in disciplines of applied mathematics, computer science, data science, operations research, physics, electrical engineering, and economics.
Professional Skills - Science Communication (UvA) - 1.5EC
Lecturer: Dr. Nicos Starreveld (UvA)
Course Content: How would you explain your academic research to your neighbours at a party? How can you make laypeople understand, or even be interested in science?
Science communication aims to bridge the gap between academia and the outside world. Science communicators want to explain, spark interest or even convince their audience of the relevance of scientific research.
In this course, you will gain an understanding of the Science communication principles and write a popular scientific article. You will learn to write in a clear, understandable way about a research topic you are interested in. In this course we will focus on making a clear reader’s profile, how to apply writing techniques and narrative forms to make your explanations more accessible and appealing, read each other’s work and exchange feedback.
Professional Skills - Science Communication (UvA) - 1.5EC
Lecturer: Prof. Dr. Noushine Shahidzadeh (UvA)
Course Content: This course is designed for anyone who wants to improve his/her/their communication skills. This course is designed to teach you how to communicate and cooperate effectively and professionally, to be more confident, and become a more influential speaker, both in private and professional life.
The content of the course consists of A) going through the PCM (Process Communication Model) model which allows you to understand the depth of personality structure and to combine behavioral analysis and typology of personality, along with adaptive communication techniques. B) learning and implementing tricks for effective spoken communication; the importance of body language combined with non-verbal communication. C) learning about confident and charismatic speakers D) learning to adapt the way you communicate in conflict situations.
2023 Spring
Algebraic Methods in Combinatorics (Mastermath and DIAMANT) - 8EC
Grade: 7.0 / 10.0
Lecturers: Dr. Guus Regts (UvA) and Dr. Krystol Guo (UvA)
Course Content: The application of algebraic methods in combinatorics has been remarkably successful in recent years. Examining combinatorial problems from an algebraic perspective also yields connections to combinatorial geometry, probability theory and theoretical computer science. In this course we study some of the most important of these methods and their applications. The two main themes are spectral graph theory and the polynomial method.
Spectral graph theory reveals fundamental information about a graph from its spectrum, i.e., the eigenvalues of the graph's adjacency matrix (or sometimes its Laplacian). We will study three important notions in graph theory from the spectral perspective: graph expansion and the Cheeger inequality, quasirandomness, and graph partitioning. Graph expansion plays an important role in the analysis of random walks which in turn leads to fast algorithms in computer science. Quasirandomness is a measure of how random-like a graph is (a fundamental concept in extremal combinatorics which is at the heart of Szemeredi's regularity lemma for example).
The polynomial method is a relatively recent innovation in combinatorics borrowing some of the philosophy of algebraic geometry. We cover Hilbert's Nullstellensatz and Alon's combinatorial Nullstellensatz and apply these to problems in additive number theory and graph colouring. We treat some very recent applications of the polynomial method to cap set problems. We will also look at stable polynomials to give a construction of Ramanujan graphs (an important family of expander graphs).
Computational Complexity (UvA - Master of Logic) - 6EC
Grade: 8.5 / 10.0
Lecturer: Dr. Ronald de Haan (UvA)
Course Content: Computational complexity theory deals with the fundamental question of how many resources (such as time, memory, communication, randomness, etc.) are needed to perform a computational task. A fundamental open problem in the area is the well-known P versus NP problem, one of the Clay Millennium problems. In this course we will treat the basics of complexity theory, including: NP-completeness, diagonalization, non-deterministic computation, alternation, Boolean circuits, randomized computation, approximation algorithms, subexponential-time complexity.
Writing in the Mathematical Sciences (UvA) - 3EC
Grade: 9.0 / 10.0
Lecturer: Prof. Dr. Jo Ellis-Monaghan (UvA)
Course Content: This workshop-style course shares a 10-step framework for writing papers in the mathematical sciences. The steps begin with articulating the initial idea and literature search, and progress through to the publication process. The course covers technical points of mathematical writing, creating a concept map to organize content into a good narrative flow, and using story-telling strategies to engage readers, as well as tips for time management and collaborative writing. We will also discuss identifying publication venues and what happens after submitting a paper to a peer-reviewed journal. Peer review among students is a component of the course.
While the primary focus is on writing research papers, theses, and dissertations, the general process also applies to grant proposals, expository articles, and pedagogical papers. This is a 'hands on' course, and participants are encouraged to bring any writing projects they may have, at any stage of development, to work on during the course.
Academic Year 2022/2023
Master Seminar in Discrete Mathematics and Quantum Information (UvA) - 6EC
Coordinator: Dr. Guus Regts (UvA)
Course Content: The Master Seminar consists of lectures on a broad range of research topics in Discrete Mathematics and Quantum Information. There will be four types of lectures:
by the participating first-year students on research literature. Students will have a choice from a list, and can propose their own subject;
by staff members and PhD students on topics closely related to their research;
by alumni no longer working in academia on their career and their work;
by second-year students on their ongoing master thesis projects.
The students will have the opportunity to suggest topics and speakers to invite at the Master Seminar.
For the first semester we will go through (a few chapters of) the book Analysis of Boolean functions by Ryan O' Donnel. For the second semester we have compiled a list of topics that will give you a feel for the richness of the fields of Discrete Mathematics, Quantum Information and their interactions.
Second Trisemester 2022/2023
Networks and Semidefinite Programming (LNMB PhD Course) - 4EC
Lecturers: Prof. Dr. Monique Laurent (CWI, Amsterdam and Tilburg University) and Dr. Sven Polak (CWI, Amsterdam)
Textbook: A Course on Combinatorial Optimization by Alexander Schrijver and the Lecture Notes: Networks and Semidefinite Programming.
Course Content: Combinatorial optimization problems are concerned with the efficient allocation of limited resources to meet desired objectives when the values of the variables are restricted to be integral. Such problems arise in various applications, e.g., airline crew scheduling, manufacturing, network design, cellular telephone frequency design, and they can often be modeled as optimization problems on graphs. The course deals with several basic combinatorial optimization problems. While these problems are intrinsically hard to solve in general, we will present polynomial-time solvable instances. Algorithms use combinatorial tools, linear and semidefinite programming. The following subjects are discussed:
- problems, algorithms and running time; basics of semidefinite programming;
- cliques, cocliques and colouring in graphs; Lovász theta number;
- maximum cuts in graphs; Goemans and Williamson approximation algorithm.
2022 Fall
Continuous Optimization (Mastermath and LNMB/4TU) - 6EC
Grade: 9.0 / 10.0
Lecturer: Dr. Daniel Dadush (CWI, Amsterdam)
Textbook: Convex Optimization by Stephen Boyd and Lieven Vandenberghe
Course Content: Continuous optimization is the branch of optimization where we optimize a (differentiable) function over continuous (as opposed to discrete) variables. Here the variables can be constrained by (differentiable) equality and inequality constraints as well as by convex cone constraints. Optimization problems like this occur naturally and commonly in science and engineering and also occur as relaxations of discrete optimization problems. Differentiability of the functions defining the problems allows for the use of multivariable calculus and linear algebra techniques to study the problems and to design and analyze efficient algorithms.This course aims to provide a concise introduction into the basics of continuous unconstrained, constrained and conic optimization.
Discrete Optimization (Mastermath and LNMB/4TU) - 6EC
Grade: 9.0 / 10.0
Lecturer: Prof. Dr. Marc Uetz (University of Twente)
Course Content: Discrete Optimization addresses structural and algorithmic questions for finding the "best" among a set of feasible solutions. Often, this set of feasible solutions is finite, such as the set of maximal matchings in a finite graph, the set of vertices of a polytope, etc. Only since around the 1960s, starting with groundbreaking work of Jack Edmonds, researchers started to realize that the quality of procedures to solve such problems should be measured in terms of the algebraic dependence of computation time on problems size. Discrete Optimization has since then evolved into a rich mathematical area that connects to many other areas in mathematics but also computer science. Throughout the course, we consider several fundamental problems from this area and develop efficient algorithms to solve them.Specifically, we cover the following topics: Shortest Path Algorithms, Spanning Trees, Matroids and the Greedy Algorithm, Maximum Flows & Minimum Cuts, Minimum Cost Flows, Matchings in Graphs, and Matroid Intersection, Integer Linear Programming & Total Unimodularity, Basic Computational Complexity (P vs. NP), Approximation Algorithms, Inapproximability & Approximation Schemes.
Stochastic Networks (UvA) - 6EC
Grade: 8.0 / 10.0
Lecturers: Prof. Dr. Michel Mandjes (UvA) and Prof. Dr. Rudesindo Núñez Queija (UvA)
Textbook: Lecture Notes on Stochastic Networks by Frank Kelly and Elena Yudovina
Course Content: Stochastic Networks theory is a subdomain of Applied Probability. While elementary Queueing Theory studies single service systems, the theory of Stochastic Networks is concerned with the interaction of networks of queueing and service systems and the impact thereof on the performance of the entire network. As the name suggests, the main tool in the understanding of Stochastic Networks are Stochastic Processes, and in particular Markov processes. The aim of the course is to discuss in detail several topics on Stochastic Networks that have been subject of much research since the late 1970s. The course will start with a brief look back at the theory of Markov Chains and elementary queueing theory and then covers migration processes, Little's law, reversibility and loss networks, decentralized optimization in networks (with applications to electrical and road traffic networks), random access networks (capacity of distributed protocols using Foster-Lyapunov criteria), effective bandwidth of multiplexed traffic streams (large deviations), distributed traffic control for small-scale dynamics in networks, large-scale dynamics and bandwidth sharing networks. If time permits we will briefly touch on fluid and diffusion limits in stochastic networks.
Most of my courses are provided by Mastermath (Dutch Master’s degree programme in Mathematics), UvA (University of Amsterdam), LNMB (Dutch Network on the Mathematics of Operations Research), DIAMANT (Discrete, Interactive and Algorithmic Mathematics, Algebra and Number Theory), STAR (Stochastics - Theoretical and Applied Research) and 4TU (The four universities of technology in the Netherlands), I appreciate them so much for the high-quality courses.