École Polytechnique Féraletedérale de Lausanne, Switzerland
Differential Privacy and Hypothesis Testing
Differential privacy (DP) is a widely studied worst-case privacy measure which provides strong guarantees on the ability to distinguish between two neighboring databases. It has emerged in recent years as a go-to measure of privacy and has been widely adopted in industry and beyond. In this mini-course we begin with an introduction to DP, as well as an overview basic DP mechanisms and properties. We then shift gears and look at the theory of hypothesis testing and its applications to the study of statistical properties of DP. Time permitting, we end with some advanced topics such as, for example, Gaussian-DP and central-limit-like behavior of DP-mechanisms.
Technische Universität Darmstadt, Germany
Algorithms and Combinatorics through the Lens of Graphs and Their Symmetries
In this course, we take a guided tour through the landscape of algorithms, combinatorics, and computational complexity. Using questions about symmetry and graph isomorphism as a guiding thread, we explore a range of fundamental concepts and techniques.
The course is divided into five lessons, each centered around a key question:
What are symmetries of graphs?
How can we verify algorithmic solutions?
What does graph isomorphism have to do with symmetry?
How hard are symmetry-related computational problems?
When can we compute symmetries efficiently?
Along the way, we encounter groups as the mathematical language of symmetry, computational models and their robustness, complexity-theoretic tools for understanding the difficulty of problems, and structural graph concepts that help tame combinatorial complexity. We also discuss certification as a means of gaining confidence in algorithmic results.
Technion - IIT, Haifa, Israel
Information Theory
Information theory quantifies, in various ways, the amount of information that some "object" conveys. The primary example is communication between two or more parties, but it is so fundamental that it appears in many other contexts as well. We shall start with the basics and describe applications in computer science and mathematics. Along the way, we will introduce key concepts such as entropy and mutual information, which form the backbone of the many applications. Participants will also gain intuition for how information-theoretic ideas can help in understanding complex systems.
Justus-Liebig-Universität, Gießen, Deutschland
Introduction to martingales in discrete time, theory and applications
The two basic structural properties of stochastic processes are the Markov property and the martingale property. Roughly speaking, a stochastic process (X_n, n∈N) is a martingale if the conditional expectation given the entire past equals the current state, i.e.
E [ X_n | X_{n−1}, ..., X_1] = X_{n−1}.
Such processes are often interpreted as “fair games”, which also motivates their role in mathematical finance.
In this course, we introduce the definition and basic properties of discrete-time martingales, with a strong emphasis on intuition and examples. We discuss fundamental results such as the optional stopping principle and illustrate how martingales can be used to analyze stochastic systems.
As a main application, we explore the connection between martingales and the absence of arbitrage in discrete-time financial markets, highlighting the role of martingale measures in pricing.