Many problems that mathematicians and physicists encounter in their work require complicated calculations. However, sometimes this tiresome work can be simplified with the use of the proper methods. During the lecture, I will present several techniques that often allow to quickly obtain the answers we are interested in, along with examples of their use in mathematics, physics, and everyday life. In particular we will look at such concepts like symmetry or scaling and at methods such as the dimensional analysis or the use of various diagrams to present problems in a more clear way. We will also try to understand why some approximation techniques are able to give results so close to the actual value.

Filip Ficek is a prolific young researcher working in the Gravitational Physics Group of the University of Vienna. He combines physical insights, numerical experiments, and sophisticated mathematical methods to a wide array of problems, including the study of supercritical nonlinear Schrödinger equations, the behavior of fields and fluids in blackhole spacetimes, and beyond standard models in atomic systems. 

We will discuss several counterintuitive facts (»paradoxes«) that arise when studying even rather simple random phenomena.

Martin Hairer is a professor of mathematics at École polytechnique fédérale de Lausanne and at Imperial College London. He is one of the pioneers of the field of stochastic partial differential equations. For his landmark contributions, Martin has been awarded the Fields Medal in 2014 and the Breakthrough Prize in Mathematics in 2021. He has a great affinity with music and has created the audio editing software Amadeus.

Let μ > 0. We will study sequences obtained by iterating the quadratic map

x → μ⋅x⋅(1-x).

We will discover fixed points, periodic sequences, and chaotic dynamics. You may find some illustrations under this link. A small prerequisite to follow this talk is the intuitive intermediate value intermediate value theorem: A continuous function f : [a, b] → R takes on every value between f(a) and f(b).

Herbert Koch is a professor of mathematics at the University of Bonn. His research focuses on partial differential equations and their applications to physical problems. By himself and with collaborators, he has made seminal contributions to the theory of the Navier-Stokes equations, i.e., the equations that govern the motion of fluids.

Consider the question of how many shuffles one needs to thoroughly shuffle a deck of cards. We will translate this question into a mathematics problem, and then solve it by introducing the concepts of Markov chains and mixing times. Time permitting, I will mention a few other situations (in computer science, physics, etc.) where these concepts play a key role.  

Mykhaylo Shkolnikov is an associate professor at Princeton University, where he applies the theory of stochastic partial differential equations to solve problems in financial mathematics, mathematical physics, and even neuroscience. Mykhaylo was born in Kharkiv, where he attended the Kharkiv Physics and Mathematics Lyceum No. 27 as well as the math circle «Evrika». Later, he moved to Germany with his family, and then to the United States to work on his PhD thesis at Stanford University.

One of the most illuminating examples of chaotic dynamical systems is Arnold's cat map. It was introduced by Vladimir Arnold about 60 years ago. You can read about it under this link. In this lecture, we will see how to use so-called Markov partitions to encode the orbits of the cat map, to find an exact formula for the number of periodic orbits of a given period, and to compute the entropy of the cat map, which is a measure of its chaos. These techniques turn out to apply to a large class of what is called hyperbolic systems. 

Vadim Kaloshin is professor at the Institute of Science and Technology Austria. He has made spectacular contributions in the fields of dynamical systems and celestial mechanics. Recently, he has settled the so-called local version of conjecture due to George Birkhoff on the shape of integrable, strictly convex billiard tables. This conjecture had stood for almost a century! Vadim was born and raised in Kharkiv, where he attended Kharkiv Physics and Mathematics Lyceum No. 27. 

The lecture will start by discussing the popular card game SET and in particular the question of how many cards one can have in this game without creating a so-called «SET». Considering this question for extended versions of the game, we will see a connection to a recent breakthrough result of Ellenberg and Gijswijt on a famous problem in the area of extremal combinatorics. The talk will also discuss related results in extremal combinatorics, some of which are of a geometric flavor.

Lisa Sauermann is professor of mathematics at Rheinische Friedrich-Wilhelms-Universität Bonn. She is a leading expert on extremal and probabolistic combinatorics. After her undergraduate studies at Bonn, she moved to the United States, where she obtained her PhD from Stanford University. After prestigious postdoctoral positions at Stanford, the Institute of Advanced Study, and the Massachusetts Institute of Technology, she has now returned to her alma mater. Lisa is also a legend of high school science competitions, winning 4 gold medals — once as the only contestant obtaining the highest score — and a silver medal at the International Mathematical Olympiad.

In this lecture, you will learn about what is called »continuous optimization« and how it is applied in machine learning. We will see a supervised learning problem based on continuous optimization in action. It will be helpful if you have some experience in computer programming, but that's not a requirement to follow and hopefully enjoy my lecture.

Yurii Malitsky is currently assistant professor at the Faculty of Mathematics at the University of Vienna and a leading researcher in the field of continuous optimization and its applications to machine learning. He finished his bachelor, master, and doctoral studies in applied mathematics at Taras Shevchenko Kyiv National University and is a graduate of the Ukrainian Physics and Mathematics Lyceum in Kyiv.

In the lecture, I will give a gentle introduction to a modern and fascinating branch of dynamical systems called complex dynamics. This branch studies iterations of functions over complex numbers, and this is where some of the most famous and beautiful fractals, such as the Mandelbrot set, appear. We will explore a connection of this field to numerical root finding methods, and also to Kharkiv. Finally, I will describe some of the active research directions in complex dynamics. There will be a lot of pictures, and the lecture will not require knowing complex numbers. However, you will understand things better if you know them. 

Kostiantyn Drach is an outstanding young researcher who works in the fields of dynamical systems, geometric inequalities, and topological data analysis. He is currently a postdoc at the Institute of Science and Technology Austria. Starting in September, Kostya will be an assistant professor at the Universitat de Barcelona in Spain. 

Waves of all sorts permeate our world: light (electromagnetic waves), sound (acoustic waves) and mechanical vibrations. Quantum mechanics revealed that, at the atomic level, all matter has a wavelike character, and, very recently, classical gravitational waves have also been detected. Simultaneously, at the cutting edge of today’s science it has become possible to map a material atom-by-atom and to manipulate individual atoms, providing us with precise measurements of a world that exhibits myriad irregularities—dimensional, structural, orientational and geometric—simultaneously. For waves, such disorder changes everything. In complex, irregular or random media, waves frequently exhibit the astonishing and mysterious behavior known as «localization»—instead of propagating over extended regions, they remain confined in small portions of the original domain. The Nobel Prize–winning discovery of the Anderson localization in 1958 is only one famous case of this phenomenon. Yet, 60 years later, despite considerable advances in the subject, we still notoriously lack tools to fully understand localization of waves and its consequences. We will discuss modern understanding of the subject, recent results, and the biggest open questions.

Suppose you want to color faces of a cube in two colors -- red and blue. How many indistinguishable colorings (i.e., ones you cannot turn into each other by rotation) do you have? You may try the brute force approach – draw them all and then count, and if you are careful, you will succeed - there are 10. But what about three colors (red, blue and green)? Now this is pretty confusing, and if you are not extra careful, you are likely to make a mistake (the answer is 57). And if instead of a cube you have a dodecahedron, this becomes totally impossible - for three colors there are thousands of possibilities (the answer is 9099). The main challenge is to account for the symmetry – some colorings are very symmetric and others less so, which makes the process messy. Nevertheless, there is a technique, called Burnside’s lemma, which allows you to count colorings precisely and efficiently without enumerating all possibilities (and you can do all the required calculations in your head!) Not surprisingly, this is based on a mathematical theory designed to study symmetry, called group theory. I will explain the basics of group theory, then derive Burnside’s lemma, and then consider examples of its use, in particular obtaining the numbers 57 and 9099. The lecture should be accessible to students with a good background in high school algebra ((pre)calculus is not required).

The story told in this lecture starts with an innocuous little geometry problem, posed in a September 2006 blog entry by R. Nandakumar, an engineer from Calcutta, India: «Can you cut every polygon into a prescribed number of convex pieces that have equal area and equal perimeter?». This little problem is a «sparrow», tantalizing, not as easy as one could perhaps expect, and Recreational Mathematics: of no practical use. 

I will sketch, however, how this little problem connects to very serious mathematics, including Computational Geometry:  

For the modelling of this problem we employ insights from a key area of Applied Mathematics, the Theory of Optimal Transportation, which leads to weighted Voronoi diagrams with prescribed areas.

This will set up the stage for application of a major tool from Very Pure Mathematics, known as Equivariant Obstruction Theory. This is a «cannon», and we'll have fun with shooting it at the sparrow.

Classical cryptography uses a shared secret which needs to be exchanged over a secure channel before two parties can communicate in a secure way over an insecure channel. Public key cryptography provides a mean to exchange this shared secret over the insecure channel. It hence is one of the cornerstones of the internet as we know it today. The currently used algorithms are based on two mathematical problems which are easy to compute in one direction, but are (computationally) difficult to invert. One is the discrete logarithm problem (it is easy to compute b= a^x (mod p), but it is difficult to find x given a and b) and the other is the factorization problem (it is easy to multiply two numbers n =p*q, but it is difficult to factor n). If the involved numbers are sufficiently large, the direct problem can still be handled even by weak computers, while the inverse problem cannot be solved even by the most powerful computers available today. However, a sufficiently large quantum computer would be able to solve both problems efficiently using Shor's algorithm. Hence the current algorithms need to be replaced in the foreseeable future. The most promising candidates for «Post Quantum Cryptography» are bases on lattice problems. The aim of the present lecture will be to give a glimpse into this current area of mathematics.