SPECIAL COURSES AND SEMINARS

FIRST SEMESTER OF 2018-2019 year

**«Neural networks for working with audio», November 29, 2018**

Instructor: *Nadezhda Zueva (MIPT)*

The lecture discusses the use of neural network methods for solving various problems connected to audio: from noise removal and sound recognition to neural transfer of musical style and speech generation.

**«Modern neural networks», November 27, 2018**

Instructor: *Nadezhda Zueva (MIPT)*

The lecture discusses neural networks and the reasons why they have become so popular now. It also discusses the main architectures that are used today in unmanned vehicles, voice assistants, pattern recognition systems. It also discusses how the artificial intelligence managed to beat a person in Go.

**«Discrete logarithm problems», November 22, 2018**

Instructor: *Dmitry Ilinsky (MIPT)*

The discrete logarithm problem is one of the main tasks on which public key cryptography is based. Namely, using a simple exponentiation operation, one can transmit information without worrying about it being intercepted. The lecture discusses the transmission protocol, as well as the basic concepts and problems arising from the operation of this algorithm.

Video lecture “Discrete logarithm problems” (in Russian)

**«Theory of elections», November 21, 2018**

Instructor: *Dmitry Ilinsky (MIPT)*

The theory of public elections emerged in the late 18th century thanks to the work of the French Condorcet and Borda. The main problem was to get the perfect election system. In 1950, an article appeared that changed the perception of a just structure of public institutions. In 1972, its author Kenneth Arrow was awarded the Nobel Prize in Economics, at the age of 51 becoming the youngest winner among scientists. He proved that with reasonable restrictions, the ideal electoral system does not exist. The lecture discusses basic concepts of this theory and provides the proof of Arrow’s theorem.

Video lecture “Theory of elections” (in Russian)

**«How does game theory help to better understand the causes of wars?», November 15, 2018**

Instructor: *Konstantin Sorokin (HSE)*

Experts in international relations and experts in the field of armed conflicts have long appreciated the practical application of game theory for developing a strategy of crisis negotiations and analysis of the defenses of countries. The need for a systematic understanding of conflict situations arose in the mid-twentieth century in connection with the creation and proliferation of nuclear weapons. A powerful impetus to the development of this theory was the Caribbean crisis and the subsequent negotiations on nuclear disarmament. Game theory allows people to understand more deeply the nature of conflicts, the options for the actions of the parties, as well as to find the best solutions for resolving the conflict. This lecture discusses game-theoretic analysis of international conflicts and related negotiations, as well as why politicians threaten the use of weapons, exactly how they do it, and when it is useful to have a reputation as an irrational and unpredictable commander in chief.

**«General preferences of groups of people: paradoxes and perspectives», November 13, 2018**

Instructor: *Konstantin Sorokin (HSE)*

The lecture discusses the derivation of common interests of groups of people, based on the preferences of individual individuals. The main task is to show that modern mathematics makes it possible to argue strictly about the initially purely humanitarian problem of the relationship between the individual and the collective. This tradition began with the pioneering ideas of Marquis de Condorcet (late 18th century) and took shape in the full-fledged scientific field by the works of Kenneth Arrow in the 1950s. Now serious scientific discussions of the most diverse social topics (from the problems of radicalization of politics to the general fight against global warming) are largely based on the mathematical apparatus of the theory of collective choice. This lecture is a brief introduction to this area.

Video lecture “General preferences of groups of people: paradoxes and perspectives” (in Russian)

**«Spectral analysis and combinatorics of self-similar dynamical systems», November 1, 2018**

Instructor: *Alexander Prikhodko (MIPT)*

One of the most non-trivial questions of analyzing the dynamics of random processes is how fully does the spectral analysis of process correlations reflect the combinatorial structure of a dynamic system? The lecture considers a simple classical example of a dynamical system with a self-similar structure and shows that the asymptotic behavior of correlations in this system is characterized by a special system of polynomials indexed by points of the Cayley’s graph of a certain discrete group. These polynomials are an interesting family of polynomials with integer coefficients, some properties of which (irreducibility, the realness of roots (the Lee-Yang property))are still just hypotheses.

**«Modern applied cryptography», October 30, 2018**

Instructor: *Alexander Prikhodko (MIPT)*

Let us consider several key paradigms of modern applied information – digital signatures, asymmetric encryption, creation of secure information systems and communication channels, electronic authentication. Mathematical ideas and designs which are embedded in each of these mechanisms are discussed during the lecture as well. The lecture mainly focuses on the following technologies: asymmetric cryptography (public key encryption) and PKI infrastructure, algorithms based on RSA, Diffie-Hellman algorithm, modern approaches including encryption using elliptic curves.

**«You encountered data analysis every day. Do you know how?», October 23, 2018**

Instructor: *Alexander Dainyak (MIPT)*

Video lecture “You encountered data analysis every day. Do you know how?” (in Russian)

**«Data visualization», October 22, 2018**

Instructor: *Alexander Dainyak (MIPT)*

Video lecture “Data visualization” (in Russian)

**«Applied analysis of networks», October 22, 2018**

Instructor: *Alexander Dainyak (MIPT)*

**«Why do we need probability and statistics?», October 16 2018**

Instructor: *Maxim Zhukovsky (MIPT)*

In the XXth century, the probability theory and mathematical statistics became mathematical disciplines. Since then, the technological progress has accelerated due to these disciplines. Thus, for example, mathematical statistics is one of the main tools of machine learning which is used in various industries. The probability theory is used in building models of financial market analysis, biological systems, social systems, etc.

**«Flexural polyhedra and related problems», October 17, 2018**

Instructor: *Alexander Gaifullin (MSU)*

The following topics are discussed:

1) Examples of flexible polyhedra with a small number of vertices.

2) Multidimensional flexible polyhedra and their parameterization. The question of the existence of non-self-intersecting multidimensional flexible polyhedra.

3) Hypothesis about blacksmith’s bellows in multidimensional spaces and non-Euclidean spaces.

4) The strong hypothesis about blacksmith’s bellows is the problem of equal composition of flexible polyhedra.

5) Cayley-Menger determinants. Relationships to sets of distances between points in space.

**«Homogenous polyhedra», October 16, 2018**

Instructor: *Alexander Gaifullin (MSU)*

The classical Boyai – Gervin theorem (1830s) states that any two polygons of equal area are homogenous: the first polygon can be cut into a finite number of polygonal parts and then folded out of these parts into the second polygon. Gauss posed the question of whether a similar statement is true for polyhedra. Namely, he was interested in whether it is possible to prove the standard formula for the volume of the pyramid (one third of the product of the length of height by the area of the base) without using the limit transition, that is, by breaking the pyramid into a finite number of pieces from which you can fold a rectangular parallelepiped. Later, David Gilbert included the problem of the homogeneity of polyhedra into his famous list of problems under number three. The funny fact is that by this time the problem had already been solved by his student Max Dehn (which Hilbert did not know about). Dehn built a series of invariants of homogeneity, and now they are called Dehn invariants. They are associated with a remarkable algebraic object, namely additive functions. After that, Dehn showed that, for example, a cube and a regular tetrahedron of equal volume are not comparable, since their invariants are different. The theorem of Jean-Pierre Sidler (1965) asserts that the equality of the volumes and the Dehn invariants of two three-dimensional polyhedra is not only necessary but also sufficient condition for their homogeneity.

The lecture discusses the proof of the Bolyai–Gerwien theorem, the Dehn invariants, and the ideas underlying the proof of the Sidler theorem.

**«Inversions, projective transformations and Lobachevskian geometry», October 12, 2018**

Instructor: *Alexander Gaifullin (MSU)*

Inversions and projective transformations are two types of geometric transformations of the plane which are quite often used in solving Olympiad problems in geometry. Although, at first glance, there is very little in common between these two types of transformations, in fact, they are interconnected much more closely than it seems. First, in a sense, inversion is a complex analog of the projective transformation. Secondly, it turns out that if we draw a circumference on the plane, then there is a very important correspondence between projective transformations and inversions (more precisely, compositions of several inversions) which leave this circumference in place. This correspondence plays a key role in establishing the connection between two famous Lobachevskian plane models: the Beltrami-Klein model and the Poincaré model.

The listeners are not expected to be familiar either with the concepts of inversion and projective transformation, or with Lobachevskian geometry.

Video lecture «Inversions, projective transformations and Lobachevskian geometry» (in Russian)

**«Quaternions, space rotations, and regular polyhedra», October 11, 2018**

Instructor: *Alexander Gaifullin (MSU)*

In many problems of mathematics and its applications it is very useful and absolutely inevitable to use complex numbers – a natural generalization of ordinary real numbers, obtained with the help of the special operation of joining the root of -1. In 1843, the Irish mathematician and physicist William Rowan Hamilton discovered that if we repeat the same operation again, applying it to complex numbers, we get another absolutely remarkable class of numbers which are now called quaternions. In contrast to real and complex numbers, for quaternions there is no commutativity property, that is, the product of ab in general is not equal to ba. This feature leads to interesting and beautiful both algebraic and geometric properties. For example, in the language of quaternions, it is very convenient to describe orientation-preserving movements (rotations) of three-dimensional space.

The lecture discusses this description and shows that it can be used to understand something not only about three-dimensional, but also about four-dimensional geometric objects, in particular about four-dimensional regular polyhedra.

The listeners are not expected to be familiar with complex numbers or with the concept of four-dimensional space.

**«Fair division», October 10, 2019**

Instructor: *Dmitry Shvartz (HSE)*

Probably, any school student knows how to divide a cake between two people: the first one divides it into two parts, the second one chooses which part he wants. But it is unfair because it is better to be the second than the first one.

The lecture discusses:

– How to divide fairly and what it means;

– What will change if you need to divide something more important than a cake;

– What to do if you think you are the smartest one and how it will end.

**«Theory and practice of auctions», October 11, 2018**

Instructor: *Alexander Filatov (FEFU)*

More than half a century ago, auctions were a marginal area, known to the public solely thanks to the Sotheby’s auctions and the book “12 Chairs.” Then professional specialists in game theory took things into their own hands and created a theory that has already given the world several Nobel laureates and is probably the most obvious success story of applying theoretical constructs of economics to practice. Currently, one third of world GDP is being redistributed at auctions. Before its division in 2015, Hewlett Packard alone purchased through auctions components worth $ 60 billion. Auctions sell flowers in Holland, cars in Japan and fish in Honolulu. Auctions are an electronic platform “eBay,” contextual advertising in Yandex and Google, tenders and privatization deals. Finally, it is a trillion dollars of daily trading in Forex, implemented in the format of a double auction. At the lecture, students learn what the designers of mechanisms do; who opened the closed auctions; how the English auction differs from the Dutch; how to identify the value of the lot for each of the participants, even if they tend to hide it; what the curse of the winner is; how to earn 40 billion euros by selling air.

**«Introduction to Experimental Economics», October 9, 2018**

Instructor: *Alexander Filatov (FEFU)*

Experimental economics is a branch of economics dedicated to the study of human behavior and testing of predictions of economic theory in a controlled experiment. Economics experiments are similar to those conducted in physics, chemistry, and other natural sciences, with the only difference being that they are conducted on people who make economic decisions in an experimental laboratory or a computer class where people play games. As a part of the lecture, students learn the basics of experimental economics, learn a lot of interesting examples, and participate in several real-life experiments.

**«Introduction to the Economics of Information: Strategic Management of Information Flows», October 2, 2018**

Instructor: *Alexander Tonis (NES)*

The lecture discusses how and why when transmitting or receiving information there can be incentives for its deception (“telling the truth and only the truth, but not the whole truth”) if the interests of the parties do not coincide.