SPECIAL COURSES AND SEMINARS

Section: RESEARCH

SECOND SEMESTER OF 2017-2018 year

**«Optimal Transport and Its Application», May 28-31, 2018**

Instructor: *Darina Dvinskikh (MIPT, SkolTech)*

In this short course, the problem of the optimal transport of Monge-Kantorovich, its history and development will be presented. This problem determines the optimal planning of transportation. Using this problem as a basis, we will construct a theory that will allow us to use the optimal transport to compare complex objects (probability measures). The problem has wide potential applications in many areas, in particular, in image processing.

**«Principle of Maximum of Entropy and Equilibrium of Macrosystems», May 28, 2018, «Correspondence Matrix Calculation Model», May 30, 2018, «The Problem of the Particular Bride», May 31, 2018**

Instructor: *Alexander Gasnikov (MIPT)*

We will try to explain from the mathematics point of view what “the equilibrium of the macrosystem” is. We will use the task of ranking web pages, the Ehrenfest paradox and the kinetics of social inequality as examples.

The lecture on the “Principle of maximum entropy and equilibrium of macrosystems” will describe how the entropy model of the correspondence matrix calculation appears. This model was proposed in the late 1960s by A.J. Wilson and to this day is an invariable attribute of any transport packages.

In a lecture, a non-married speaker will teach others at what point they should marry 🙂 The general strategy is like this: skip a third of brides, and then marry the first one who will be better than all previous ones (if there will be one…). This strategy allows you to choose the best bride with a probability close to 1/3. It turns out that this strategy will be the optimal one. The lecture will be devoted to why it is so.

**«Helly’s Theorem and Its Environs», May 23 and May 25, 2018**

Instructor: *Alexander Polyansky (MIPT)*

The lectures will be devoted to the classical theorem of convex geometry: Helly’s theorem. Helly’s theorem asserts that if there are a hundred convex polygons (a union of boundaries and integers is meant by a polygon) and it is known that any three of them have a common point, then all polygons will have a common point. Helly’s theorem plays an important role in various areas of mathematics (convex analysis, optimization, etc.). We discuss some beautiful applications of this theorem, as well as some nontrivial generalizations.

**«Influence in Elective Bodies», May 14-18, 2018**

Instructor: *: Dmitry Schwartz (HSE)*

The basic idea of proportional representation is that the number of seats in parliament received by the party as a result of elections should be proportional to the number of voters who voted for this party. It can be assumed that the influence of the party should also be proportional to the number of seats it receives in the parliament. However, in contradiction with intuition, influence cannot be proportional to the number of votes.

Let’s consider two variants of the distribution of votes between three parties in the parliament of 100 deputies (51 votes are required for making a decision). In the first case, all parties will have an almost equal number of seats (say, 34, 33, and 33), in the second – 49, 48, and 3 seats in the parliament. Then, in both cases, the support of two or three parties is needed to make a decision, thus, the influence of all parties is the same. But the distribution of votes is strikingly different.

The purpose of the mini-course is to tell what the influence of the party is, how to count it, how it is possible to take into account the relations between parties. It turns out that the influence has many unexpected properties, and we can see them not only “on paper”, but also in “real” parliaments, including the State Duma of the Russian Federation.

**«Mechanisms of Public Choice», May 14-18, 2018**

Instructor: *Daniil Musatov (MIPT)*

Since the Great French Revolution, scientists have been looking for an ideal way to make democratic decisions. The recipe was presented in the form of a mathematical function that translates the preferences of individual members of the group into the solution of the whole group. In the twentieth century, it turned out that there was no ideal function, each had its drawbacks. In the course, we will study various procedures for selecting one alternative from several, ordering the list of alternatives or determining the norm of representation. In each case, we prove a theorem on the impossibility of constructing an ideal procedure.

**«Combinatorial Game Theory», May 7-10, 2018**

Instructor: *Daniil Musatov (MIPT)*

Many mathematical games are built on a simple principle: two people take turns to make a move. The one, who cannot make a move, loses. How to understand who wins the game and what is the winning strategy? In general, this problem is computationally complex but it is possible to construct a universal mathematical theory of such games. It turns out that each game is associated with a certain value, and these values themselves are combined according to certain rules. It is possible to construct large arithmetic which includes not only real numbers but also ordinals, infinitesimal values, fuzzy values, etc. In this mini-course, we will discuss the basics of such a world.

**«Polyhedra», May 4-5, 2018**

Instructor: *Aleksey Savvateev (Dmitry Pozharsky University)*

1. What is a soccer ball construction? Euler’s formula and 12 pentagons.

2. What kinds of polyhedra exist if each side borders each edge? The combinatorial formula Г (Г-7) = 12 (g-1), where g is the number of holes inside the polyhedron.

3. What kinds of regular polyhedral exist? Adjacent classes, actions of groups on a set, finite subgroups, formula 2 (1-1 / n) = \ sum (1-1 / n_i).

Video lecture “Polyhedra” part 1 (in Russian), Video lecture “Polyhedra” part 2 (in Russian)

**«Cake-cutting Problem», May 3-5, 2018**

Instructor: *Daniil Musatov (MIPT)*

Suppose that several agents divide an equally heterogeneous resource, for example, a piece of land. The problem is that the site is diverse, and the agents have different preferences: someone needs a lake, someone wants a forest, etc. How to divide the land so that everyone is happy? This task has two basic statements: proportional division where everyone should get at least 1 / n and sharing without envy where everyone thinks that his piece is the best. This course will provide an overview of the methods of such sharing as well as related problems.

**«Dynamic systems. Combinatorics. Information. Complexity», April 23-26, 2018**

Instructor: *Alexander Prikhodko (MIPT)*

During this course, we will discuss the directions of the modern theory of dynamic systems and their applications to mathematics and computer science. The course is accompanied by tasks for self-study, and at the end of the course, topics of possible research will be offered.

**«Elementary Combinatorics», April 23-28, 2018**

Instructor: *Dmitry Schwartz (HSE)*

Many people are searching for an answer to the question: “How many options?” This person can be anybody: a senior citizen who has forgotten the code to the entrance of the building, an analyst or an insurance agent calculating the risk probability. Sometimes, it is not easy to find an answer and it requires knowledge of science, which is called combinatorics. But in most cases, it is enough to know a few elementary techniques, how to combine them, and how to choose the right one.

**«Formal Languages», April 21-27, 2018**

Instructor: *Maxim Zhukovsky (MIPT)*

Mathematicians, when writing a set of even numbers or, for example, a set of powers of 2, have agreed to use a formal entry. Similar records (which can be used for the symbolic notation of some less natural sets) are subject to certain strict rules. The course discusses what kind of formal languages for such records exist and how they can make life easier for mathematicians.

**«Variation Principles», April 18-20, 2018**

Instructor: *Alexey Shabat (Institute of Theoretical Physics, Karachay-Cherkess State University)*

In the course, we will discuss the Dirichlet problem, harmonic polynomials, and Poisson brackets in classical mechanics.

**«Instruments of Scientific Computing», April 17-18, 2018**

Instructor: *Daniil Merkulov (MIPT, SkolTech)*

The course discusses:

- how to do scientific calculations using Python, Jupyter, Docker, etc .;
- how to report scientific calculations using Latex, Markdown, Rise, etc .;
- how to make the mathematical problem clear on examples of real optimization problems, linear algebra, and machine learning.

Video lecture on Latex , video lecture on Python (in Russian)

**«Introduction to Modern Numerical Methods of Convex Optimization», April 17-19, 2018**

Instructor: *Alexander Gasnikov (MIPT), Daniil Merkulov (MIPT, SkolTech)*

The course examines the materials on effective modern numerical methods for solving large optimization problems that appear when analyzing data and modeling of large networks. The course will have practical application tasks. In particular, we will cover the examples in Python.

**«Around the Exponent, or How to Teach Higher Mathematics?», April 17, 2018**

Instructor: *Аlexey Savvateev (rector of the University of Dmitry Pozharsky, deputy head of the CMC ASU)*

Higher mathematics and mathematical analysis contain an abundance of new and unusual concepts for an ordinary school student. There is not often enough inner motivation not to get confused and organize it in one’s head. Perhaps, it is easier to acquire knowledge if there is some kind of idea which goes through the entire program, and other components are just added to it. A.V. Savvateev says that such an idea exists, in the form of a constructed exponent that transfers addition to multiplication.

Video lecture “Around the Exponent, or How to Teach Higher Mathematics?”

**«Game Theory», 2March 29 – April 3, 2018**

Instructor: *Alexander Tonis (New Economic School)*

The new mini-course on “Game Theory” is focused on the initial familiarization with the game theory. Game theory is a discipline at the intersection of mathematics and economics which studies the interaction of rational subjects each of which has its own interests. During the course, we learn how to move from an informal verbal description of the game to its mathematical model, how to predict the most likely actions of rational players – their strategies which are optimal answers to each other. Most of the content of the course is made up of examples that illustrate the basic concepts and results of the game theory. The theory itself is presented only in the minimum necessary volume.