Dear All,
there will be a lecture series on "Classical and Modern Mathematical Problems" for students and also a general audience (5 lectures) next week (28.-30. Sept.).
Time: Tuesday, 28. Sept., 13:15-15:00 (1. lecture), 15:15-17:00 (2. lecture), Wednesday, 29. Sept., 10:15-12:00 (3. lecture), 12:15-14:00 (4. lecture), Thursday, 30. Sept., 10:15-10:45 (5. lecture).
Place: Room 1000, 10. floor, NHA, MI, Blindern
Description:
The "training camp" Classical and Modern Mathematical Problems is organized in the framework of a Norwegian-Ukrainian project (SIU, project number: CPEA-LT-2016/10139, Eurasia 2016-Long term). Ukrainian teams at International Mathematical Olympiads for both school and university students have been quite successful. For example, five students from the National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute“ are granted with silver and bronze medals at the International Mathematics Competition for University Students 2021 (IMC 2021). The head of this team, Vladimir Pavlenkov, is one of the speakers of the "training camp".
The aim of the "training camp" is to share Ukrainian mathematical Olympiad experience with Norwegian colleagues and deliver several talks on classical and modern mathematical problems that form a basis when compiling the set of problems for each Olympiad.
The topics of the "training camp" are rater wide: from finding solutions of polynomial equations by various methods to some actual questions of genetics. The prerequisites for the majority of the talks at the "training camp" are minor, some talks can even be understood by students of secondary schools. Some topics sound rather advanced, say "Grassmann Algebra", but, in fact, they are quite simple and easy to understand (say, "Grassmann Algebra" is a nice method to evaluate determinants).
In general, the "training camp" could be of interest for both students and teachers and not only for those who are interested in solving advanced mathematical problems.
Lectures:
Tuesday, 28. Sept., room 1000, NHA
Lecture 1: 13:15-15:00, Prof. Natalia Konovalova (National Technical University of Ukraine, Kiev)
Title: Quaternions. Grassmann Algebra and Determinant Theory
Abstract: Inspired by a relation between algebra of complex numbers and plane geometry, William Hamilton has been searching an analogue of algebra for triples in order to apply it to three-dimensional geometry. Unable to multiply and divide triples, he invented a non-commutative division algebra of quadruples. In this talk I will try to provide a motivated introduction to quaternions and discuss what they are, why they are useful, how to use them, and where to use them.
New attempts to generalize the concept of complex numbers led to the creation of new algebras in XIX century. Hermann Günter Grassmann in “Theory of Extensive Magnitudes” developed a calculus, which is associated with geometric representations. In the talk I will give only basic definitions of Grassmann Numbers that are necessary to present the theory of determinants.
Lecture 2: 15:15-17:00, Prof. Anton Syrotenko (National Technical University of Ukraine, Kiev)
Title: Vieta’s formulas and their applications
Abstract: The polynomials are one of the most popular mathematical objects. They are often used in diverse mathematical fields such as analysis, algebra, geometry, probability theory etc. Various problems about the polynomials are proposed often in numerous Olympic competitions. Thorough knowledge of important theorems and formulas about the polynomials as well as ability to apply them allow students to find nontrivial ways of the solutions.
This talk is devoted to Vieta’s formulas. Vieta’s formulas for the polynomials of arbitrary degree are simple but powerful mathematical tool. Correct applications of these formulas allows us to make complicated things much simpler and find the beauty in Math.
Wednesday, 29. Sept., room 1000, NHA
Lecture 3: 10:15-12:00, Prof. Igor Orlovskyi (National Technical University of Ukraine, Kiev)
Title: Functional Equations
Abstract: Functional equations is one of the most beautiful fields of Mathematics. Such equations arise in many fields, such as Mechanics, Statistics, Economics, Engineering, etc. This two lecture mini-course is introductory to the field. Some types of functional equations and the methods for solving them will be considered during the talk.
Lecture 4: 12:15-14:00, Prof. Vladimir Pavlenkov (National Technical University of Ukraine, Kiev)
Title 1 (1 hour): Pick’s formula
Abstract: This talk deals with classical Pick's formula for the area of polygons with integer vertices coordinates. Some new tricky method of proving this formula will be discussed.
Title 2 (1 hour): Polynomials of a special form and their roots
Abstract:We will consider different polynomials with some special conditions on their coefficients. These conditions allow one to use non classical methods of finding the roots. Different examples of applying these methods will be presented.
Thursday, 30. Sept., room 1000, NHA
Lecture 5: 10:15-10:45, Prof. Andrey Pilipenko (National Technical University of Ukraine, Kiev)
Title: Mathematical problems in genetics
Abstract: The flowers of some plant may be red, pink, and white. The seeds of all flowers are sown every year on the same field. Suppose that the initial proportions of colors of red, pink, and white are equal to x, y and z (x + y + z = 1).
We discuss the following questions of population genetics:
a) What is the proportion of white flowers grown in the n-th year?
b) What is the proportion of white flowers grown in the n-th year, if we discard all seeds of white flowers every year?
c) What is the proportion of white flowers grown in the n-th year, if we discard half of the seeds of white flowers every year?
Welcome to the lecture series!