2018/2019

Research Seminar "Introduction to Combinatorial Theory"

Type: Optional course (faculty)

Delivered by: Faculty of Mathematics

Where: Faculty of Mathematics

When: 1, 2 module

Instructors: Yuri M. Burman

Language: English

ECTS credits: 5

Contact hours: 60

Course Syllabus

Abstract

Combinatorics is a part of mathematics studying finite sets. The question to answer is usualy «how many»: how many are there connected graphs with n numbered vertices containing no cycles? how many are there ways to draw diagonals of a convex n-gon so as to cut it into triangles? etc. This question is answered by a multitude of methods from real and complex analysis, number theory, geometry, and more. We do not expect, however, that the students are familiar with all these areas: the necessary techniques will be explained in the course. Combinatorics is very rich in applications, ranging from mathematical physics to algebraic geometry to finance, including topology and dynamical systems on the way. Very often questions from various sciences eventually turn to be combinatorial problems. Combinatorics is an indispensable part of every mathematician's education.

Learning Objectives

• The seminar is intended to give a broad background in modern combinatorial theory.

Expected Learning Outcomes

• Successful participants will be able to apply combinatorial methods to various enumerative problems arising in mathematics, physical and social sciences, engineering.

Course Contents

• Formal power series
• Linear recurrence.
• Lagrange inversion theorem.
• Transfer matrix.
• Binomial coefficients.
• Lattice paths.
• Catalan numbers.
• Partitions and compositions.
• Trees.
• Parking functions.
• Hurwitz numbers.
• Tutte polynomial.

Assessment Elements

• Cumulative grade
  cumulative grade is proportional to number of tasks solved.
• Final exam

Interim Assessment

• Interim assessment (2 module)
  0.3 * Cumulative grade + 0.7 * Final exam

Bibliography

Recommended Core Bibliography

• Edward A. Bender, & S. Gill Williamson. (2013). Foundations of Combinatorics with Applications. [N.p.]: Dover Publications. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1152860

Recommended Additional Bibliography

• Gnedenko, B. V., & Ushakov, I. A. (1997). Theory of Probability (Vol. Sixth edition). Amsterdam, the Netherlands: Routledge. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1835590