ATMW Topological Combinatorics (2014)

Brief Description: The topics covered will be on application of Topological Dynamics, Ergodic Theory and Algebraic Topology in the areas of Combinatorics, Diophantine approximation and Combinatorial Number Theory.

Objective:To introduce advanced topics in ‘Topological and Ergodic methods’ in Combinatorics and Number Theory to young researchers in India. We give the details below.

Introductory lectures on Topological Dynamics, Ergodic Theory, Diophantine approximation and Combinatorics, followed by interaction of these areas of mathematics. Each speaker will give two to four talks and there will be some discussion hours.

Titles and abstracts of the topics covered are as follows:

Introduction to Diophantine approximation by S. D. Adhikari and R. Thangadurai:

1. Abstract: We shall first discuss some basic facts about Diophantine approximation. Afterwards, we shall see its connection with solutions of Diophantine equations.
2. Application of topological dynamics in Diophantine approximation by S. G. Dani:
   Abstract: We discuss certain general principles of applying topological dynamics to address questions in Diophantine approximation, relating especially to studying values of linear and quadratic forms at integer points. The case of binary quadratic forms will be considered in detail.
3. Fractal methods in Diophantine approximation by Anish Ghosh
   Abstract: In this course, we will discuss techniques to compute the Hausdorff measure and dimension of sets of arithmetic interest. We will also discuss Diophantine properties of a large class of measures including fractal measures, a subject which has seen rapid advances recently.
4. Ordered permutations in topological dynamics, Counting conjugacy classes and Partitions for classification by V. Kannan.
5. G- indices and the Topological Tverberg Theorem by Satya Deo:
   Abstract: Starting from the classical Tvergerb Theorem we will introduce the notion of G-indices for a finite group G. With some review of algebraic topology,
   will will state and prove the Topological Tverberg Theorem and the BMZ colored Tverberg Theorem.
6. large sets in amenable group and mixing properties by Dibyendu De:
   Abstract: Mixing properties for dynamical systems via largeness properties of various “typical” return times sets or independent return times sets, have been studied by various authors.
   In the current discussions we consider a hierarchy of notions of largeness for subsets of arbitrary amenable groups (such as thick sets, syndetic sets, IP-sets, etc., as well as some new classes) and study them in conjunction with recurrence in topological dynamics and ergodic theory. We use topological dynamics and topological algebraic properties of the Stone-Cech compactification (βG, ·) where G is an amenable group, to establish the connections between various notions of largeness.
   List of speakers.
   1. Prof. S. G. Dani, IIT Bombay, Mumbai
   2. Prof. Anish Ghosh, TIFR and University of East Anglia
   3. Prof. V. Kannan, University of Hyderabad.
   4. Prof. Satya Deo, Harish - Chandra Research Institute, Allahabad.
   5. Prof. S. D. Adhikari, Harish - Chandra Research Institute, Allahabad.
   6. Prof. R. Thangadurai, Harish - Chandra Research Institute, Allahabad.
   7. Prof. Dibyendu De, Kalyani University, Kolkata

   Lecture Notes. We shall be giving lecture notes, related materials to the participants during the school.

Time Table (Tentative) :In the morning two and after noon one 1 and 1/2 hours talks meant for main topics.the speakers will cover their topics. The last one hour talk is meant for Discussion and Tutorials. We shall give the tentative time-table as below.

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