Name of the speaker with affiliation

No. Of Lectures

(90 minutes each)

Detailed Syllabus

Amitabh Triparthi

IIT Delhi

6

Ramsey Theory

Lecture 1. What is Ramsey Theory? Notations, conventions and prerequisites. Compactness principle. Introduction to Ramsey type theorems - Schur's Theorem (1916), van der Waerden's Theorem (1927), Ramsey's Theorem (1930), Rado's Theorem (1934), Hales-Jewett Theorem (1963), Graham-Leeb-Rothschild Theorem (1972).

Lecture 2. Integer Ramsey Theory: van der Waerden's Theorem, Hilbert's Cube Lemma.

Lecture 3. Integer Ramsey Theory: Schur's Theorem, Rado's Theorem, nonlinear equations. 

Lecture 4.Finite Sums: Arnautov-Folkman-Rado-Sander's Theorem, Hindman's Theorem. Density results: Roth's Theorem, Szemeredi's Theorem. 

Lecture 5.Graph Ramsey Theory: complete graphs, other graphs, hypergraphs.

Lecture 6. Probabilistic Methods: Lower boundson Ramsey, van der Waerden and Hales-Jewett numbers. Turan's Theorem. Lovasz Local Lemma. 

 References

1. N. Alon and J. H. Spencer, The Probabilistic Method, Second Ed., Wiley-Interscience Series in Discrete Mathematics and Optimization, New York, 2000.

2. P. Erdos and J. H. Spencer, Probabilistic Methods in Combinatorics, Probability and Mathematical Statistics, Vol. 17, Academic Press, New York-London, 1974.

3. R. L. Graham, B. L. Rothschild and J. H. Spencer, Ramsey Theory, Second Edition, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons Inc., New York, 1990.

4. B. M. Landman and A. Robertson, Ramsey Theory on the Integers, Second Edition, Student Mathematical Library, 24, American Mathematical Society (AMS), Providence, RI, 2014.

5. J. Nesetril, Ramsey Theory, Handbook of Combinatorics, Vol. 1, 2 (R. L. Graham et al.,Eds.), Elsevier, Amsterdam, 1995, 1331--1403.

6. S. P. Radziszowski, Small Ramsey Numbers,Dynamic Survey 1, 16th Revision, Electron. J.Combinatorics, 2021.

7. A. Robertson, Fundamentals of Ramsey Theory,Discrete Mathematics and Its Applications Series, CRC Press, 2021.

8. A. Soifer, Ramsey theory:yesterday, today,and tomorrow, Progress in Mathematics, Boston, Mass., Birkhauser, New York, 2011.

Amitav Bhatacharya

TIFR Mumbai

6

Percolation and optimal transport

Lecture 1. What is percolation ? Percolation models and basic probability results.

Lecture 2. Existence of nontrivial critical value. Elementary properties of percolation on Z^d.

Lecture 3. Uniqueness of infinite cluster.

Lecture 4. Inequalities (FRG, Harris etc) and tools from probability (second moment method).

Lecture 5. Margullis-Russo Formula and critical value of Z^2 is ½.

Lecture 6. Completing proof of p_{v}(2)=1/2 and proof of RSW.

References

1. G. Grimett, Percolation

2. B. Bellabos, Percolation

3. G. Grimett, Probability on graphs, Random processes on graphs and lattices

4. Hugo Duminil Copin, Online lecture notes, Introduction to Bernoulli percolation

5. Feller, Probability, Vol 1 and 2

Ambat Vijayakumar

Cochin University of Science and Tech. Kerala

4

Graph spectra and distance based characterizations of some graph classes

Lecture 1. Spectral Graph theory- adjacency spectra, Basic properties, basic theorems, Examples, Co- spectral graphs.

Lecture 2. Graph products, Spectra of graph products, Integral graphs, graph energy.

Lecture 3. Strongly regular graphs, its spectra, Moore graphs, Shrikhande and his graph ,Distance spectra of some graph classes.

Lecture 4. Equienergetic graphs, Some recent results etc.

References.

1. R. Balakrishnan, K.Ranganathan , A Text book of Graph theory, Springer

2. R. B. Bapat- Graphs and Matrices, Springer

3. F.Harary, F.Buckley , Distance in Graphs ,Addison Wesley, 1989.

4. Chris Godsil and Gordon Royle , Algebraic Graph Theory.  Springer-Verlag. (2004), 

Murali Srinavasan

IIT Bombay

4

Applications of group actions

Lecture 1. Introduction to Group representations. Statement of the spectral theorem for *-algebras of matrices.

Lecture 2. Delsarte bound for binary code size.

Lecture 3. Schrijver Bound for binary code size.

Lecture 4. Unimodality of the q-binomial coefficients.

References

1. G. James and M. Liebeck  Representations and Characters of Groups. Cambridge University Press (2001).

2. R. P. Stanley  Algebraic Combinatorics.  Springer (2018).

3. 3. A. Schrijver New code upper bounds from the Terwilliger algebra and semidefinite programming. IEEE Transactions on Information Theory  51 (2005), no. 8, 2859-2866.

S. Pirzada

University of Kashmir

2

Matchings in graphs

Lecture 1. Matchings, Berge theorem, Hall’s theorem for complete matchings, Hall’s SDR theorem, Konig’s theorem

Lecture 2. Tutte’s 1-factor theorem, Cubic graphs without cut edges has 1-factor

References

1. S. Pirzada, An Introduction to Graph Theory, Universities Press, Orient Blackswan, 2012

S. Krishnan

IIT Bombay

2

Distance matrices of graphs

Lecture 1.  Distance matrices of graphs and trees.  The Graham-Pollak and the Graham-Hossman-Hosoya theorem.

Lecture 2.  Steiner distance matrix of trees and the Four point condition.

References

1. R. B. Bapat - Graphs and Matrices, TRIM series (2nd Ed).