Name of the Speaker with affiliation

No. of Lectures (90 minutes each)

Detailed Syllabus

Professor S. Sivaramakrishnan,
IIT Bombay

6

Some basic algebraic graph theory. Start the addressing problems. Isometrically embedding graphs into the squashed cube and Winklers theorem. Witsenhausen's theorem connecting eigenvalues of the distance matrix of a graph and embedding dimension. Hermitian rank of a Hermitian matrix. Distance matrices of tree, its determinant and inverse. Embedding finite metric spaces in the \ell_1 metric space. The Graham-Hoffman-Hosoya (GHH) theorem. q-analogues of distance matrices, exponential distance matrices, immanants of matrices and some basic representation theory of the symmetric group S_n. Second-immanantal analogue of the GHH Theorem for exponential distance matrices. Steiner distances in graphs and the Four point condition (4PC in short) for tree distances (due to Buneman). Results on the 4PC matrix of a tree.

References:

1. Graphs and Matrices, R. B. Bapat, Springer, 2014.

2. Spectra of Graphs, A. E. Brouwer and W. H. Haemmer, Springer, 2012.

Professor Arvind Ayyer,
IISc Bangalore

6

Formal power series, ordinary and exponential generating functions with examples, Sets, Multisets and Permutations; basic properties and combinatorial techniques,Permutation statistics, and connections with other combinatorial objects, Counting with symmetry & Combinatorial Identities, Bijective Combinatorics, Posets and Lattices

References:

1. Enumerative Combinatorics, Vol. 1, 2nd. ed., Richard Stanley, Cambridge University Press, 2011.

2. Enumerative Combinatorics, Vol. 2, Richard Stanley, Cambridge University Press, 1999.

Professor Murali K. Srinivasan, IIT Bombay

6

An introduction to algebraic combinatorics through significant results involving little in way of background. A subset of the following topics will be discussed:

1. Unimodality of the q-binomial coefficients.

2. Solution of Erdos-Moser conjecture.

3. Delsarte bound on binary code size.

4. Schrijver bound on binary code size.

5. Terwilliger algebra of the n-cube.

6. Differential posets.

References:

1. Algebraic Combinatorics: Walks, Trees, Tableaux, and More, Richard P. Stanley, Springer, 2013.

Professor Niranjan Balachandran,
IIT Bombay

6

The Second Moment Method: Variance of a Random Variable and Chebyshev’s theorem, The Erdos-Ginzburg-Ziv theorem : When do we need long sequences? Distinct subset sums, The space complexity of approximating frequency moments, UniformDilations, Resolution of the Erd˝os-Hanani Conjecture: The Rodl ‘Nibble’.

Concentration Inequalities: A Simple Random Walk, Why are these bounds so important? The Johnson-Lindenstrauss Lemma, The Azuma-Hoeffding Inequality, McDiarmid’s Inequality, Janson’s Inequality. The Lovasz Local Lemma and Applications.

References:

1. The Probabilistic Paradigm: A Combinatorial Perspective, Niranjan Balachandran, lecture notes: https://homepages.iitb.ac.in/~niranj/Probabilistic_Paradigm_Combin-chap_1-6.pdf

Professor Eshita Mazumdar, Ahmedabad University

6

Plünnecke's Inequality : Plünnecke's Graphs and Some Examples, Multiplicativity of magnification ratio, Menger's Theorem, Plünnecke's Inequality, Application: Estimates for sumsets in groups, Application: Essential Components; Sumset estimates: Sum sets, Doubling constant, Rusza distance and additive energy, Covering Lemma, The Balog-Szemeredi-Gowers theorem and its uniformity version, Symmetry sets and imbalanced partial sum sets, Non-commutative analogues, Elementary sum-product estimates

References:

1. Additive Combinatorics, Terence Tao and Van H. Vu, Cambridge University Press, 2006.

Professor Manjil Saikia, Ahmedabad University

6

Introduction to q-analogs, inversions, integer partitions, Elementary theory of integer partitions, Euler's theorem, Glashier's bijection, Pentagonal Number Theorem, Durfee squares, Franklin's map, Divisor sums, q-Binomial Theorems, Jacobi’s Triple Product Identity, Ramanujan’s congruences, other arithmetic properties of integer partitions, Rogers-Ramanujan identities, analytic proof, combinatorial counterparts, Gollnitz-Gordon identities, tilings and partitions, computer algebra and partitions

Reference:

1. The Theory of Partitions, George Andrews, Cambridge University Press, 1997.

2. An introduction to q-analysis, Warren P. Johnson, American Mathematical Society, 2020.