Titles and Abstracts
(including student presentations)


Abstract.
In this lecture, I intend to introduce the basic framework of game
theoretic choice problems and combinatorial games. I am also planning to
discuss some open problems, particularly the ones that involve
probability (mainly discrete). The objective of this lecture is to give
students an idea of some research areas in Game Theory.


Abstract.
Random walks on networks are a way to model reversible Markov chains.
This point of view makes several notions more natural and these notions
can be used to get quite powerful results about the probabilities or even
rates of escape of a random walk. We shall discuss the
Nash–Williams inequality which will eventually help us prove an
interesting and somewhat counterintuitive result about a knight's random
walk on a biinfinite chessboard. We would also survey a few broad
classes of techniques which can be helpful to understand not just the
probability of escape but even the rate of escape of random walks.


Abstract.
We shall discuss a fascinating and fundamental result of probability
theory, the Lovász Local Lemma, which roughly says that a set of
"bad" events do not occur with positive probability provided the
dependence between them is restricted. Thereafter, we shall explore its
farreaching applications in solving various deterministic problems
arising in combinatorics, graph theory, and algorithms. Finally, since
the LLL is nonconstructive by nature, we shall also briefly discuss
modern research on this topic that aims at finding efficient algorithms
for putting this mighty probabilistic tool in action.


Abstract.
In this talk, we will discuss certain special proprieties of polynomials
with nonnegative coefficients but with only real zeros. We will sketch
that for such polynomials with high degree, the asymptotic limits of the
coefficients can easily be derived using certain probabilist techniques.
We will illustrate with examples of few important combinatorial
sequences, namely,
(i) Binomial Coefficients:
the number of subsets of a set with $n$ objects which are exactly of the
size $k$;
(ii) Stirling's Number of Ist Kind:
the number of permutations of $n$ objects with exactly $k$ cycles; and
(iii) Stirling's Number of IInd Kind:
the number of partitions of a set with $n$ objects into exactly $k$
parts.
We will show that they all have an "universal" asymptotic limit and
discuss the the relation with one of the most celebrated result in
Probability Theory, namely, the Central Limit Theorem (CLT). In
the process, we will outline a very powerful combinatorial technique,
known as the Harper's Method.
[The talk will only need basic concepts from probability theory.]


Abstract.
A geometric graph is a graph drawn in the plane such that its vertices
are points in general position and its edges are straightline segments
between these points. There are many interesting results and open
problems about geometric graphs that are relevant to well known problems
in discrete and computational geometry, such as the halving line problem
(Erdős–Lovász–Simmons–Straus), questions
on repeated distances and incidences (Erdős,
Szemerédi–Trotter), etc. We survey some basic results in
geometric graph theory and highlight some open problems whose solution
may be within reach.


Abstract.
There are several questions in mathematics which are unrelated to
probability theory where bringing in ideas from probability theory leads
to alternate proofs, which may be shorter or easier to understand,
Weierstrass's result on approximating continuous functions on $[0,1]$ by
polynomials is one such.
In this talk, we will discuss one such numerical question involving large
finite sets where bringing in Markov chains and the related Monte Carlo
technique seems to be the only way to approximate the answer. We will
also introduce the required background on Markov Chains.


Abstract.
Branching random walk is a system of growing particles that starts with
one particle. This particle branches into a random number of particles,
and each new particle makes a random displacement independently of each
other and of the branching mechanism. The same dynamics goes on and gives
rise to a branching random walk. This model arises in statistical
physics, mathematical biology, ecology, etc. It is important because it
has connections to various probabilistic objects. In this overview talk,
we shall try to answer the following question: if we run a branching
random walk for a very long time and take a snapshot of the particles,
how would the system look like? We shall investigate how the tails of the
progeny and displacement distributions change the answer to this
question.
This talk is based on a series of joint work with Ayan Bhattacharya,
Rajat Subhra Hazra, Krishanu Maulik, Zbigniew Palmowski, Souvik Ray and
Philippe Soulier.


Abstract.
We discuss several results on proper graph coloring and graph
choosability by a polynomial method of Alon and Tarsi and its further
development.

