Venue: Lecture Hall I, Department of Mathematics
NOTE: THIS TALK WAS CANCELLED.
Speaker: Chaitanya Tappu
We will quickly recap the last talk (hyperbolic space, discrete group of hyperbolic motions, limit sets). We will show the construction of measures on the boundary which possess nice ergodicity properties with respect to the action of the group on the boundary. Specifically, if the group is non-elementary or has bounded fundamental domain, then the action is ergodic on the conical limit set.
- Nicholls, P.J. (1989). The Ergodic Theory of Discrete Groups, volume 143 of the London Mathematical Society Lecture Note Series. Cambridge University Press.
Speaker: Pramath Anamby
Modular forms are interesting number theoretic objects and are used widely to study arithmetic functions. In this talk I will briefly introduce modular forms, mainly through examples.
Speaker: Chaitanya Tappu
We will define the n-dimensional hyperbolic space. An object of study in hyperbolic geometry is the group of isometries, i.e. distance preserving transformations of the hyperbolic space, and its action on the hyperbolic space. These correspond to Mobius transformations, and their action extends onto the boundary of the hyperbolic space. The action of discrete subgroups of this group has orbits that accumulate on the limit set, which lies on the boundary. We will show the construction of measures on the limit set which possess nice ergodicity properties with respect to this action.
Prerequisites: Familiarity with linear algebra and metric space topology will be assumed.
- Nicholls, P.J. (1989). The Ergodic Theory of Discrete Groups, vol. 143 of London Mathematical Society Lecture Note Series, Cambridge University Press
Speaker: Ritwik Pal
Various summation formulas are very important tools in analytic number theory. We here wish to discuss two elementary (they don’t need complex analysis) summation formulas, namely, Abel’s summation formula and Dirichlet’s hyperbola method. We will also show some applications of them to estimate sums. I hope that the talk will be accessible to students who have had a course in real analysis in one variable.
Prerequisites: Basic real analysis
Speaker: Subhajit Ghosh
In this talk, I shall discuss discrete time Markov chains with finite state space briefly. Mainly, I’ll prove a version of the Perron-Frobenius theorem, which ensures the existence and uniqueness of the stationary distribution for irreducible chains; moreover, if the chain is aperiodic, then the distribution after the nth transition will converge to the stationary distribution.
Prerequisites: Basic Probability and Linear Algebra
Speaker: Nimisha Pahuja
ASMs are a class of square matrices in combinatorics. The formula enumerating all nxn ASMs was conjectured in 1983 and it took nearly two decades to find a bijective proof for the identity. In this talk, I will briefly go over the proof and will discuss a few of the many objects like plane partitions, fully packed loops etc. which are in bijection with ASMs. Further, I will give some of their applications.
Prerequisites: Basic Linear Algebra and elementary ideas about bijections and recursions.