Sans titre

Some lectures related to the course

Proofs without words: the example of the Ramanujan continued fraction

abstract:

Visual proofs of identities were common at the Greek time, such as the Pythagoras theorem. In the same spirit, with the renaissance of combinatorics, visual proofs of much deeper identities become possible. Some identities can be interpreted at the combinatorial level, and the identity is a consequence of the construction a weight preserving bijection between the objects interpreting both sides of the identity.

In this lecture, I will give an example involving the famous and classical Ramanujan continued fraction. The construction is based on the concept of "heaps of pieces",

which gives a spatial interpretation of the commutation monoids introduced by Cartier and Foata in 1969.

Ramanujan Institute, Chennai, India, 10 January 2017

Twenty Second Srinivasa Rajan Memorial Lecture

Amrita Vishwa Vidyapeetham, Amrita University,

Coimbatore, 7 March 2017

Indian Institute of Sciences, Bangalore, 9 March 2017

colloquium IMSc, Chennai, February 21, 2019 (lecture related to Part IV of the ABjC course

slides (pdf, 28 Mo) video link to YouTube (from IMSc Matsciencechanel Playlist)

Zeta function on graphs revisited with the theory of heaps of pieces

ICGTA19, International Conference on Graph Theory and Applications, Amrita Vishwa Vidyapeetham Coimbatore, India, 5th January 2019

slides (second version after the talk, pdf, 39Mo)

abstract

An identity of Euler expresses the classical Rieman zeta function as a product involving prime numbers. Following Ihara, Selberg, Hashimoto, Sunada, Bass and many others, this function has been extended to arbitrary graphs by defining a certain notion of « prime » for a graph, as some non-backtracking prime circuits. Some formulae has been given expressing the zeta function of a graph. We will revisit these expressions with the theory of heaps of pieces, initiated by the speaker, as a geometric interpretation of the so-called commutation monoids introduced by Cartier and Foata. Three basic lemma on heaps will be used: inversion lemma, logarithmic lemma and the lemma expressing paths on a graphs as a heap. A simpler version of the zeta function of a graph proposed recently by Giscard and Rochet can be interpreted as heaps of elementary circuits, in relation with MacMahon's Master theorem.