Name of the Speaker with their Affiliation

No. of Lecture Hours

Detailed Syllabus

Dr. Santosh Nadimpalli,
IIT Kanpur

&

Dr. Amiya Mondal IISER Berhampur

6*1.5h = 9h

Title: Induced representations and Intertwining operators.

Topics:

• Formalism of induced representations, intertwining operators in the setting of finite groups with finite groups of low rank as principal example.

• Induced representations in the context of real groups such as SL2(R), SL2(C) and SO(n,1). We will introduce the setting in the p-adic case as well.

• Intertwining operators of the above induced representations and their analytic continuation

• Properties of intertwining operators such as, admissibility, unitarity, poles, reducibility points, and complementary series.

Prof. C. S. Rajan,
Ashoka University

4*1.5h = 6h

Title: Eisenstein series and L-function for rank one groups.

Topics: Eisenstein series on GL2(R) for a lattice Γ and definition of constant term.

Eisenstein series for rank one groups, constant term, mass Selberg relations, analytic continuation. Relation with L-functions.

Prof. Pralay Chatterjee,
IMSc Chennai

3*1.5h=4.5h

Title: Word maps on Lie groups and algebraic groups

Topics: We will introduce the problems associated to the word maps in Lie groups with many examples. We will discuss the remarkable theorem of A. Borel on the Zariski density of images of word maps on semisimple algebraic groups, and the contrasting (relatively recent) results of A. Thom on compact Lie groups. We will also sketch some open problems in this area.

Prof. T. N.Venkataramana,
ICTS, Bengaluru

3*1.5h=4.5h

Title: Arithmeticity of higher rank lattices.

Topics: In 1984, a translation into English of a paper of Margulis on arithmeticity of lattices in higher rank groups was published in the Inventiones. The proof there is quite accessible and self-contained. It uses measure theory in an essential way. We review the proof developing the necessary background.

Dr. Sandip Singh,
IIT Bombay

3*1.5h=4.5h

 Title: Hypergeometric Groups and their Arithmeticity and Thinness

Topics: A hypergeometric group Γ(f, g) is a subgroup of GLn(C) generated by the companion matrices of two monic coprime polynomials f and g of degree n. It arises as the monodromy group of a hypergeometric differential equation, and if the defining polynomials f and g are self-reciprocal and form a primitive pair, then the Zariski closure G of Γ(f, g) inside GLn(C) is either the symplectic group Spn or the orthogonal group On. It is immediate that when the defining polynomials f, g also have integer coefficients (in particular, when they are products of the cyclotomic polynomials), the associated hypergeometric group Γ(f, g) ⊆ G(Z). In this case, the hypergeometric group Γ(f, g) is called arithmetic if it has finite index inside G(Z), and thin, otherwise. The question of determining the arithmeticity and thinness of the hypergeometric groups has been of great interest since the beginning of the last decade. In this mini-course, we will provide all the details about the hypergeometric groups and discuss the progress in answering the arithmeticity and thinness question for hypergeometric groups.

Dr. Pranab Sardar,
IISER Mohali

2*1.5h=3h

Title: Aspects of discrete subgroups of SO(n,1) and some questions

Topics: We will discuss various propertiesofandquestions on discrete subgroups of SO(n,1) starting with the motivating examples from the cases n = 2.

Dr. Subhabrata Das, Presidency University

2*1.5h = 3h

Title: Property Rapid Decay for Discrete Groups

Topics:

• Equivalent formulations of Property Rapid Decay (P(RD)).
• Brief outline of the theorems: (a) Free groups have P(RD) (Haagerup), (b) co-compact lattices in SL(3,R) and SL(3,C) have P(RD) (Lafforgue).
• A short survey of results around Valette's conjecture: co-compact lattices in semisimple Lie groups have P(RD).

Prof. Krishnendu Gongopadhyay,
IISER Mohali

3*1.5h=4.5h

Title:Reversibility in Lie groups

Topics: Reversible or real elements in a group are those elements that are conjugate to their own inverses. Such elements appear naturally in different branches of mathematics. They are closely relatedtostronglyreversible or strongly real elements, which can be expressed as a product of two involutions. Classifying reversible and strongly reversible elements in groups has been a problem of broad interest.We will discuss some recent developments on this topic.

In addition to the above main courses, we shall also have the following research expository lectures to expose the participants to some of the current developments invarious directions on the subject.

Prof. Anish Ghosh, TIFR, Mumbai

1.5h

Dr. Arunava Mandal, IIT Roorkee

1.5h

Dr. Arghya Mondal, Krea University

1.5h

Topics: Property (T) of a group has found many applications. We will discuss why and where it is useful.

Dr. Tanusree Khandai, IISER Mohali