Name of the Speaker with affiliation

No. of Lectures

Detailed Syllabus

Soumya Bhattacharya,
IISER,Kolkata.

3 hours

Introduction to Modular Forms.

Definition of ζ, Dirichlet series, Euler product, functional equation.

• Zeros of ζ : distribution, count in rectangles, zero-free regions, Riemann Hypothesis.
• Estimations of ζ, convexity bounds.
• Proof of the Prime Number Theorem, the possible error terms.

Lucile Devin, Universite de Littoral, calais, France

3 hours

Aditya Karnataki , CMI, Chennai.

3 hours

1. Fourier analysis in number fields
2. Topological groups
3. Duality for locally compact abelian groups
4. Adeles, ideles, and class groups
5. Tate's thesis and applications

Ritabrata Munshi, ISI, Kolkata.

3 hours

1. Convexity of L-functions
2. Sub-convexity of L-Functions
3. Generalized Riemann Hypothesis
4. Generalized Lindelöf Hypothesis

Emmanuel Royer, Université Clermond- Auvergne, France.

3 hours

1. Trace formula on GL(1): Poisson summation formula
2. Trace formula on GL(2): a general principle
3. The Eichler-Selberg trace formula
4. The Petersson trace formula
5. Equidistribution of Hecke eigenvalue

Leila Schneps, Institut Mathématiques Jussieu ,France

3 hours

• Introduction to multizeta functions, above all their special values at positive integers and the Q -algebra formed by them; its conjectured dimension, weight-grading, depth filtration, the relations with cusp forms on SL2(Z) and period polynomials;
• The absolute Galois group of the rationals acting on braid groups as fundamental groups, and the Grothendieck-Teichm¨uller group;
• The Grothendieck-Teichm¨uller Lie algebra and the conjectured isomorphism with multizeta Values.

Masatoshi Suzuki,Tokyo Institute of Technology,Japan

4.5 hours

• Definition of the Selberg class. L-functions in the sense of Iwaniec-Kowalski.
• Examples of L-functions in the Selberg class.
• Major conjectures on L-functions in the Selberg class.
• Analytic properties of L-functions that follow from axioms. Explicit formulas.
• Prime Number Theorem for L-functions in the Selberg class.
• Other applications of explicit formulas. (optional)
• Similarities between the explicit formulas and the Selberg trace formulas. (optional)

Masao Tsuzuki, Sofia University, Japan

3 hours

Montgomery’s theorem (1973): computation of the correlation of order 2 of the zeroes of the zeta function

1. proof of Riemann’s explicit formula for the ζ function
2. proof of Montgomery’s theorem

• Consequences of Montgomery’s theorem on the order of the zeroes of the ζ
• Higher level correlation (theorem of Rudnick & Sarnak (1996), probably  without proof
• Statistics on eigenvalues of matrices

IlaVarma, University of Toronto,Canada

3 hours

• Adeles and ideles
• Automorphic Functions and Forms
• Mellin Transform of an automorphic form to a zeta function
• How a Dirichlet L-function Lχ is associated to a Dirichlet character χ (which is an automorphic form on the (abelian) idele group).
• Connection with reciprocity statements