AIS Analytic Theory of Algebraic Numbers (2016)

 The event is Cosponsored by The Institute of Mathematical Sciences, Chennai

 Kaneenika Sinha : The purpose of these lectures was to introduce students to fundamental concepts in algebraic number theory. As such, the chief goals of these lectures were as follows:
1) A study of the notions of divisibility and primes in integers and modular arithmetic.
2)An introduction to basic properties of integral domains,
3) An investigation into generalising the divisibility properties of integers (most notably, the unique factorization into product of prime powers) to other domains,
4) A study of various kinds of domains, their algebraic structures and how they help to find solutions in integers of Diophantine equations.
5) A detailed study of the properties of rings of integers in algebraic number fields as special examples of Dedekind domains.
K. Srinivas : The lectures started with the basic summation techniques in analytic number theory; applications of Euler summation formula, partial summation were extensively discussed, convolution method and hyperbola method was discussed. The analytical properties of the Riemann zeta function, growth estimates of the zeta function in the critical strip,zero-free regions, prime number theorem, equivalent formulations of prime number theorem, Dirichlet L-function and Dirichlet theorem on the primes in arithmetic progression were some of the topics extensively covered in the programme.
R. Thangadurai : The following topics were covered: Introduction to ideal class groups, finiteness of ideal class groups for number fields, Geometry of Numbers and Minkowaski first theorem, Explicit bounds for an ideal in an equivalence class in terms of discriminant, degree etc. Dirichlet Unit Theorem for number fields, Quadratic reciprocity laws, splitting of primes in quadratic fields over Q
R.Munshi : The main aim of the course was to introduce the students to the circle method. The first two lectures were on the paper of Hardy and Ramanujan on the partition function, where the circle method was used for the first time. The focus of the third and the fourth lectures was the work of Hardy and Littlewood on the application of the circle method to the Waring problem. The Kloosterman’s version of the circle method was introduced in the fifth lecture. In the last lecture a brief introduction was given of the delta method, and it was shown how this method simplified several intricacies of the circle method.
Stephan Baier : In 6 lectures, the essential parts of the theory of exponential sums which is of great importance in analytic number theory, were covered. As applications, the Dirichlet divisor problem, the Gauss circle problem and the growth of the Riemann zeta  function on the critical line were considered. After explaining the notion of an exponential sum and looking at simple examples, a link between the average behaviour of the divisor function and exponential sums via Fourier analysis was established. Then a first non-trivial bound for exponential sums, the van der Corput bound was proved and from this a non-trivial estimate for the error term in the Dirichlet divisor problem was deduced. In the following lectures, the A process (Weyl di↵erencing) and the B process, which are processes that transform given exponential sums into new ones, were developed. Then the theory of exponent pairs which provides a set of bounds for exponential sums based on the A and B processes were derived. Finally, as applications, the bound for the error term in the Dirichlet divisor problem was sharpened and a similar bound for the error term in the Gauss circle problem was discussed. A subconvexity bound for the Riemann zeta function on the critical line was established. In the last lecture, a summary of the Bombieri-Iwaniec method which goes beyond the A and B processes was discussed.
Kumar Murty :The lectures began with briefly recalling prime number theorem and primes in arithmetic progression. Then Chowla’s problem of least prime in arithmetic progression was discussed. A short introduction to algebraic number fields was given, splitting of primes in Galois extensions were discussed. A crash course on basic representation theory of finite abelian groups was given, Artil L-function was introduced. Chebotarev density theorem and its applications were studied in great detail.
Course associates : Kashi Viswanadham, Saurab Singh, Subramani M, Usha K Sangale, Senthil Kumar, Brundaban Sahoo, Jaban Meher and Kamalakshya Mahatab did wonderful job of handling the tutorial sessions.

Time Table (AIS ATAN - 2016, KIIT BBS, 13 June - 2 July, 2016)

Speakers

Course Associates

How to reach

Bhubaneswar, the capital city of Odisha state is located in the East part of India and is well connected through Air, Rail and Road Network. KIIT University is located at a distance of 16 km from Bhubaneswar Biju Patnaik Airport and at a distance of 13 km from Bhubaneswar Railway Station.

Air Connection:
The Biju Patnaik Airport of Bhubaneswar is the domestic airport and is well linked by air to New Delhi, Mumbai, Chennai, Kolkota, Hyderabad, and Bangalore.

Rail Connection:

The Bhubaneswar City is well connected by Indian Railway to major cities of India. The East Coast Railway of Indian Railway has provided the fast and superfast train links to New Delhi, Kolkota, Mumbai, Chennai, Bangalore, Hyderabad, Ahmedabad, 
Thiruvannathapuram and other important cities of India as well as within the state.

Road Connection:

Bhubaneswar is situated on the National Highway no.5 that runs between Kolkata and Chennai. Bhubaneswar is well linked to the rest of India by the national highways. The Bus Stand (Baramunda Bust Stand) is located on NH5. Bhubaneswar is on the golden quadrilateral on NH-5 between Kolkata and Chennai. DTS city bus service which is Dream Team Sahara city bus service operates around the city with 15 minutes difference. Getting around in Bhubaneswar does not take much time, due to the perfect town planning and well laid roads.
By Taxi Auto-rickshaw drivers in Bhubaneswar are courteous and helpful. Still, negotiate a rate with the driver beforehand, and make sure that the driver understood your destination. From airport to KIIT University taxi charge is approximately Rs. 250 - Rs. 300.

Climate

Bhubaneswar is located on the coastal plains of Odisha, it has humid, warm and mixed weather during March to October. The average temperatures range between 15°C in winter to a maximum of 45°C in the summer. The summer months from March to May are hot and humid, and temperatures often shoot past 40° C in May. The south west monsoon lashes Odisha in June, bringing relief to the parched environs of Bhubaneswar.