## Pictures

The Madhava Competition Nurture Camp, Chapter 1, will be held at ISI Bangalore during June 05-June 10, 2017. This camp is a part of the Madhava Mathematics Competition 2017. Participation in the Nurture Camp is only by invitation, based on the performance of the Madhava Mathematics Competition 2017. See the website for more details: Madhava Mathematics Competition.

Coordinator, organizer and contact: Jaydeb Sarkar (jaydeb AT gmail DOT com) from Indian Statistical Institute, Bangalore.

#### List of participants

Accommodation And Direction: Arrival: On June 04. Departure: On June 10. Accommodation has been arranged at the ISI Guest House, ISI Hostels and ISI quarters, inside ISI campus. Participants may report to the main security desk at the main entrance of ISI campus for further instructions upon arrival.

Taxi from Airport or Rail station to ISI and back: (A) One can book a taxi in advance by calling one of the taxi operators: (1) Meru Cabs (080--44224422), (2) Easy Cabs (080--43434343) and (3) Karnataka Taxi (080--43464346). Taxis are also available outside the airport (after exiting the terminal, turn left to find the "Airport Taxi" sign. The taxi ride to ISI should cost you around Rs 1000-1200) and Bangalore City railway station and Yeswanthpur rail station. One can hire a prepaid (recommended) auto from the rail stations as well. Bangalore airport has a very convenient app based taxi service (like Ola and Uber) located just next to the airport bus terminus. (B) Vayu Vajra Service is bus a service provided by KSRTC from airport to the town. There is no direct bus from the airport to ISI but the nearest place is Mysore Road Bus Stand (AKA: Bapujinagar). Get KIAS-10 from airport to Bapujinagar (the last stop) and from the same terminal, you can ride for onward trip to ISI (Jairamdas bus stop).

## Schedule


9.30-11.00 11.00-11.30
TEA
11.30-1.00 13.00-14.30
LUNCH
14.30-15.30 15.30-16.00
TEA
16.00-17.00
Monday
June 05
9.15-9.30- Registration
& Welcome
Sury Raghavan Tirthankar Tutorial
Tuesday, June 06 Sridharan Raghavan Bhat Tutorial
Wednesday, June 07 Sridharan Sury Shreedhar Tutorial
Thursday, June 08 Manjunath Bhatia Siva Tutorial
Friday, June 09 Manjunath Bhatia Manish Tutorial

## Speakers

Thematic Talks: Rajendra Bhatia (ISI Delhi), Manjunath Krishnapur (IISc), K. N. Raghavan (IMSc, Chennai), Raja Sridharan (TIFR Mumbai), B. Sury (ISI Bangalore).

Special Talks: Siva Athreya (ISI Bangalore), B.V. Rajarama Bhat (ISI Bangalore), Tirthankar Bhattacharyya (IISc), Manish Kumar (ISI Bangalore), Shreedhar Inamdar (ISI Bangalore).

Tutors: Geetanjali Phatak (S. P. College, Pune) and S Nanda Kishore Reddy (ISI Bangalore).

Lecture Notes: (1) K. N. Raghavan (A) Note 1. (B) Note 2. (C) Note 3. (D) Note 4. (2) Raja Sridharan .

## Titles and Abstracts

Siva Athreya (ISI Bangalore)
Title: "Shape of the Earth and the method of least squares".
Abstract: In the 18th century, while dealing with astronomical and geodesic measurements,the scientists confronted "the problem of combining inconsistent equations". People who worked on this problem and contributed towards its solutions include Euler, Laplace, Gauss and Legendre among many others. I shall discuss the history of the (Statistics and Linear Algebra) problem.

B.V. Rajarama Bhat (ISI Bangalore)
Title: "Differentiators of linear maps and critical points of polynomials.".
Abstract: Consider a linear map A on C^n with characteristic polynomial p. A differentiator for A is an (n-1) dimensional subspace of C^n such that A compressed to this subspace has characteristic polynomial equal to \frac {1}{n}p^{\prime}, where p^{\prime} is the derivative of p. Roots of p^\prime are called critical points of p. Rajesh Periera found interesting properties of critical points of polynomials making use of the notion of differentiators. We explore these ideas and state an open problem known as Sendov’s conjecture.

Rajendra Bhatia (ISI Delhi)
Title: "Averages and means".
Abstract: We will discuss different notions of averages of numbers, vectors, and matrices. The reasons for considering these notions, their properties and applications will be presented.

Tirthankar Bhattacharyya (IISc)
Title: "The Schur Cohn Theorem".
Abstract: Schur's theorem and Cohn's later generalization address the question of finding a criterion to decide how many roots of a polynomial (of degree n, say) are inside the unit disc {z ∊ C : |z| < 1 } and how many are in the set {z ∊ C : |z| > 1}. They accomplish it by using elementary matrix theory. We shall talk of positive definite matrices of defect rank one and associated quadratic forms and then will build towards the proof of the theorem.

Shreedhar Inamdar (ISI Bangalore)
Title: "Group Actions and Applications".
Abstract: I will prove that every action decomposes in to disjoint orbits and explain all the terms involved and also give applications of group actions. This talk will be accessible to all first year B.Sc. students.

Manjunath Krishnapur (IISc)
Talk 1 - Title: "Dimension reduction". Abstract: Data such as images are stored in a computer as vectors in a high dimensional space. Is it possible to keep only a part of the vector so as to reduce the dimension and hence the storage space required? We discuss algorithms that do precisely this. Notions that one learns in linear algebra such as eigenvalues and eigenvectors appear naturally here.
Talk 2 - Title: "Eigenvalues of a symmetric random matrix". Abstract: Let A be an nxn symmetric matrix whose upper triangular entries are i.i.d Normal random variables with zero mean and variance 1. What can we say about the eigenvalues of A? We show a celebrated theorem of Wigner that the eigenvalues are distributed according to the semi-circle density. Techniques will use basic linear algebra, probability and some topology.

Manish Kumar (ISI Bangalore)
Title: "Solving polynomial equations".
Abstract: There is a beautiful relation between solving polynomial equations and subgroups of symmetric groups, discovered by Galois. This led to an answer to a long standing problem of solving a quintic equation. We will discuss these questions and delve in to some topics in Galois theory.

K. N. Raghavan (IMSc, Chennai)
Talk 1 - Title: "Ruler and compass construction". Abstract: Can a given angle be trisected (in some way analogous to the way we bisect it using a ruler and compass)? This and other geometric questions which were open since antiquity were solved by Gauss in the last decade of the 18th century (Gauss was a teenager at that time). The idea behind the stunningly simple solution is something that we now take for granted: to represent points in the plane as complex numbers! After translating the problems to algebra by means of this idea, some rudimentary field theory is all that is needed to finish them.
Talk 2 - Title: "Heaps and applications". Abstract: The combinatorial notion of a "heap" has many varied applications. It is useful to think in terms of heaps, for instance, in the passage to what is called the "thermodynamic limit" in statistical mechanics. We will illustrate the elegance and usefulness of heaps by means of some applications to graph theory (since these require minimal background to appreciate).

Raja Sridharan (TIFR Mumbai)
Title: "Symbolic Methods".
Abstract: In my two lectures, I will talk about the Symbolic Methods used by George Boole in the development of logic and the symbolic methods used by Grassmann in the development of Linear and Multilinear Algebra. I will discuss some applications of Symbolic methods to Geometry, Algebra and Topology.

B. Sury (ISI Bangalore)
Title: "Primes and Integer Polynomials".
Abstract: Several aspects of the theory of polynomials with integer coefficients are related to properties of prime numbers. We discuss some of these.