FACETS 2017 : Schedule

Combinatorics is fundamental to most of mathematics; we often need to count precisely how many objects there are of a certain kind. Algebra is broadly the study of abstract structures and rules for their manipulation. In this talk, we will consider many examples of interesting algebraic functions which arise as solutions to counting problems. Many beautiful algebraic identities turn out to have combinatorial explanations, and the search for such explanations has often led to new insights and discoveries.

Gauss's Theorema Egregium tells us that the sphere and the plane have different curvatures, and as a consequence there can never be a perfect map of the surface of the Earth. However, there are projections which manage to preserve important features of our planet's surface, such as infinitesimal surface area, geodesic curves, and angles of intersection. We will look at some classical examples of maps that perfectly capture the above properties, and discuss some of the underlying geometry. Then we will take a look at a few modern innovations, including a tetrahedron that can be "unfolded" into an infinitely repeating, center-less map of Earth.

Modern democracies rely on general elections to form governments. People elect a candidate from a choice of alternatives, and the one declared elected is supposed to "represent" all the voters. This also holds for voting in committees to arrive at a decision.

The world over, there are many ways to conduct elections. Our Presidential election is very different from that of France or the USA. Is there a mathematical way to decide which is the best election procedure ? Are there ways to ensure that nobody can manipulate elections to achieve the results they want ?

Attempts to answer such questions bring together ideas and techniques from not only politics, economics and sociology, but also mathematics and computer science. The talk is meant to introduce the basic ideas of this fast-growing field of research.

(Working on these problems could get you a Nobel Prize in Economics; there have been three such winners, including a famous Indian.)

A linkage is a mechanism built with rigid bars that are connected using freely revolving joints. We come across many linkages in our daily life, for example, a desk lamp, deployable mirror, vehicle suspensions etc. Even a robot arm is an example of a linkage. The configuration space of a linkage is the set of all its possible states or the places it can reach. A clear and concise understanding of this set is needed in several real life applications. In this talk I will explain how one comes up with a simple mathematical model called a flexible polygon in order to 'visually' describe the configurations of closed linkages.

1. Alok Laddha (Physics, CMI)
2. Gautam Menon (Physics and Biophysics, IMSc)
3. Priyavrat Deshpande (Mathematics, CMI)
4. Shubashree Desikan (Science Desk, The Hindu)