Mathematical Sciences 2015

“The instinct that there is a kind of structure underlying • M.Sc. in Mathematics Prof. Mahan has made a substantial impact is what we call instinct or intuition. That’s probably from Indian Institute of in the fields of geometric group theory, the most powerful thing in the doing of mathematics. Technology, Kanpur low-dimensional topology and complex Without that I don’t think a person can be a practicing • Ph.D. in Mathematics from geometry. His work in all these fields is mathematician really. The formal reasoning comes University of California, characterized by its creativity and clever use Berkeley of delicate geometric arguments. afterwards. The intuition tells you that this is the direction that I ought to look in. Otherwise a computer would be doing mathematics!” Mahan Mj Professor, School of Mathematics, Tata Institute of Fundamental Research, Mumbai

Uncovering structures beyond the observable universe

While Euclid only worked assuming all five postulates, mathematicians through the centuries have been trying to prove or disprove the fifth Mahan’s work lies at the One way that mathematicians postulate assuming only the first four. While complicated and intersection of hyperbolic study 3- manifolds is to study Prof. Mahan Mj works in the A consequence of this quest was the complex, hyperbolic geometry and topology, and them by considering the field of hyperbolic geometry Topology is often discovery of several non-Euclidean geometry and topology also specifically low-dimensional special ‘surfaces’ embedded on and topology. Most of us called ‘rubber sheet geometries, including hyperbolic translate into hypnotically topology—the study of abstract them. A ‘surface’ in topology are familiar with Euclidean th geometry’. Topologists geometry in the 19 century. beautiful visual patterns and mathematical shapes called is a 2- dimensional topological geometry—the geometry of tend to see the world Hyperbolic geometry is a geometry shapes. Among these are manifolds in four or lower manifold. Mahan’s work was flat surfaces. When Euclid of in terms of stretchy, on a surface that is everywhere fractals which are never- dimensions. Mahan was able to proof that every Kleinian Alexandria first formulated the twisty objects i.e. shapes saddle-shaped. Shapes behave in ending self-similar patterns establish a central conjecture surface group admits a Cannon- principles of what is now called that can be twisted and peculiar and particular ways in the which repeat infinitely. They in a program which was Thurston map (mapping in Euclidean geometry around glued together without hyperbolic plane. In mathematical are found everywhere around established in the 1970’s by topology is a function with 300 BCE, he came up with five tearing. terms, while on a flat surface, the us in nature, in geometry the theoretical mathematician a special structure). He has postulates. Of these, the fifth sum of angles of a triangle is always and even in the visualization William Thurston to study recently extended this to postulate stated that “In a 180 degrees, in a hyperbolic plane, of algebraic formulae. The hyperbolic 3-manifolds to include all finitely generated plane, through any point not on the sum of angles of the triangle principles of hyperbolic complement his approach to Kleinian groups. Mahan’s work a given line only one new line will always be less than 180 degrees. geometry has also been his famous Geometricization has many applications in the can be drawn that is parallel to Around the same time in the used in art by artists like M.C. Conjecture—which says that all study of hyperbolic manifolds. the original one.” th 19 century, the field of topology Escher in his paintings such possible 3- dimensional spaces developed as a way of trying to as Snakes, Circle Limit III and are made up of eight types of understand geometry and set theory. Ascending and Descending. geometric pieces.

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