Student Seminar
Speaker: Aryan Gupta, IIT Bombay
Host: Manas Rachh
Title: Finding Structure with Randomness: The Randomized Rangefinder
Time, day and date: 10:00:00 AM – 11:00:00 AM, Thursday, April 23
Venue: Ramanujan Hall
Abstract: Computing the singular value decomposition of a large dense matrix costs O(n³) operations which is prohibitive when only the leading rank-k structure is needed. The randomized rangefinder (Halko–Martinsson Tropp, 2011) replaces this with an O(n²k) algorithm: multiply the matrix by a short, random Gaussian test matrix, orthogonalize, and project. The resulting approximation is provably close to the Eckart–Young optimum, and a single step of power iteration makes it near-optimal even on slowly decaying spectra.
This talk presents the algorithm, its error guarantee, and three experiments run on the IIT Bombay math cluster: (1) wall-clock scaling showing a 759× speedup at n = 15,000; (2) spectral-norm near-optimality across power iteration depths; and (3) a free adaptive stopping criterion via an exact Frobenius-norm identity. The talk closes with a brief look at how the same ideas extend to kernel matrices through randomly pivoted Cholesky.
No background beyond linear algebra is assumed.

Topology and Geometry seminar
Speaker: Dr. Sumanta Das, IIT Bombay
Host: Rekha Santhanam
Title: Mapping Class Groups: Teichmüller Theory and Geometry
Time, day and date: 11:30:00 AM – 12:30:00 PM, Thursday, April 23
Venue: Ramanujan Hall
Abstract: he problem of parametrizing all possible complex structures on a given surface dates back to Riemann and can be studied via Teichmüller spaces. This subject lies at the intersection of complex manifolds, Fuchsian groups, differential geometry, algebraic geometry, and several other areas.
In this and subsequent talks, we will define and explore Teichmüller space through several equivalent points of view: complex structures, hyperbolic metrics, representation varieties, and quasiconformal deformations.
Moreover, we will discuss the moduli space of Riemann surfaces of genus g, defined as the set of all biholomorphism classes of closed Riemann surfaces of genus g. We will explain how this moduli space is obtained as the quotient of Teichmüller space by the action of the Mapping Class Group.