Commutative Algebra seminar
Speaker: Sourjya Banerjee (IMSc)
Host: Manoj Keshari
Title: From Unimodular Rows to Zero Cycles over Real Varieties
Time, day and date: 4:00:00 PM - 5:00:00 PM, Monday, September 1
Venue: Ramanujan Hall
Abstract: A unimodular row of length $n$ over a commutative Noetherian ring $R$ (with $1 \neq 0$) is a row vector $(v_1,\ldots,v_n) \in R^n$ such that the ideal generated by $v_1,\ldots,v_n$ is the whole ring $R$. We discuss unimodular rows and their connections with projective modules, specifically addressing when a unimodular row of length $n$ can be completed to a row of an invertible matrix and how this question arises naturally in the study of projective modules. We present a classical example of a unimodular row of length $(d+1)$ over a $d$-dimensional smooth real variety that cannot be completed to such a row. We then describe a class of $d$-dimensional real varieties where every unimodular row of length $d+1$ is completable to a row of an invertible matrix. We discuss some applications to the study of the $d$-th Euler class group $\mathrm{E}^d(R)$ defined by Bhatwadekar and Raja Sridharan, the Levine--Weibel Chow group of zero cycles $\mathrm{CH}_0(\mathrm{Spec}(R))$, and the natural maps between them. If time permits, we discuss unimodular rows of length $d$ over $d$-dimensional smooth real varieties. The final part is based on ongoing joint work with Jean Fasel.
 

Statistics and Probability seminar
Speaker: Chirag Modi (New York University)
Host: Manas Rachh
Title: ATLAS: Adapting Trajectory Lengths and Step-Size for Hamiltonian Monte Carlo
Time, day and date: 3:00:00 PM - 4:00:00 PM, Tuesday, September 2
Venue: Ramanujan Hall
Abstract: Hamiltonian Monte Carlo (HMC) is the most widely used Markov chain Monte Carlo technique for Bayesian inference in high dimensions. However, the performance of HMC is sensitive to several tuning parameters like step size and trajectory lengths for leapfrog integration that can be difficult to set. As a result, it can still struggle to accurately sample distributions with complex geometries, e.g., varying curvature, due to the constant step size. In this talk, I will present a strategy to locally adapt the step size parameter of HMC at every iteration by evaluating a low-rank approximation of the local Hessian and estimating its largest eigenvalue. I will then combine it with a strategy to similarly adapt the trajectory length by monitoring the no U-turn condition, resulting in an adaptive sampler, ATLAS: adapting trajectory length and step-size. I will further use a delayed rejection framework for making multiple proposals that improve the computational efficiency of ATLAS, and develop an approach for automatically tuning its hyperparameters during warmup. Finally, I will compare ATLAS with NUTS on a suite of synthetic and real-world examples, and show that i) unlike NUTS, ATLAS is able to accurately sample difficult distributions with complex geometries, ii) it is computationally competitive to NUTS for simpler distributions, and iii) it is more robust to the tuning of hyperparameters.
 

Mathematics Colloquium
Speaker: Manoj K Keshari (IIT Bombay)
Title: Positive polynomials and sums of squares
Time, day and date: 4:00:00 PM - 5:00:00 PM, Wednesday, September 3
Venue: Ramanujan Hall
Abstract: Given a closed set K in R^n, where R denote the real numbers, it is an important problem to decide whether a given polynomial in n variables is non-negative on K. We will start with two motivations to study non-negative polynomials. Then we will discuss the following descent problem: If f has rational coefficients and there is a certificate over R that f is non-negative over K, can one find such a certificate over rational numbers.
 

Commutative Algebra seminar
Speaker: Tony Puthenpurakal (IIT-B)
Title: F-modules- IV
Time, day and date: 4:00:00 PM - 5:00:00 PM, Thursday, September 4
Venue: Ramanujan Hall
Abstract: We continue our discussion on F-modules