Vigyan Pratiba Teachers Workshop

Vigyan Pratiba Teachers Workshop

We study the problem of testing whether a function $f: \reals^n \to \reals$ is a polynomial of degree at most $d$ in the distribution-free testing model. Here, the distance between functions is measured with respect to an unknown distribution $D$ over $\reals^n$ from which we can draw samples. In contrast to previous work, we do not assume that $D$ has finite support. We design a tester that given query access to $f$, and sample access to $D$, makes $\poly(d/\eps)$ many queries to $f$, accepts with probability $1$ if $f$ is a polynomial of degree $d$, and rejects with probability at least $2/3$ if every degree-$d$ polynomial $P$ disagrees with $f$ on a set of mass at least $\eps$ with respect to $D$. Our result also holds under mild assumptions when we receive only a polynomial number of bits of precision for each query to $f$, or when $f$ can only be queried on rational points representable using a logarithmic number of bits. Along the way, we prove a new stability theorem for multivariate polynomials that may be of independent interest. This is a joint work with Arnab Bhattacharyya, Esty Kelman, Noah Fleming, and Yuichi Yoshida, and appeared in SODA’23.

Given a symmetrizable Kac–Moody Lie algebra $\mathfrak{g}$, it is natural to ask what are all the symmetrizable Kac–Moody Lie algebras $\mathfrak{g}'$ so that we have a graded embedding $\mathfrak{g}' \hookrightarrow \mathfrak{g}$? In this talk, we will focus on embeddings where the image of the derived ideal of $\mathfrak{g}'$ is isomorphic to a subalgebra of $\mathfrak{g}$ generated by real root vectors, known as root-generated subalgebras. Dynkin proved that for finite-dimensional semisimple Lie algebras, there exist bijections between root-generated subalgebras, closed subroot systems, and $\pi$-systems containing positive roots. Therefore, $\pi$-systems and closed subroot systems arise naturally in the embedding problem. In a joint work with Dipnit Biswas and R. Venkatesh, we classify the symmetric real closed subsets of affine root systems and study the (Dynkin's) correspondence between symmetric real closed subsets and the regular subalgebras they generate. We prove that the exact analogue of (extended) Dynkin's result does not hold in the affine case. The main obstacle is that, unlike the finite case, there are symmetric real closed subsets which are not subroot systems. For symmetrizable Kac–Moody Lie algebras, Naito and Morita proved in the early 1990s that linearly independent $\pi$-systems lead to embeddings. We prove that every real closed subroot system admits a unique $\pi$-system containing positive roots, although the associated $\pi$-system can be infinite (hence linearly dependent). Nonetheless, we prove that the exact analogue of Dynkin's result holds and describe a presentation of root generated subalgebras. This implies that only those closed subroot systems with an associated linearly independent $\pi$-system appear in the embeddings. This work is joint with Deniz Kus and R. Venkatesh. Finally, for rank-2 Kac–Moody algebras, we show that every closed subroot system leads to embedding by proving that the associated $\pi$-system is linearly independent. Moreover, we classify all $\pi$-systems in the rank-2 case, proving they can contain at most two elements. We also precisely describe the types of embedded subalgebras. This work is joint with Chaithra P.

Amorphous materials, such as silica glasses, metallic glasses, colloids, foams, and granular materials, exhibit heterogeneous patterns of dynamics, often referred to as dynamical heterogeneity, particularly at lower temperatures or under external driving forces. Dynamical heterogeneity consists of mobile and immobile spatial regions, suggesting the presence of non-trivial correlations. Recently, machine learning techniques have been applied to predict and forecast future dynamics, as well as the patterns of dynamical heterogeneity, using only static snapshots of the system. In this talk, I will introduce the basics of glassy dynamics and dynamical heterogeneity, along with the relevant machine learning techniques. I will then discuss our recent work on predicting glassy dynamics, with a focus on fluctuations in particle sizes.