Seminars

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.

Vigyan Pratiba Teachers Workshop

Vigyan Pratiba Teachers Workshop

We prove a bijection between the branching models of Kwon and Sundaram, conjectured by Lenart-Lecouvey. To do so, we use a symmetry of the Littlewood-Richardson coefficients in terms of the hive model. Along the way, we introduce a new ranching rule in terms of flagged hives. This talk is based on a joint work with Dr. Jacinta Torres.

Colloidal open crystals — sparsely populated periodic structures, comprising low-coordinated colloidal particles — are attractive targets for self-assembly because of their variety of applications, for example, as photonic materials, phononic and mechanical metamaterials, as well as porous media [1-4]. Colloidal particles in their primitive form offer short-range isotropic interactions, and thus tend to form close-packed crystals. Despite the advances over the last two decades in the synthesis of colloidal particles, endowed with anisotropic and/or specific interactions [5-7], programming self-assembly of colloidal particles into open crystals has proved elusive. In this presentation, I will first talk about a series of computational studies that establish facile bottom-up routes for rationally designed patchy particles to self-assemble into a variety of colloidal open crystals, especially those much sought-after as photonic crystals [8-12]. The strategies include encoding hierarchical self-assembly pathways and ring size selection, in close connection with advances in colloid synthesis. I will also talk about how hierarchical self-assembly of designer patchy particles can instead be exploited to develop a colloidal model of water – a classic example of empty liquids [13]. I will demonstrate how this colloidal model unravels a novel topological distinction in terms of entanglement between the two liquids of different densities involved in the liquid-liquid phase transition (LLPT) [13] – originally hypothesised in connection with the host of anomalous thermodynamic properties in water [14]. Finally, I will illustrate how entanglement can emerge as a general mechanism for densification with a hierarchy of topological transitions in a network liquid, which is known to densify via two successive LLPTs [15]. References [1] X. Mao, Q. Chen and S. Granick, Nat. Mater. 2013, 12, 217. [2] J. D. Joannopoulos, P. R. Villeneuve and S. Fan, Nature, 1997, 386, 143. [3] K. Aryana and M. B. Zanjani, J. Appl. Phys., 2018, 123, 185103. [4] X. Mao and T. C. Lubensky, Annu. Rev. Condens. Matter Phys., 2018, 9, 413. [5] S. C. Glotzer and M. J. Solomon, Nat. Mater., 2007, 6, 557. [6] W. B. Rogers, W. M. Shih and V. N. Manoharan, Nat. Rev. Mater., 2016, 1, 16008. [7] T. Hueckel, G. M. Hocky and S. Sacanna, Nat. Rev. Mater., 2021, 6, 1053. [8] D. Morphew, J. Shaw, C. Avins and D. Chakrabarti, ACS Nano, 2018, 12, 2355. [9] A. B. Rao, J. Shaw, A. Neophytou, D. Morphew, F. Sciortino, R. L. Johnston and D. Chakrabarti, ACS Nano, 2020, 14, 5348. [10] A. Neophytou, V. N. Manoharan and D. Chakrabarti, ACS Nano, 2021, 15, 2668. [11] A. Neophytou, D. Chakrabarti and F. Sciortino, Proc. Natl. Acad. Sci. USA, 2021, 118, e2109776118. [12] W. Flavell, A. Neophytou, A. Demetriadou, T. Albrecht and D. Chakrabarti, Adv. Mater., 2023, 35, 2211197. [13] A. Neophytou, D. Chakrabarti and F. Sciortino, Nat. Phys., 2022, 18, 1248. [14] P. H. Poole, F. Sciortino, U. Essmann and H. E. Stanley, Nature, 1992, 360, 324. [15] A. Neophytou, F. W. Starr, D Chakrabarti and F. Sciortino, Proc. Natl. Acad. Sci. USA, 2024, 121, e2406890121.

Health is arguably one of the most crucial aspects of life, and advancements in medicine have significantly extended human lifespans over the centuries. Since the sequencing of the first human genome over two decades ago, genomic technologies have rapidly evolved, enabling earlier disease detection. In this presentation, I will provide a brief overview of how genomics and machine learning are advancing precision medicine. I will also discuss the limitations of these approaches and highlight our limited understanding of human biology, particularly female biology. Additionally, I will present a few cases where mathematical modeling can potentially enhance our understanding of human physiology and pathology. Finally, I will share recent, concerning health demographic data from India, which have significant policy implications.

Fluctuation analysis is an experimental design in molecular biology where fluctuations across independent clones grown from genetically identical single cells allow to draw inferences about heritable mechanisms. Originally invented by Luria and Delbrück in the 1940s to demonstrate that in bacteria, genetic mutations arise in the absence of selective pressure, fluctuation analysis has been used more recently in mammalian cells to demonstrate the existence of transcriptional memory, that is, slow fluctuations in gene expression that persist for multiple cell divisions but are ultimately transient. Here I will present ongoing work on a mathematical model for fluctuation analysis of gene expression using stochastic processes on trees, in which genes exhibit transcriptional memory as soon as the half-life of their fluctuations exceeds twice the cell cycle time. I will show that in equilibrium, causal links between genes can be reconstructed from correlations across independent clones, a surprising example of a fluctuation-dissipation relation that does not require two-time correlations. I will conclude by speculating on a potential role of transcriptional memory in establishing long-range spatial correlations in gene expression that have been observed in spatial transcriptomics data.

Shock waves, whether triggered by localized energy perturbations or continuous energy input, are essential for understanding non-equilibrium dynamics in phenomena like atomic blasts and supernova explosions. In the case of a blast in a homogeneous medium, the system’s dynamics are described by the exact solution of the Euler equation, known as the Taylor-von Neumann-Sedov (TvNS) solution. However, molecular dynamics simulations reveal significant inconsistencies with this theoretical framework. In this talk, I will present our investigation into shock propagation in both homoge- neous and inhomogeneous media, considering scenarios initiated by instantaneous blasts and those driven by sustained energy input. Using event-driven molecular dynamics sim- ulations, numerical solutions of the Navier-Stokes equation, and scaling solutions of the Euler equation, we explore critical questions regarding the hydrodynamic description of shocks. For blast waves, we resolve discrepancies between TvNS predictions and simulation results by emphasizing the crucial role of heat conduction at the shock center, incorpo- rating the Navier-Stokes equation. In inhomogeneous media, the Euler equation provides a valid description only at specific values of the inhomogeneity parameter, where dissi- pation are comparable to Euler terms. In contrast, for splash problems, where energy is introduced at the boundary of vacuum and an inhomogeneous medium, the Euler equa- tion suffices under particular mathematical treatments. However, for driven shocks, the Euler equation fails entirely, while the Navier-Stokes equation continues to describe the system accurately, even in this extreme non-equilibrium conditions. Our findings highlight the importance of dissipation terms in governing shock dynam- ics and reveal key crossover regimes between the Euler and Navier-Stokes frameworks in blast and driven shock problems.

TBA

Cluster cluster aggregation (CCA) is a nonequilibrium, irreversible phenomenon where particles, or clusters coalesce on contact to form larger clusters. The most common approach to study CCA is the mean-field Smoluchowski coagulation equation. However, this equation can be exactly solved only when the rate of collision is independent of the colliding masses (constant kernel), is the sum of the masses (sum kernel), or the product of the masses (product kernel). Moreover, the solution of the Smoluchowski equation describes only the typical or average trajectories and moments of the mass distribution, and does not provide any information about atypical or rare trajectories.

In this talk, I will describe a biased Monte Carlo algorithm, and the analytical formalism developed to study probabilities of rare events in CCA for arbitrary collision kernels, and the results obtained therein. The algorithm can sample probabilities as low as 10^{-40}, and obtains atypical and typical trajectories. We establish the efficacy of the algorithm by benchmarking the numerically obtained large deviation function of interest, and the trajectories, with the exact answer for constant kernel. We establish an analytical path wise large deviation principle for arbitrary collision kernels, and obtain the explicit large deviation functions, for constant, sum and product kernels, as well as the optimal evolution trajectories for constant and sum kernels, all of which show excellent agreement with Monte Carlo simulations. We use this analytical formalism to obtain the large deviation function of interest and atypical trajectories for $k-$nary coalescence, $kA —> lA$.

Insect sociality

Conversation series talk on wildlife photography

Public lecture on insect sociality

Nature and Numeracy for Chennai SCERT: 2024-25