IPDF Progress Seminar
Speaker: Dr. Utsav Dewan, IIT Bombay
Host: Mayukh Mukherjee
Title: Boundary behaviour of eigenfunctions and superharmonic functions on harmonic manifolds of purely exponential volume growth
Time, day and date: 3:00:00 PM – 4:00:00 PM, Wednesday, September 16
Venue: Ramanujan Hall
Abstract: On X, a non-positively curved harmonic manifold of purely exponential volume growth, of dimension at least 3, we study certain quantitative aspects of the boundary behaviour of eigenfunctions and superharmonic functions. We first focus on complex-valued eigenfunctions with eigenvalues lying outside the L^2-spectrum of the Laplace-Beltrami operator and obtain the almost everywhere existence of weighted non-tangential limits, sharp Hausdorff dimension and Hausdorff measure estimates of the boundary exceptional sets for radial limits. Then in the second part, we shift our attention to non-tangential and tangential boundary behaviour of positive superharmonic functions. Most of our results are new even for the homogeneous setting of rank one Riemannian symmetric spaces of non-compact type and Damek-Ricci spaces. Our arguments are based on potential theory adapted to the intrinsic Gromov hyperbolic geometry of X.

Seminar
Speaker: Dr. Smaranika Bera, Department of Mathematics Indian Institute of Engineering Science and Technology, Shibpur
Host: Siuli Mukhopadhyay
Title: Some Problems Related to Nonparametric Inference and Stochastic Ordering
Time, day and date: 4:30:00 PM – 5:30:00 PM, Wednesday, September 16
Venue: Online (https://meet.google.com/ywy-cbhx-eeg)
Abstract: My research work can be broadly divided into two parts. The first part focuses on the inferential aspects of nonparametric ageing classes of life distributions. Particularly in this side I have worked on hypothesis testing problems related to testing exponentiality against different non-monotonic ageing alternatives and on goodness of-fit assessment. The second part deals with selected probabilistic problems in reliability theory. This side mainly involves stochastic ordering, characterization of ageing classes and related probabilistic methods. In this presentation, I shall discuss two selected problems, one from each of these two areas. The first problem is from the inferential side and concerns a two-sample nonparametric testing problem for Second-Order Stochastic Dominance (SSD). We develop a large-sample testing procedure based on U-statistic theory. Since conventional U statistic methods can involve considerable computational effort, we consider alternative approaches based on Jackknife Empirical Likelihood (JEL), Adjusted JEL (AdJEL), and Balanced Adjusted JEL (BaJEL). The performance of these methods is studied through simulation experiments and their practical usefulness is demonstrated using real-life data.
The second problem comes from the probabilistic side and concerns stochastic ordering of order statistics from heterogeneous generalized Gompertz distributions. Here, we consider independent but non-identically distributed generalized Gompertz random variables. One of the main advantages of this distribution is its flexibility in describing different ageing behaviours, including increasing, decreasing and bathtub-shaped hazard rates. Using vector majorization techniques, we establish a variety of stochastic ordering results for order statistics from heterogeneous generalized Gompertz populations.
Although the main focus is on extreme order statistics, the results are also extended to general kth order statistics. We further study multiple-outlier models and obtain stochastic ordering results for extreme order statistics from two independent collections of generalized Gompertz random variables. Stronger ordering results are also obtained for second-order statistics.
Through these two problems, the presentation highlights how both nonparametric inference and stochastic ordering can be used to address interesting and practically relevant questions in reliability theory and survival analysis.

PDF extension seminar
Speaker: Dr. Cyril Jacob, IIT Bombay
Host: Ronnie Mani Sebastain
Title: The Interpolation Problem on Hirzebruch Surfaces
Time, day and date: 11:30:00 AM – 12:30:00 PM, Thursday, September 17
Venue: Ramanujan Hall
Abstract: The interpolation problem for plane curves on $\mathbb P^2$ has been extensively studied, with the SHGH Conjecture providing a conjectural answer. An analogous interpolation problem for curves on Hirzebruch surfaces $\mathbb F_e=\mathbb P(\mathcal O_{\mathbb P^1}\oplus\mathcal O_{\mathbb P^1}(-e))$ was studied by Antonio Laface, who proposed a conjecture closely related to the SHGH Conjecture. In this talk, we examine versions of the SHGH Conjecture for Hirzebruch surfaces. Finally, we prove this conjecture when the number of very general points satisfies $r\leq e+5$. This is joint work with Ronnie Sebastian.

APS
Speaker: Debapriya Ojha, IIT Bombay
Host: Manoj Kumar Keshari
Title: Positive Polynomials on a Strip
Time, day and date: 12:00:00 PM – 1:00:00 PM, Friday, September 18
Venue: Room 113 (https://meet.google.com/vid-bpnc-ujc)
Abstract: Let F be a subfield of the real numbers R. Let S = {g1, . . ., gs} ⊂ F[x1, . . . , xn] be a finite set of polynomials in n variables and let K_S be the basic closed semialgebraic set generated by the non-negativity of S in R^n. Also, let U ⊂ F[t] be the natural choice of generators of a noncompact basic closed semialgebraic set K_U in R with boundary ∂K_U ⊂ F. We show that f ∈ F[x1, . . . , xn, t] > 0 on K_S × K_U and fully m-ic on K_S implies f belongs to the preordering of F[x1, . . . , xn, t] generated by S, U and non-negative elements of F.

Spectral geometry
Speaker: Anusha Bhattacharya, IISER Mohali
Host: Saikat Mazumdar
Title: Anusha Bhattacharya, IISER Mohali
Time, day and date: 3:00:00 PM – 4:00:00 PM, Friday, September 18
Venue: Online (https://us06web.zoom.us/j/84551518386?pwd=7Kv5mWUygLqQPCDE7dpoBq3vMutUbB.1)
Meeting ID: 845 5151 8386
Passcode: 607090
Abstract: A central problem in spectral geometry is to determine how the eigenvalues of the Laplacian on a Riemannian manifold depend on geometric quantities, such as curvature, diameter, volume, and injectivity radius. While uniform spectral estimates are well understood under sectional curvature bounds, considerably less is known when only Ricci curvature bounds are imposed. We address this question by deriving uniform estimates and spectral approximation results for the eigenvalues of the Laplacian using discretizations of the Riemannian manifold.
We consider the class of closed Riemannian n-manifolds with Ricci curvature and injectivity radius bounded below and diameter bounded above. In this class, we prove uniform estimates for the eigenvalues of the Hodge Laplacian and connection Laplacian acting on differential forms. This extends earlier work of Dodziuk and Lott, which required sectional curvature bounds. As an application, we derive a uniform upper bound on the spectral gap for non-compact manifolds with Ricci curvature bounded below. Under the additional assumption of an upper Ricci curvature bound, we consider weighted proximity graphs constructed from sufficiently dense points on the manifold. We then establish uniform convergence of the spectrum of the associated normalized graph Laplacians to those of the Laplace-Beltrami operator.
We further study compact Riemannian manifolds with smooth cylindrical boundary satisfying the same geometric bounds. On these manifolds, we construct graph Laplacians from sufficiently dense point samples on appropriate truncations and establish uniform convergence of the spectrum of the truncated graph Laplacian to the Dirichlet spectrum of the Laplace-Beltrami operator.