APS
Speaker: Soumyajit Acharya, IIT Bombay
Host: Chandan Biswas
Title: Uniform estimates for Fourier restriction to polynomial curves
Time, day and date: 10:00:00 AM – 11:00:00 AM, Thursday, September 25
Venue: Ramanujan Hall
Abstract: We study uniform L p (R d ) → L q (λγ dt) bounds for Fourier restriction to degree-N polynomial curves γ equipped with affine arclength, in the range q = (d(d + 1)/2) p ′ and q > (d^2 + d + 2)/2 , with constants depending only on d, N, p. The method uses the extension formulation, a local decomposition with control of Lγ, a Littlewood–Paley square function, and almost-orthogonality across dyadic shells. We also note the link to adjoint restriction for the paraboloid.

Special Colloquium
Speaker: Venky Krishnan, TIFR-CAM Bangalore
Host: Suman Sahoo
Title: Boundary rigidity problem, tensor tomography and microlocal analysis
 Time, day and date: 12:00:00 PM - 1:00:00 PM, Thursday, September 25
Venue: Ramanujan Hall
Abstract: Boundary rigidity problem, loosely speaking, asks whether it's possible to determine a Riemannian metric in a bounded domain from the knowledge of the associated boundary distance function. This problem is closely related to the integral geometry problem of determining a symmetric 2-tensor field from the knowledge of its integrals along geodesics connecting boundary points. Microlocal analysis naturally enters into the study of integral geometry problems. The talk will survey some known results and some interesting open problems.

APS
Speaker: Adarsh Gupta, IIT Bombay
Host: Sudarshan Gurjar
Title: Homotopy groups of smooth projective surfaces
 Time, day and date: 12:45:00 PM – 1:45:00 PM, Thursday, September 25
Venue: Room 215
Abstract: In this seminar, I will talk about the second homotopy group of smooth projective surfaces. When the universal cover of these surfaces are holomorphically convex. Then these groups are free abelian groups.

APS
Speaker: Lal Bahadur Sahu, IIT Bombay
Host: Sandip Singh
Title: Bruhat Decomposition
Time, day and date: 3:00:00 PM – 4:00:00 PM, Thursday, September 25
Venue: Room 215
Abstract: In this seminar, we will discuss the Bruhat decomposition, a fundamental structural result in the theory of reductive algebraic groups. It asserts that such a group can be expressed as a disjoint union of double cosets of a chosen Borel subgroup, with each coset uniquely determined by an element of the Weyl group. As a corollary, we will also derive a normal form for elements of reductive algebraic groups, illustrating how this decomposition provides both conceptual clarity and computational tools in the study of algebraic groups.

APS
Speaker: Suraj Panigrahi, IIT Bombay
Host: Kummari Mallesham
Title: Rational Numbers represented by cubic forms in 11 variables
Time, day and date: 4:00:00 PM – 5:00:00 PM, Thursday, September 25
Venue: Conference Room
Abstract: Given a rational cubic form C(X) and a non-zero rational number r, We show that the equation C(X)=r has a rational solution provided C(X) is a non-degenerate cubic form in at least 11 Variables, This is an improvement upon earlier result by Tim Browning

Commutative Algebra seminar
Speaker: Tony Puthenpurakal, IIT Bombay
Title: F-modules 5
Time, day and date: 4:00:00 PM - 5:00:00 PM, Thursday, September 25
Venue: Ramanujan Hall
Abstract: We continue our discussion on F-modules

APS
Speaker: Mayukh Choudhury, IIT Bombay
Host: Debraj Das
Title: Asymptotics of Cross-validation in Lasso and High Dimensional GLMs
Time, day and date: 5:00:00 PM – 6:00:00 PM, Thursday, September 25
Venue: Ramanujan Hall
Abstract: In the first half of this talk, we will discuss the asymptotic theory of $K$-fold Cross Validation in Lasso. Lasso is one of the widely used regularization methods in regression. Statisticians usually implement Lasso in practice by choosing the penalty parameter in a data-dependent way, the most popular being the $K-$fold cross-validation (or $K-$fold CV). In this case, we establish that only under some moment type conditions, Lasso with $K$-fold CV based penalty is $n^{1/2}-$consistent, but not variable selection consistent. Additionally, we establish the validity of Bootstrap in approximating the distribution of the $K-$fold CV based Lasso estimator. Therefore, our results theoretically justify the use of $K-$fold CV based Lasso estimator to perform statistical inference. We validate our Bootstrap method in finite samples based on simulations.

In the later half, we shift our focus to Generalized linear models or GLMs. With continuation to earlier reports, so far, we have presented asymptotic results for properly centered and scaled Lasso estimator and its Bootstrapped version, when the underlying dimension $d$ was fixed. Now we will observe the asymptotic traits when $d$ can grow with $n$. We will first discuss the asymptotic distribution of properly centered and scaled GLM estimator when parameter dimension $d$ increases at a slower rate than $n$. In a later regime, when $d>>n$, we employ sparsity in the model, typically Lasso in GLM. When the penalty sequence $n^{-1/2}\lambda_n\to\infty$, considering Lasso to be VSC, we establish that Gaussian approximation fails to this end for properly centered and scaled Lasso estimator. As an alternative, a PB-Lasso estimator is defined and asymptotic results are established to mimic the original distribution of Lasso.