• Friday May 2, 2025

Speaker: Divyansh Agrawal

Title: d-plane transform: unique and non-unique continuation

Abstract: The d-plane transform maps functions to their integrals over d-planes (d-dimensional affine subspaces) in R^n. For d=n-1 and d=1, it coincides with the classical Radon and X-ray transforms respectively. We consider the following question: If a function vanishes in a bounded open set, and its d-plane transform vanishes on all d-planes intersecting the same set, does the function vanish identically? We give a complete answer to the above question. It turns out, surprisingly, (or not!) that the answer depends on the parity of d. For d an even integer, we show by producing an explicit counterexample that neither the d-plane transform nor its normal operator has this property. On the other hand, for d odd, the question above has an affirmative answer. The proofs use some classical tools from Harmonic analysis and integral geometry, together with some geometric and analytic tricks. Based on a joint work with Nisha Singhal.

Fridays 1:30pm in SCHM 314

• Friday April 25, 2025

Speaker: Michael Poole

Title: Analysis of Entities Less Than Any Real Magnitude

Abstract: This talk is a brief introduction to nonstandard analysis. I will discuss the motivation for the concrete construction of the infinitesimal, and its novel use in proving results from classical real analysis. We do this by extending the real numbers to the field of hyperreal numbers, after which we give a nonstandard (and more intuitive, in my opinion) proof of the intermediate value theorem using infinitesimal and infinite quantities.

• Friday April 18, 2025

Speaker: Ben Doyle

Title: On Moments of Exponential Sums over r-Free Polynomials

Abstract: The moments of exponential sums which are restricted to a certain subset of the integers can reveal interesting additive information about these sets. With this in mind, Balog and Ruzsa, and later Keil, examined the k-th moments of exponential sums over the r-free integers, which are integers not divisible by r-th powers. In particular, Keil found the exact order of magnitude of these moments for all k>0, with the exception of the case k=1+1/r, where he missed by a factor of log(X). We examine the analogous problem in the function field setting, obtaining a result analogous to Keil's, with two added improvements. In the case k=1+1/r, we obtain the exact order of magnitude, and in the case k>1+1/r, we refine the result to an asymptotic formula using the Hardy-Littlewood circle method.

• Friday April 11, 2025

Speaker: Estepan Ashkarian

Title: Integration Theories for Stochastic Analysis

Abstract: Stochastic Analysis was part of the pioneering work of Itô in the 1940s. The theory developed probabilistic methods to initiate an integration theory for stochastic processes. This theory covers various types of "integrators", namely semimartingales. However, not all interesting processes possess this property, such as fractional Brownian motion (fBm). In the late 1990s, Terry Lyons published a revolutionary paper in which he developed a pathwise/analytic theory of integration to deal with integrating with respect to non-semimartingales. This led to a large body of interdisciplinary work, which continues to this day. The purpose of this talk is to give the audience a sense of why we care about having an integration theory for more general processes, and how one can achieve this.

• Friday March 28, 2025

Speaker: General Ozochiawaeze 

Title: On the Biharmonic Scattering by Impenetrable Cavities with Limited Data

Abstract: In this talk, we will discuss extending the fairly new extended sampling method (ESM) to inverse shape problems for Biharmonic Scattering with limited aperture data. This computational simple and rigorous point sampling method defines an indicator function that recovers the location of the scatterer of interest. Using far field data, we will first analyze the ESM and show it is a stable reconstruction method. We will focus on the cases of scattering by a clamped region by a single incident wave at fixed frequency, a few incident waves at fixed frequency, and a single incident wave at multiple frequencies, showing the method's versatility.

• Friday Feb 28, 2025

Speaker: Kale Stahl

Title: A New Sampling Imaging Functional for Imaging Photonic Crystals

Abstract: Our goal is to solve the inverse problem of determining the shape of penetrable periodic scatterers from scattered field data. We propose a sampling method with a novel indicator function for solving this inverse problem, with the indicator function having a well-defined form that makes it simple to implement and robust against noise in the data. We provide detailed analysis of the resolution and stability of this functional. Our numerical study shows that the proposed sampling method is more stable than the factorization method and more efficient than the direct or orthogonality sampling method in reconstructing periodic scatterers.

• Friday March 7, 2025

Speaker: Otto Baier

Title: An application of distribution theory to entire functions on C^n

Abstract: We will start with a brief introduction to distributions (aka "generalized functions"). The Fourier transform has a natural extension to these generalized functions, and this machinery will allow us to take the Fourier transform of a much larger class of functions on R^n (including polynomials). We will cover a key result: the distributions with support only at the origin are exactly those whose Fourier transforms are polynomials. Then, we will see how this fact along with the Paley–Wiener theorems proves that entire functions on C^n subject to a particular growth restriction must be polynomials.

• Friday Feb 14, 2025

Speaker: Bilal Ahmed 

Title: Learning in Probabilistic Models

Abstract: Graphical models are a powerful framework for representing complex domains using probability distributions, with numerous applications in machine learning, computer vision, natural language processing, etc. They bring together graph theory and probability theory and provide framework for modelling large collections of random variables with complex interactions. In this talk we will cover a brief introduction to graphical models with special emphasis on their application in machine learning. We will also talk about information entropy, KL divergence and their utility for estimating the parameters of probabilistic models.

• Friday January 31, 2025

Speaker: Pedro Morales 

Title: Resonance Expansion of Waves

Abstract: In its simplest form, scattering theory refers to the mathematical study of how particles or waves interact with obstacles or potentials and how they scatter at large distances. In this context, it focuses on the asymptotic behavior of solutions to certain partial differential equations. In this talk, we will discuss how to obtain resonance expansions of solutions to the wave equation in the presence of a compactly supported potential by examining the poles of the meromorphic continuation of the resolvent of the perturbed operator.

• You can see information about talks from Spring 2023 and Fall 2023 here: https://sites.google.com/view/purdue-graduate-student-analys/