∇ Seminar (read as "nabla seminar" or "Grad seminar") is a student run Analysis and Geometry seminar at the Department of Mathematics, Rutgers University. The seminar features various expository and research talks from students meeting every Monday 4-5 PM at Graduate Student Lounge, Math Dept Hill 701, Rutgers University.
The Spring 2026 edition of the ∇ Seminar is being organized by myself, Qi Ma and Anupam Nayak.
You may find the titles and abstracts for the past and upcoming talks here.
Those who wish to receive e-mails for latest information of the seminar may subscribe to our mail-list by sending “Subscribe” to grad_analysis-join@email.rutgers.edu or you can email me at ah1531[at]math[dot]rutgers[dot]edu.
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Date: April 26th 2026
Speaker: Mark Vaysiberg
Title: Reifenberg Theorems
Abstract: Reifenberg’s topological disc theorem says that if a set is close to a k dimensional ball at all scales, then it is biholder to a k dimensional ball. This was used by Cheeger–Colding to study properties of Gromov–Hausdorff limits. After giving an outline of the proof, I will also discuss the discrete and rectifiable Reifenberg theorems of Naber–Valtorta which relax a pointwise assumption to controlling an L2 quantity. It turns out that this L2 hypothesis has many applications to studying sets arising from elliptic problems.
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Date: April 13th 2026
Speaker: Priya Kaveri
Title: The space of stationary m-dimensional surfaces
Abstract: In this talk, I will discuss Brian White’s work on the local structure of the space of immersed m-dimensional surfaces that are stationary for a broad class of elliptic parametric functionals, including the area functional. After reviewing the first and second variation formulas, I will explain how one accounts for the degeneracy caused by reparametrization invariance, leading to a Banach manifold structure on the space of stationary immersions.
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Date: April 6th 2026
Speaker: Jakub Niksinski
Title: Introduction to Decoupling Inequalities and Vinogradov's Mean Value Theorem
Abstract: Assume we are given a curved surface and a collection of Schwartz functions which are frequency supported on small disjoint neighborhoods of that surface. Decoupling inequalities measure, in terms of Lp norms, how the size of the sum of these functions relates to the sizes of the individual components.
These inequalities were introduced in 2000 by Thomas Wolff in his influential work on the local smoothing conjecture for the wave equation. In 2015, Bourgain and Demeter revolutionized the field by introducing techniques allowing one to prove decoupling in optimal ranges of exponents. Since then, decoupling inequalities and the techniques surrounding them have become a central tool in Euclidean harmonic analysis with far-reaching applications to: restriction problems, PDEs, incidence geometry, geometric measure theory (specifically Kakeya-type problems), and even analytic number theory.
We will briefly go over some of these applications and see how Vinogradov's Mean Value Theorem easily follows from the general decoupling inequality for the moment curve proved by Bourgain, Demeter, and Guth in 2016.
I intend to spend most of the talk giving a sketch of the proof of the decoupling inequality for the parabola and showcasing some of the crucial ideas involved.
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Date: Mar 30th 2026
Speaker: Owen Drummond
Title: The Alt-Caffarelli Free Boundary Problem with a Volume Constraint
Abstract: Caffarelli and Alt published their landmark paper on the Bernoulli problem in 1981. Concurrently, they had a parallel problem in mind: what if we prescribe the volume of the positivity set? As it turns out, classical methods fail to provide minimizers for the most straightforward unconstrained functional. This necessitates a more sophisticated penalized approach to recover the powerful techniques used in the original program. Instead, we require some heavier analysis, estimates, a different kind of penalized functional to apply these methods. Remarkably, all of the same regularity results for the minimizer carry over for the volume constraint problem, and the volume of the positivity set for the penalized minimizer automatically corrects to the prescribed volume in the problem.
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Date: March 23rd 2026
Speaker: Sam Wallace
Title: About Liouville’s Theorem – From harmonic function to nonlinear PDEs.
Abstract: Thin elastic structures (TES), whether 1D rods or 2D sheets, are becoming widespread in mechanical design, used to combat heart disease and explore the depths of space. The basic reason for their use is that TESs deform by local isometry, which allows for any length shortening map of a TES to be nearly obtained (Nash-Kuiper). Thus many qualitatively different shape changes can be obtained by compression.
One type of shape change, origami, is becoming a widespread design paradigm; it is characterized by flat, rigid panels with flexible joints. I will present 2-3 results that illuminate fine properties of maps modeling flat-folded origami. This work is part of my thesis.
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Date: March 2nd 2026
Speaker: Aprameya Girish Hebbar
Title: Metric Flows: Heat Kernels, F-convergence and Compactness
Abstract: In his landmark 2020 work, R. Bamler introduced metric flows which are weak synthetic analogues of smooth flows. Roughly, a metric flow consists of time-slices of metric spaces equipped with heat kernel type measures, providing a "parabolic analogue of a metric space". I will explain how the axioms of metric flows arise naturally from smooth Ricci flow, the convergence notion (“F-convergence”), and if time permits, how this yields a compactness theory for sequences of (super) Ricci flows.
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Date: Feb 16th 2026
Speaker: Qi Ma
Title: About Liouville’s Theorem – From harmonic function to nonlinear PDEs.
Abstract: Liouville’s Theorem plays a fundamental role in the uniqueness of solutions. In this talk, we will start from the most trivial case – harmonic function. Classical proof, proof by energy estimation and proof by Fourier transform will be presented. Finally, we will generalize our result to nonlinear equations and introduce the some results of removable singularities in Navier-Stokes.
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Date: Feb 9th 2026
Speaker: Samanthak Thiagarajan
Title: Schauder Estimates three different ways
Abstract: Schauder estimates are classical estimates for second order, uniformly elliptic PDEs. Naturally, there are lots of ways to prove them! We will discuss three interesting ways in this talk.
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Date: Feb 2nd 2026
Speaker: Junyoung Park
Title: On Eells and Sampson theorem
Abstract: The idea of using heat type equations to solve problems in geometry has been proven to be an extremely powerful method. In this talk, we will discuss one of the earliest success of such approach, which is the use of harmonic map heat flow to find minimizing harmonic maps in a given homotopy class, under a geometric constraint on the target manifold.
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UPCOMING!
Date: May 4th 2026
Speaker: Anupam Nayak
Title: TBA
Abstract: TBA
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