∇ 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 5-6 PM at Graduate Student Lounge, Math Dept Hill 701, Rutgers University.
The Fall 2025 edition of the ∇ Seminar is being organized by Larry Frolov, myself, Qi Ma and Anupam Nayak.
You may find the titles and abstracts for the past and upcoming talks below. For talks of the Spring 2025 edition of this seminar, see 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: Dec 8th, 2025
Speaker: Mark Vaysiberg
Title: A compactness theorem for the one phase Bernoulli problem
Abstract: Given a converging sequence of solutions to the Bernoulli problem (classical solutions, critical points of the energy functional, or even semilinear approximations) we would like to be able to pass the pde to the limit. As the energy functional is not continuous in u, the limit may apriori satisfy a more general equation. By studying the free boundary via different types of blow ups, we will succeed in obtaining compactness of solutions.
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Date: Dec 1st, 2025
Speaker: Ryan McGowan
Title: Extension Phenomena in Several Complex Variables
Abstract: The Hartogs extension phenomenon shows that a holomorphic function defined on a subset of C^n may analytically continue to a larger set than the one on which it was originally defined. This behaviour highlights the differences between the analysis of functions of one complex variable and of several complex variables, and helped initially to spur interest in this field. In this talk, we will discuss this phenomenon and its relation to pseudoconvexity, as well as other pathological examples such as the Hartogs triangle. Time permitting, we will also explore the Diederich-Fornæss worm domain.
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Date: Nov 24th, 2025
Speaker: Qiyuan Song
Title: The Differentiable Sphere Theorem and Ricci Flow
Abstract: The differentiable sphere theorem says that any closed, simply connected Riemannian manifold of dimension greater than 3 with pointwise 1/4-pinched sectional curvature(i.e. 1/4<Kmin/Kmax≤1) is diffeomorphic to a sphere. The theorem was proved by Brendle and Schoen in 2008, using Ricci flow and properties of PIC (positive isotropic curvature). In dimension 3, Hamilton proved that any closed, simply connected manifold with positive Ricci curvature is diffeomorphic to a sphere. In this talk I will briefly introduce Ricci flow and sketch the proof of the differentiable sphere theorem.
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Date: Nov 17th, 2025
Speaker: Qi Ma
Title: Stationary Navier-Stokes, open problem and some results.
Abstract: Navier-Stokes equations are the key problem in hydrodynamics. In this talk, I will introduce the stationary Navier Stokes equations to you. We will consider two open problems here: Flow past an obstacle problem in 2D case, and Liouville type problem in 3D case. The recent work by M. Korobkov, K. Pileckas, and R. Russo that the D-solutions are bounded will be presented. I’ll also address some history and other results of the problem if time permits.
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Date: Nov 10th, 2025
Speaker: Samanthak Thiagarajan
Title: The Bernoulli/Alt-Caffarelli Problem
Abstract: The Bernoulli/Alt-Caffarelli Problem is a variation of the Laplace equation in which the energy functional includes a penalty term on the positivity set. This leads to intriguing behaviors of the free boundary—the boundary of the zero set. In this talk, we discuss the optimal regularity of minimizers, showing that they are Lipschitz. We then prove a non-degeneracy estimate and, time permitting, explore density estimates and the regularity of the free boundary.
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Date: Nov 3rd, 2025
Speaker: Hussein Hijazi
Title: Heat Kernels and Trace Asymptotics
Abstract: To analysts, the heat kernel plays an important role as the propagator or fundamental solution of the heat equation, providing regularity estimates and smoothness for parabolic PDEs. However, its efficacy extends far beyond analytic questions. The heat kernel, the spectral profile of its associated Laplacian, and the asymptotic expansion of its trace provide a salient tool connecting PDEs, topology, geometry and spectral theory. My talk will be a gentle overview of its key role within several familiar phenomena, avoiding much technical detail and, hopefully, providing insight to its unification of analysis, geometry and topology rather than just an analytic trick.
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Date: Oct 27th, 2025
Speaker: Aprameya Girish Hebbar
Title: Rotational Symmetry in Riemannian Manifolds
Abstract: In this talk, we will explore the structure of n-dimensional Riemannian manifolds admitting a smooth, effective SO(n) action by isometries, aka rotationally symmetric manifolds. We first characterize this symmetry using Killing fields. We then discuss how SO(3) and O(3) actions induce a local warped product structure for the metric in dimension 3. Finally, we will motivate Brendle's ε-symmetry and its role in the classification of 3-dimensional ancient solutions to the Ricci flow.
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Date: Oct 20th, 2025
Speaker: Anupam Nayak
Title: Generic regularity of minimal surfaces
Abstract: The famous Plateau's problem asks if there is a (smooth) area minimizing hypersurface with prescribed (smooth) boundary. By the Direct Method, the existence of some (possibly unsmooth) area minimizing hypersurface is "easy". The hard part is to then show smoothness. In ambient dimensions upto 7, one (not me!) can show smoothness, but there is a counterexample to smoothness in ambient dimension 8: The Simons cone. Are such counterexamples special, that is, can they be perturbed away generically? In ambient dimension 8 (and also in 9,10, and 11), yes! I'll sketch the proof (due to Hardt-Simon) in dimension 8 (and time permitting, also in dimension 9,10, and 11 (due to Chodosh, Mantoulidis, Schulze, and Wang)).
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Date: Oct 13th, 2025
Speaker: Erik Bahnson
Title: The “simple” equation for a system of interacting Bosons
Abstract: In this talk, I will discuss the “simple” equation, a non-linear integro-differential equation which has been recently studied by Ian Jauslin, Eric Carlen and Elliot Lieb. I will show an existence result. Along the way, we will discuss tools from functional analysis, including contraction semigroups, the Hille-Yosida Theorem and the Kato-Rellich Theorem.
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Date: Sept 29th, 2025
Speaker: Nilava Metya
Title: An elementary proof of Pisier's inequality
Abstract: We'll give an elementary and constructive proof of Pisier’s inequality, which is central in the study of normed spaces and has important applications in convex geometry. Along the way, we will discuss tools of harmonic analysis.
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Date: Sept 22nd, 2025
Speaker: Junyoung Park
Title: On the interior Holder estimate for parabolic equation.
Abstract : In this talk, I will discuss the Krylov Safanov estimates for parabolic equations. The estimate gives us an interior holder continuity of solution to parabolic equations in nondivergence form. We will prove the estimate by first establishing a weak Harnack inequality. Then by applying the Harnack inequality, we will obtain an oscillation decay which leads to the desired holder estimate.
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Date: Sept 15th, 2025
Speaker: Dana Zilberberg
Title: The Fractional Helmholtz Equation.
Abstract: The main focus of my talk will be the Fractional Helmholtz Equation (a non local PDE), and in particular the question of a proper condition at infinity to ensure the uniqueness of the solution. I will briefly talk about the fractional Laplacian and a few of its properties, then I will tell you how to compute the Green's function of the fractional Helmholtz equation and finally I will spend some time sketching the uniqueness of the solution to this equation.
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