∇ 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 Friday 4-5 PM at Graduate Student Lounge, Math Dept Hill 701, Rutgers University.
The Spring 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 Fall 2024 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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Speaker: Owen Drummond
Date: April 25th 2025
Title: The monotonicity formula for energy minimizing maps
Abstract: In this talk, we will define energy minimizing maps and introduce key examples. Then, we will derive the associated variational formulae. A central theme of the talk will be the monotonicity formula: we will explore its significance and highlight its appearance in related geometric contexts such as minimal surfaces, mean curvature flow, and Ricci flow. We will then prove the monotonicity formula for energy minimizing maps. Finally, we will define the density function, introduce tangent maps, and discuss the role of monotone quantities in the study of regularity and singularities.
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Speaker: Mark Vaysiberg
Date: April 18th 2025
Title: Unique continuation for elliptic PDEs
Abstract: In complex analysis we learn that if a holomorphic function is zero on a nonempty open set, then it is identically zero on its domain. The proof is due to analyticity so the same result follows for harmonic functions. As solutions to elliptic pde mimic harmonic functions, we hope that we can prove similar results. Indeed, we will look at the Garofalo Lin proof of strong unique continuation for solutions to divergence form elliptic pde with lipschitz coefficients by means of an Almgren frequency formula. Like many monotonicity formulas, this proof was then built upon to answer many questions relating to the geometric structure of solutions.
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Speaker: Aprameya Girish Hebbar
Date: April 11th 2025
Title: The Li-Yau Harnack Inequality
Abstract: The Harnack inequality, in its classical form, provides a way to compare the values of the positive solution at different points. In their 1986 seminal paper, Peter Li and Shing-Tung Yau introduced a revolutionary differential form of the Harnack inequality for the heat equation on Riemannian manifolds, now known as the Li-Yau Harnack (LYH) inequality. This differential inequality is very important in geometric analysis and they crop up in Hamilton's analysis of the Ricci flow on surfaces as well. In this talk I will discuss the proof of LYH inequality first on closed manifolds, then on open subsets of R^n and finally on a complete non-compact manifold, and will obtain classical Harnack inequality as a corollary.
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Speaker: Junyoung Park
Date: April 4th 2025
Title: Huisken’s theorem on the evolution of closed convex solutions under mean curvature flow
Abstract: Mean curvature flow is a geometric evolution equation which deforms a given hypersurface by its mean curvature vector, and has been extensively studied in the past several decades in the hopes of understanding the long time behavior of the flow. In this talk, we discuss Huisken’s theorem which completely describes the evolution of closed convex hypersurfaces in Euclidean space. We will discuss several proofs of this classical result, and highlight the key ideas behind each method.
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Speaker: Lawrence Frolov
Date: March 28th 2025
Title: Elliptic Boundary Value Problems
Abstract: In this talk we will study linear elliptic boundary value problems in their most general sense. We will answer questions like “what does it even for a distributional solution of Laplace psi=f to satisfy a boundary condition, when distributions may fail to have a trace!” I will also give some explicit applications of the closed graph theorem
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Speaker: Anupam Nayak
Date: February 28th 2025
Title: Minimizers for the Area Functional (continued)
Abstract: The area functional is of linear growth in the gradient, so (cf. my previous talk) the natural space to pose the corresponding minimization problem is BV. We'll now try to show that this minimization problem is actually well posed by proving the lower semicontinuity theorem by Ambrosio-Dal Maso.
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Speaker: Owen Drummond
Date: February 21st 2025
Title: Curvature, Stability, and Scalar Rigidity in Minimal Surface Theory
Abstract: This talk will explore the interplay between energy estimates, curvature bounds, and rigidity results for minimal surfaces. The starting point will be Bernstein's theorem, which provides sufficient condition for a minimal graph of a function in $R^3$ to be a flat plane. Then, we will examine the relationship between energy bounds and curvature, including a key theorem of Schoen and Choi, which provide critical analytic control over the geometry of minimal surfaces. Building on this foundation, we will discuss Daniel Stern’s recent work, which reframes stability in terms of harmonic maps and provides a novel perspective on scalar curvature and minimal surface analysis. Finally, we will examine the famous theorem of Schoen and Yau, which reveals topological constraints on stable minimal surfaces. We will compare two distinct proofs of this theorem: one using the stability inequality and the other relying purely on Stern’s harmonic map framework. Together, these results highlight the deep connections between stability, curvature, and the analytic methods that underpin our understanding of minimal surfaces.
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Speaker: Ryan Mc Gowan
Date: February 14th 2025
Title: Generalising the Riemann Mapping Theorem to Several Complex Variables
Abstract: The Riemann Mapping Theorem is a fundamental result of the theory of one complex variable that classifies all simply connected domains up to biholomorphism. However, once one moves to just two complex variables, it is a famous result due to Poincaré that there cannot exist a biholomorphism between the unit ball and the unit polydisc. In this talk, we will describe some notions specific to several complex variables and introduce the Caratheodory and Kobayashi metrics. These distance-decreasing metrics, as we shall see, allow us to "measure" the failure of the Riemann Mapping Theorem. To finish, we will look at some generalisations of the mapping theorem to the higher dimensional setting.
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Speaker: Sam Wallace
Date: February 7th 2025
Title: Compactness and Limits of Energy for Origami
Abstract: How does material choice affect the work needed to fold origami? In this talk, I'll present the basic modeling for elastic materials, and the progress I've made towards a Gamma convergence result. I'll show some theorems and open problems in my project. This talk will show some commonalities and differences between variational and classical PDE methods, and review some BV theory and GMT in service of the problem.
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Speaker: Qi Ma
Date: January 31st 2025
Title: Introduction to Sationary Navier-Stokes equation and Leray's Theory.
Abstract: Navier-Stokes Equation describes the flow of incompressible vicous fluid. In this talk, I will introduce the stationary Navier-Stokes equation and some classical work in the study of Navier Stokes equation, such as Helmoholtz decomposition, Leray-Schauder Theorem, and invading domain method. I will try to prove some basic existence theorems from Leray's Thm. Some recent results and open problems will also be presented if time permitted.
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Speaker: Lawrence Frolov
Date: January 24th 2025
Title: Must there be a boundary condition?
Abstract: When solving parabolic/hyperbolic evolution equations in some region \Omega of \mathbb{R}^n, we are typically equipped with two things: initial data which holds at t=0 and a boundary condition which holds for all t>0. It's no surprise that we need initial data to solve for the evolution. The purpose of this talk is to ask: must there be a boundary condition? Given a one-parameter family of operators W(t):L^2(\Omega) to L^2(\Omega) with W(t)\psi satisfying the evolution equation in \Omega for all \psi, can we say that W(t)\psi must satisfy a boundary condition for all t>0? Does it even make sense to ask this question, given that these are L^2 functions and the boundary condition takes place on a measure zero subset of \Omega? In this talk we will answer this question and more for certain linear evolution equations motivated by quantum mechanics.
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