Nabla Seminar
Grad Analysis Seminar at Rutgers University
Grad Analysis $\nabla$ Seminar is a graduate seminar at Rutgers University, Department of Mathematics, Orgnized by Qi Ma since Nov. 2024.
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 could contact Qi Ma at qi.ma@rutgers.edu.
2025 Apr.25 , Speaker: Owen Drummond
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.
2025 Apr.18 , Speaker: Mark Vaysiberg
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.
2025 Apr.11 , Speaker: Aprameya Girish Hebbar
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.
2025 Apr.4 , Speaker: Junyoung Park
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.
2025 Mar.28 , Speaker: Lawrence Frolov
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.
2025 Feb.28 , Speaker: Anupam Nayak
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.
2025 Feb.21 , Speaker: Owen Drummond
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.
2025 Feb.14 , Speaker: Ryan McGowan
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.
2025 Feb.07 , Speaker: Sam Wallace
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.
2025 Jan.31 , Speaker: Qi Ma
Title: Introduction to Stationary Navier-Stokes Equation and Leray's Theory.
Abstract: Navier-Stokes Equation describes the flow of incompressible viscous 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 Helmholtz 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.
2025 Jan.24 , Speaker: Lawrence Frolov
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.
2024 Dec.09 , Speaker: Anupam Nayak
Title: Minimizers for the Area Functional
Abstract: The area functional is of linear growth in the gradient, so a natural space in which to pose the corresponding minimization problem is the W^1,1 Sobolev space. As it turns out, this space is too small, and we should instead consider the larger space of BV functions, and then extend our area functional to this space. I'll explain what this means, and state some results about structure of BV functions and their singularities that help us study this extended problem.
2024 Dec.02 , Speaker: Yuan Gao
Title: Infinitely many bubble solutions for the prescribed scalar curvature problem on $\mathbb{S}^N$
Abstract: The prescribed scalar curvature problem on $\mathbb{S}^n$ asks if one can find a conformally invariant metric such that the curvature becomes a fixed smooth function. In the case that the given function is positive, rotationally symmetric and has a local maximum point between the poles, one can construct infinitely many non-radial positive solutions to this problem by applying Lyapunov-Schmidt reduction arguments. The energy of these solutions can be made arbitrarily large.
2024 Nov.25 , Speaker: Ryan McGowan
Title: Regularity Theory of Energy Minimizing Maps between Riemannian Manifolds
Abstract: By analysing the Dirichlet energy of harmonic functions, one can apply variational methods to get regularity of solutions even from weak solutions. However, in the case of harmonic maps, i.e. maps onto Riemannian manifolds, the situation is decidedly different as one has to contend with the geometry of the target. In this talk, we examine harmonic maps and elucidate how a result due to Schoen and Uhlenbeck gives us analogous regularity results.
2024 Nov.18 , Speaker: Aprameya Girish Hebbar
Title: Pseudo-differential operators and regularity for elliptic PDEs with smooth coefficients.
Abstract: We will start by introducing the Kohn-Nirenberg symbols. We will then introduce Pseudodifferential operators (PsiDO) which enlarges the class of differential operators. We will then study the action of the PsiDOs on Schwartz functions and tempered distributions which allow us to understand certain smoothing properties. Then we will construct elliptic parametrix for elliptic PsiDOs which will in particular show regularity for any elliptic PDE with smooth coefficients, including Sobolev regularity for PDEs in the same class.
2024 Nov.11 , Speaker: Baozhi Chu
Title: Liouville theorems for second order conformally invariant equations and their applications
Abstract: I will present optimal Liouville-type theorems for second order conformally invariant equations. A crucial new ingredient in proving these theorems is our enhanced understanding of solution behaviors near isolated singularities of such equations. These Liouville-type theorems lead to optimal local gradient estimates for a wide class of fully nonlinear elliptic equations involving Schouten (Ricci) tensors. As an application of these Liouville-type theorems and gradient estimates, we establish new existence and compactness results for conformal metrics on a closed Riemannian manifold with prescribed symmetric functions of the Schouten (Ricci) tensor, allowing the scalar curvature of the conformal metrics to have varying signs.