In the 2024 Spring semester, the seminars will be held both online and in-person (CU 215).  Online talks are usually on Thursdays starting at 4:30 PM, and in-person talks will be on Mondays at 4:20 PM.

Organizers: Huai-Dong Cao, Ao Sun, Xiaofeng Sun.

If you are interested in participating in the seminar, please email Ao (aos223 at lehigh dot edu).

Thursday 01/25/24 4:30-5:30 (Zoom)

Speaker: Zhenhua Liu (Princeton University)

Title: General behavior of area-minimizing subvarieties

Abstract: In this talk, we will review some recent progress on the general geometric behavior of homologically area-minimizing subvarieties, namely, objects that minimize area with respect to homologous competitors. They are prevalent in geometry, for instance, as holomorphic subvarieties of a Kahler manifold, or as special Lagrangians on Calabi-Yau, etc. A fine understanding of the geometric structure of homological area-minimizers can give far-reaching consequences for related problems. 

De Lellis and his collaborators have proven that area-minimizing integral currents have codimension two rectifiable singular sets. A pressing next question is what one can say about the geometric behavior of area-minimizing currents beyond this. Almost all known examples and results point towards that area-minimizing subvarieties are subanalytic, generically smooth, and calibrated. It is natural to ask if these hold in general.

In this direction, we prove that all of these properties thought to be true generally and proven to be true in special cases are totally false in general cases. We prove that area-minimizing subvarieties can have fractal singular sets. Smoothable singularities are non-generic. Calibrated area-minimizers are non-generic. Consequently, we answer several conjectures of Frederick J. Almgren Jr., Frank Morgan, and Brian White from the 1980s.

Monday 01/29/24 4:20-5:20 (in-person)

Speaker: Tianyue Liu (UPenn)

Title: Toric Einstein 4-manifolds with nonnegative sectional curvature 

Abstract: We study some interesting curvature properties of Einstein 4 manifolds in the presence of symmetry. In particular, we show that the only $T^2$ invariant Einstein metrics with nonnegative sectional curvature on closed 1-connected four manifolds are the known examples: the round metric on $S^4$, the Fubini-Study metric on $\C P^2$, or the product metric on $S^2\times S^2$.

Thursday 02/01/24 4:30-5:30 (Zoom)

Speaker: Zihui Zhao (JHU)

Title: Boundary unique continuation of harmonic functions 

Abstract: Unique continuation property is a fundamental property for harmonic functions, as well as a large class of elliptic and parabolic PDEs. It says that if a harmonic function vanishes at a point to infinite order, it must vanish everywhere. In the same spirit, we are interested in quantitative unique continuation problems, where we use the local growth rate of a harmonic function to deduce some global estimates, such as estimating the size of its singular or critical set. In this talk, I will talk about some recent results together with C. Kenig on boundary unique continuation.

Thursday 02/08/24 4:30-5:30 (Zoom)

Speaker: Tristan Ozuch (MIT)

Title: Selfduality along Ricci flow and instabilities of Einstein metrics 

Abstract: Einstein metrics and Ricci solitons are the fixed points of Ricci flow and model the singularities forming. They are also critical points of natural functionals in physics. Their stability in both contexts is a crucial question, since one should be able to perturb away from unstable models. I will present new results and upcoming directions about the stability of these metrics in dimension four in joint work with Olivier Biquard. The proofs rely on selfduality, a specificity of dimension four.

Thursday 02/15/24 4:30-5:30 (Zoom)

Speaker: Jian Song (Rutgers-New Brunswick)

Title: Geometric analysis on singular complex spaces 

Abstract: We establish a uniform Sobolev inequality and diameter bound for Kahler metrics, which only require an entropy bound and no lower bound on the Ricci curvature. We further extend our Sobolev inequality to singular Kahler metrics on Kahler spaces with normal singularities. This allows us to build a general theory of global geometric analysis on singular Kahler spaces including the spectral theorem, heat kernel estimates, eigenvalue estimates and diameter estimates. Such estimates were only known previously in very special cases such as Bergman metrics. As a consequence, we derive various geometric estimates, such as the diameter estimate and the Sobolev inequality, for Kahler-Einstein currents on projective varieties with definite or vanishing first Chern class.

Thursday 02/16/24 1:30-2:30 (in-person) Please note the special time!

Speaker: Ernani Ribeiro Jr (Federal University of Ceará)

Title: Rigidity of compact quasi-Einstein manifolds with boundary 

Abstract: In this talk, we discuss the geometry of compact quasi-Einstein manifolds with boundary. This topic is directly related to warped product Einstein metrics, static spaces and smooth metric measure spaces. We show that a 3-dimensional simply connected compact quasi-Einstein manifold with boundary and constant scalar curvature must be isometric to either the standard hemisphere $S^3_{+},$ or the cylinder $I\times S^2$ with product metric. For dimension n=4, we prove that a 4-dimensional simply connected compact quasi-Einstein manifold with boundary and constant scalar curvature is isometric to either the standard hemisphere $S^4_{+},$ or the cylinder $I\times S^3$ with product metric, or the product space $S^2_{+}\times S^2$ with the doubly warped product metric. Other related results for arbitrary dimensions are also discussed. This is a joint work with J. Costa and D. Zhou.

Monday 02/19/24 4:20-5:20 (in-person)

Speaker: Jingwen Chen (UPenn)

Title: Mean curvature flow with multiplicity $2$ convergence

Abstract: Mean curvature flow (MCF) has been widely studied in recent decades, and higher multiplicity convergence is an important topic in the study of MCF. In this talk, we present an example of an immortal MCF that converges to a plane with multiplicity $2$. Additionally, we will compare our example with some recent developments on the multiplicity one conjecture, and the min-max theory. This is based on joint work with Ao Sun.

Thursday 02/29/24 4:30-5:30 (Zoom)

Speaker: Otis Chodosh (Stanford)

Title: On minimizers in Gamow's liquid drop model 

Abstract: The physicist George Gamow proposed a model of the nucleus called the "liquid drop model" which combines the standard isoperimetric problem with an additional repulsive term. I will describe some recent work with Ian Ruohoniemi concerning the shape of minimizers in this problem for a definite range of volumes.

Monday 03/11/24 - Friday 03/15/24

Spring Break

Monday 03/18/24 (in-person)

Speaker: Yang Yang (JHU)

Title: The anisotropic Bernstein problem 

Abstract: The Bernstein problem asks whether entire minimal graphs in R^{n+1} are necessarily hyperplanes. It is known through spectacular work of Bernstein, Fleming, De Giorgi, Almgren, Simons, and Bombieri-De Giorgi-Giusti that the answer is positive if and only if n < 8. The anisotropic Bernstein problem asks the same question about minimizers of parametric elliptic functionals, which are natural generalizations of the area functional that both arise in many applications and offer important technical challenges. We will discuss the recent solution of this problem (the answer is positive if and only if n < 4). This is joint work with C. Mooney.

Thursday 03/21/24 4:30-5:30 (zoom)

Speaker: Peter McGrath (NC State)

Title: New Minimal Surfaces via Equivariant Eigenvalue Optimization

Abstract: I will discuss joint work with Karpukhin, Kusner, and Stern on the application of equivariant optimization of Laplace and Steklov eigenvalues on surfaces to constructions of embedded minimal surfaces in the 3-sphere and 3-ball.  Riemannian metrics which maximize such normalized eigenvalues are known to give rise to branched minimal immersions by first eigenfunctions into spheres and balls, with codimension in general expected to grow with the topology of the surface.  Nonetheless, there is a class of group actions on surfaces which allow each topological type and satisfy optimal eigenvalue bounds, ensuring that any branched minimal immersion by first eigenfunctions is a codimension-1 embedding, which doubles a minimal 2-sphere (or 2-disk) and has area less than 8\pi (or 2\pi).  We show maximizing metrics can be found for these actions, leading in particular to the existence of orientable, embedded minimal surfaces with free boundary in the 3-ball with arbitrary topological type, answering a question of Fraser and Li.

Monday 03/25/24 (in-person)

Speaker: Keaton Naff (MIT)

Title: Area estimates and intersection properties for minimal hypersurfaces in space forms

Abstract: In this talk, I wish to discuss recent work (joint with Jonathan J. Zhu) on area estimates for minimal submanifolds and ``half-space" Frankel properties for minimal hypersurfaces in space forms. In the first setting, we will discuss sharp area estimates for minimal submanifolds in the curved space forms which pass through a prescribed point (building on work of Brendle-Hung). Our work settles the question in the hyperbolic setting, but leaves open an interesting outstanding question in the sphere. This leads naturally to the question of stability of minimal submanifolds in the hemisphere and in this direction we will demonstrate a Frankel property for the hemisphere (and other related settings).

Thursday 03/28/24 4:30-5:30 (zoom)

Speaker: Nicolò De Ponti (University of Milano-Bicocca)

Title: Unique continuation at infinity: Carleman estimates on general warped cylinders

Abstract: We discuss vanishing results for solutions of $|\Delta u| \leq q_1 |u| + q_2 |\nabla u|$ that decay to zero at infinity on a Riemannian manifold. The main new ingredient that we present is a Carleman estimate suitable for dealing with general warped cylindrical ends and potential functions $q_1, q_2$. We also discuss some geometric applications to conformal deformations and to minimal graphs. 

Joint work with Stefano Pigola and Giona Veronelli.

Monday 04/01/24 (in-person)

Speaker: Teng Fei (Rutgers-Newark)

Title: The Type IIA flow and its applications

Abstract:  The equations of flux compactifications of Type IIA superstrings were written down by Tomasiello and Tseng-Yau. To study these equations, we introduce a natural geometric flow called Type IIA flow on symplectic Calabi-Yau 6-manifolds. We prove the wellposedness of this flow and establish the basic estimates. We show that the Type IIA flow can detect canonical geometric structures on certain symplectic manifolds. We also prove its dynamical stability and apply it to prove a stability result about symplectic deformations of Calabi-Yau manifolds. This is based on joint work with Phong, Picard and Zhang.

Thursday 04/11/24 (Zoom)

Speaker: Nick Edelen (Notre Dame)

Title: Improved regularity for minimizing capillary hypersurfaces

Abstract: We give improved estimates for the size of the singular set of minimizing capillary hypersurfaces: the singular set is always of codimension at least 4, and this estimate improves if the capillary angle is close to $0$, $\pi/2$, or $\pi$. For capillary angles that are close to $0$ or $\pi$, our analysis is based on a rigorous connection between the capillary problem and the one-phase Bernoulli problem.  This is joint work with Otis Chodosh and Chao Li.

Everett Pitcher Lectures

Richard Schoen

Monday, April 15, 2024 Lecture I (Public Lecture) Geometry is a singular subject Lewis Lab 270; 7:30 PM; Lobby Reception at 6:45 PM 

Wednesday, April 17, 2024 Lecture II Geometric aspects of General Relativity Neville 1; 4:30 PM 

Thursday, April 18, 2024 Lecture III Minimal surfaces in higher codimension Chandler-Ullmann 218;  4:30 PM

Lecture 1 Title: Geometry is a singular subject 

Abstract: A great deal of current research in differential geometry has to do with what mathematicians call singularities. These range from corners which arise in optimal networks and compound soap bubbles to infinite curvature regions in geometric flows to black holes in general relativity. This talk will attempt to explain in a non-technical way what a singularity is and how they arise in some examples. In addition we will give a broad outline of the role of singularities in proofs of major theorems including the Poincare conjecture, differentiable sphere theorem, and the Penrose singularity theorem in general relativity. 

Lecture 2 Title: Geometric aspects of General Relativity 

Abstract: We will introduce the Einstein equations of general relativity which describe how the spacetime geometry interacts with matter fields. The energy conditions on matter fields induce constraints on the geometry. In this lecture we will explain in a general way the nature of these constraints and introduce some of the mathematical tools which can be used to study such geometric problems. We will especially focus on the importance of minimal hypersurfaces in the study. Finally we will briefly describe how some theorems in the subject are affected by singularities which can arise in these hypersurfaces in higher spatial dimensions. The talk will be aimed at a general mathematical audience. 

Lecture 3 Title: Minimal surfaces in higher codimension

Abstract: In the previous lecture we saw the relationship of stable minimal hypersurfaces to the geometry of Riemannian manifolds with non-negative scalar curvature. If we ask how stable minimal submanifolds in higher codimensions relate to Riemannian geometry we find many open problems. For two dimensional surfaces in higher codimension there are complex analytic methods which yield some information. For higher genus surfaces such methods are not well understood. We will introduce this subject and describe recent progress on the structure of stable minimal surfaces and applications to manifolds with positive isotropic curvature. The talk will be aimed at a general mathematical audience.

Monday 04/22/24 (in-person, 3:00-4:00, CU 218) Please note the special time and location!

Speaker: Ruobing Zhang (Princeton)

Title: Metric geometric aspects of Einstein manifolds 

Abstract: This lecture concerns the metric Riemannian geometry of Einstein manifolds. We will exhibit the rich geometric/topological structures of Einstein manifolds and specifically focus on the structure theory of moduli spaces of Einstein metrics. My recent works center around the intriguing problems regarding the compactification of the moduli space of Einstein metrics, which tells us how Einstein manifolds can degenerate. The metric geometry of Einstein manifolds is a central theme in modern differential geometry, and it is deeply connected to fundamental problems in algebraic geometry, analysis of nonlinear PDEs, and mathematical physics. We will introduce recent major progress and propose several open questions in the field.

Monday 04/22/24 (in-person, 4:20-5:20, CU215)

Speaker: Bing Wang (University of Science and Technology of China)

Title: On Kähler Ricci shrinker surfaces

Abstract: We prove that any Kähler Ricci shrinker surface has bounded scalar curvature. Combining this estimate with earlier work by many authors, including the curvature estimate of Munteanu-Wang, the volume estimate of Cao-Zhou, the construction and classification of Bamler, Conlon-Cifarelli-Deruelle and Sun,  we provide a complete classification of all Kähler Ricci shrinker surfaces. This is joint work with Yu Li.

Thursday 04/25/24 (in-person, 4:00-5:00, CU215) Please note the special time!

Speaker: Yi Lai (Stanford)

Title: Riemannian and Kahler flying wing steady Riccisolitons 

Abstract: Steady Ricci solitons are fundamental objects in the study of Ricci flow, as they are self-similar solutions and often arise as singularity models. Classical examples of steady solitons are the most symmetric ones, such as the 2D cigar soliton, the O(n)-invariant Bryant solitons, and Cao’s U(n)-invariant Kahler steady solitons. Recently we constructed a family of flying wing steady solitons in any real dimension n≥3, which confirmed a conjecture by Hamilton in n=3. In dimension 3, we showed all steady gradient solitons are O(2)-symmetric. In the Kahler case, we also construct a family of Kahler flying wing steady gradient solitons with positive curvature for any complex dimension n≥2, which answers a conjecture by H.-D. Cao in the negative. This is partly collaborated with Pak-Yeung Chan and Ronan Conlon.

Monday 05/06/24 (in-person, 4:20 - 5:20, CU 215)

Speaker: Costante Bellettini (University College London)

Title: PDE analysis on stable minimal hypersurfaces: curvature estimates and sheeting

Abstract: We consider properly immersed two-sided stable minimal hypersurfaces of dimension n. We illustrate the validity of curvature estimates for n \leq 6 (and associated Bernstein-type properties). For n \geq 7 we illustrate sheeting results around "flat points". The proof relies on PDE analysis. The results extend respectively Schoen-Simon-Yau's estimates (obtained for n \leq 5) and Schoen-Simon's sheeting theorem (valid for embeddings).

Thursday 05/09/24 (in-person)

Joint Geometry & Probability Seminar

Speaker: Chih-Wei Chen (National Sun Yat-sen University)

Title: How to compute the hessian of a function based on random samples on a manifold?

Abstract: We provide a systematic convergence analysis of the Hessian operator estimator from random samples supported on a low dimensional manifold. We show that the impact of nonuniform sampling, curvature, and boundary effect on the widely applied Hessian operator estimator is asymptotically negligible. In particular, our result justifies the key step in the algorithm Hessian Eigenmap for generic data manifolds, and clarifies the convergence rate of local quadratic regression on a Riemannian manifold. This is joint work with Hau-Tieng Wu (Courant Institute, NYU).