Mathematical teasers (answers at the bottom of the page, proof available in my work)

1. Can you encode everything on this page into the singular set of a 4-D area-minimizer of a smooth metric on a 7-D tori? 

What about encoding it into the singular set of a 5-D area-minimizer of a smooth metric in mod 2 and integral homology on an 8-D real projective space, so that the info is stable under perturbations of metrics? 

2. Suppose one has an immersed submanifold that is area-minimizing in a metric g. Which of the following about the moduli space of area-minimizers near g is impossible?

I. For generic metrics near g, area minimizers are smooth.

II. For all metrics near g, area-minimizers are singular. 

III. For open set A of metrics, minimizers are smooth. For open set B of metrics, minimizers are singular. For a residual set C of metrics, minimizers possess fractal singularity. The closure of A, B, and C each contains g.

What problems do I work on and what I have done?

For my grandma: I study soap films in higher dimensional analogs of donuts. I experimentally verified that these soap films have many unexpected properties, in sharp contrast to all previous knowledge about soap films.

For the general mathematical audience: I work on the geometric behavior of homologically area-minimizing subvarieties. Namely, objects that minimize area with respect to homologous competitors. They are just analogs of soap films in higher dimensions. Interestingly, 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. For instance, by using classical results about such area-minimizing subvarieties, one can prove a very weak formulation of the Hodge conjecture is true: rationally the (p,p) classes of a projective manifold are spanned by area-minimizing subvarieties as close to holomorphic as one wants, in an Euler–Maclaurin sense. There are also direct calculus of variation criteria for determining if a (p,p) class admits holomorphic chains. (Unfortunately, it is hard to compute, otherwise one can solve the Hodge conjecture this way.) 

My Ph.D. work unveils surprising behaviors of these subvarieties in general, challenging established beliefs and opening the door to a wealth of uncharted territories. I prove that many properties thought to be true generally and proven to be true in special cases are totally false in general cases, including but not limited to subanalytic singular set, generic smoothness, calibrated minimizers, L^2 curvature bounds, etc. 

For mathematicians who are interested in a more detailed explanation: I mainly study area-minimizing currents in integral, finite coefficient, and real coefficient homology settings. In view of the pioneering work of Almgren, 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 fine geometric behavior of area-minimizing currents beyond this. The main goal of my Ph.D. thesis is to show that the fine geometric behavior of area-minimizers is way more thrilling and uncharted than people have previously imagined. I will list them in chronological order.

Fractal singular sets

1. On a conjecture of Almgren: area-minimizing surfaces with fractal singularities, submitted, preprint available at https://arxiv.org/abs/2110.13137

2. Area-minimizing surfaces with fractal singular sets: prevalence, moduli space, and refinements on strata, preprint available at https://arxiv.org/abs/2306.07271

3. Every finite graph arises as the singular set of a compact 3-d calibrated area minimizing surface, submitted, preprint available at https://arxiv.org/abs/2106.03199

Comment: It has been an open problem since the early days of geometric measure theory whether area-minimizing currents can possess fractal singular sets, and Almgren wrote this conjecture down explicitly in the early 1980s. Before my work, all area-minimizing integral currents had at least a subanalytic singular set. Together with the groundbreaking constructions of Leon Simon, Almgren's conjecture is settled completely with the following features:

a. Fractal singular sets of area-minimizers exist with any real Hausdorff dimension allowed by the regularity theorems, in both Z and mod 2 coefficient homology. (Theorem 1.2 and 1.3 in 1) Also, for stable stationary varifolds, a similarly sharp statement holds. (Theorem 1.4)

b. They are prevalent on the homology level, even in metrics arbitrarily close to the flat torus. (Theorem 1.1 and 1.3 in 2)

c. The moduli space of area-minimizers in nearby metrics is wilder than expected. Even in open sets of metrics arbitrarily close to the flat torus, one can simultaneously have open sets of metrics with non-smoothable singular sets, open sets of metrics with smooth minimizers, and residual sets of metrics with fractal singular sets, the closure of each scenerio containing the flat metric. (Theorem 1.4, 1.5 and Corollary 1 of 2)

d. Different fractal strata can limit to each other.  (Theorem 1.6 of 2)

e. For concrete illustration, for any 6-dimensional manifold M and any integral homology class Sigma on M, there is a smooth Riemannian metric on M, so that the unique area-minimizer in Sigma has a singular set K of almost any prescribed closed subset of any finite graph.  (3 and Theorem 1.7 of 2)

Non-smoothable singular sets

4. Homologically area-minimizing surfaces with non-smoothable singularities, preprint available at https://arxiv.org/abs/2206.08315

Comment: Since area-minimizers can possess singularities, a fundamental question is whether generic perturbations of metric can smooth out the singularities. This is a conjecture raised by Brian White in the 1980s. Before my work, all general results available, which are in dimension 2 and codimension 1, proved the generic smoothness of minimizers and the disappearance of singularities. It is widely expected that this is true in general. I settle White's conjecture in almost all dimensions and codimensions with a negative answer. Singularities can always persist, with the following features.

f. For a smooth d+c dimensional manifold  with d at least 3, c at least 3, define the singularity dimension Mu_d to be the infimum of real number mu, so that there exists a metric g, near which in open sets of metrics, a chosen homology class has area-minimizers with singular sets of Hausdorff dimension at least mu. I prove that Mu_d is at least max{d-c,d-5} (Theorem 1.1 of 4). In other words, if one wants to cover all generic metrics, one cannot get lower than Mu_d=max{d-c,d-5} for the possible size of the singular set.

h. The conclusion holds for both integral and mod 2 homology. As a corollary, every mod 2 and integral homology class of dimension at least 5 and codimension at least 3 must have open sets of metrics with singular area-minimizers. 

i. I use two obstructions for smoothing singularities. One relies on transverse intersections of not-so-small angles. The other relies on the bordism ring. Both obstructions are present, even in the setting of special Lagrangians. In the former obstruction, I completely determine the local moduli space. (Theorem 1.2 and Section 4.2 of 4). Due to the defining nature of Mu_d in f, even restricting to analytic metrics cannot dispense with the singularities.

Non-calibratable minimizers

5. Homologically area-minimizing surfaces that cannot be calibrated, available at https://arxiv.org/abs/2310.19860

Comment: Since the 1970s, almost all area-minimizing currents known to date are proven to be area-minimizing using calibrations. In the codimension 1 case, Federer proved that all area-minimizers are calibrated. In sharp contrast to the known literature, I show that in codimension larger than 1, for every homology class, on every manifold, calibrated area-minimizers are non-generic, with the following feature.

j. For a smooth d+c dimensional manifold with d at least 1, c at least 2, and any d-dimensional integral homology class, there exist open sets of metrics so that no area-minimizers in the class can be calibrated by weakly measurable closed forms.  (Theorem 1.1 in 5) 

k. On complex projective space, the closure of such open sets of metrics with non-calibratable minimizer includes the Fubini-Study metric. (Theorem 1.2 in 5)

l. In codimension 1, even if smooth area-minimizers exist, the calibration form can be forced to be singular. This answers a question of Michael Freedman. (Theorem 1.3 in 5)

m. The ratio of the minimal area in integral and real coefficient can always be made unbounded. One can always fill a multiple of a homology class much more efficiently than the class itself (Theorem 1.4, 1.5 in 5). This answers several questions by Frank Morgan, Brian White and Robert Young. 

Mod v minimizers are asymptotically integral currents

6. Homologically area-minimizing surfaces mod $v$ have at worst codimension 2 singular sets asymptotically, available at https://arxiv.org/abs/2401.18074

Comment: The regularity of area-minimizers in Z/vZ coefficient homology has been an open problem since the 1970s. Only recently did De Lellis and coauthors completely settle this problem and prove the sharp result that singular sets of d-dim mod v area-minimizers are of dimension at most d-1. Classical examples like triple junctions testify to the sharpness of the result. However, almost seemingly impossible, I show that if one instead considers any fixed homology class or a fixed boundary, then for large enough v, the regularity can be drastically improved to d-2 in general and d-7 in the hypersurface case.

n. Let [Σ] be a d-dimensional integral homology class on a d+c dimensional smooth compact closed Riemannian manifold M, with d, c ≥ 1. Then for v large enough, any area-minimizing mod v current in [Σ mod v] is an integral current, thus smooth outside of a singular set of codimension at least 2 for c ≥ 2 and codimension at least 7 for c ≥ 1. (Theorem 1.1 in 6) 

o. For any smooth submanifold Gamma of Euclidean space, for v large enough, any area-minimizing mod v current with boundary Gamma is an integral current with boundary Gamma, thus smooth outside of a singular set of codimension at least 2 for c ≥ 2 and codimension at least 7 for c ≥ 1. (Theorem 1.4 in 6)

p. Both results are sharp. Concrete examples show that for most v, there exist examples where mod v minimizers do not have such improvement in regularity (Theorem 1.5,1.6 in 6)

P.S. There is some other research work available on arxiv: https://arxiv.org/a/liu_z_17.html. They either are superseded by the works of others or are projects in my undergraduate years.

Research acknowledgment: I want to thank my advisor Camillo De Lellis for giving the questions in 1,3,4. Camillo De Lellis and Robert Bryant pointed out the ray construction, which formed the basis of 3. The main construction in 1 dates back to Frank Morgan and Dana Mackenzie's work. Leon Simon's work convinced me that 1 can be done. Frank Morgan, Mark Haskins, and Tommaso Pacini first pointed out the two types of obstructions in 4 in the Euclidean space. Gary Lawlor's work set the analytic foundation for 2 and 4.  Camillo De Lellis suggested using Yongsheng Zhang's work to deal with 4. Communications with Michael Freedman inspired 5. Almgren's example convinced me that 5 is true. Frank Morgan's works convinced me that 6 is true. Yongsheng Zhang's work has remained foundational to all of 1,2,3,4,5. Last but not least, William Allard, Hubert Bray, and Robert Bryant taught me the basics of geometric measure theory and Riemannian geometry, all of which are used in 1,2,3,4,5. Donghao Wang and Kai Xu are instrumental in helping me navigate differential and algebraic topology, which are essential to 1,2,3,4,5.  This leaves open the question of what is my contribution to the work above.

Selected seminar talks

Columbia University Analysis Seminar, March 2024

University of Pennsylvania Geometry Seminar, February 2024

Lehigh University Geometric Analysis Seminar, January 2024

University of Maryland Geometric Analysis Seminar, January 2024

Stanford Geometric Analysis Seminar, January 2024

Cornell Analysis Seminar, January 2024

Yau Mathematical Sciences Center, January 2024

The UC-Berkeley Differential Geometry Seminar, May 2023

Caltech Geometry & Topology Seminar, April 2023

University of Chicago - Geometric Analysis Seminar, October 2022

UMN PDE seminar, January 2022

Geometric Analysis and Topology Seminar, NYU Courant, January 2022

Simons Collaboration on Special Holonomy in Geometry, Analysis, and Physics, September 2021

Fellowships

Centennial Fellowship, Princeton, 2019-current

Answer to the teasers

1. Yes to all three. 

2. All three exist on any manifold, even the counterintuitive III.