# Workshop on Monge-Ampere equations: in celebration of Professor John Urbas’s 60th birthday

19–23 August 2019

The aim of this workshop is to bring together international leading mathematicians and provide participants an opportunity to exchange ideas and foster/enhance collaborations. It will focus on new advances and strengthen connections between Monge-Ampere equations and their applications, in particular in optimal transportation.

### Overseas speakers

• Gerard Awanou, Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago
• Jean-David Benamou, Institut National de Recherche en Informatique et en Automatique (Inria), Rue Simone Iff,  Paris 12e, France
• Huai-Dong Cao, Department of Mathematics, Lehigh University
• Mikhail Feldman, Department of Mathematics, University of Wisconsin
• Xianfeng David Gu, Department of Applied Mathematics, State University of New York at Stony Brook
• Bo Guan, Department of Mathematics, Ohio State University
• Yong Huang, Institute of Mathematics, Hunan University
• Emanuel Indrei, Department of Mathematics, Purdue University
• Hitoshi Ishii,  Institute for Mathematics and Computer Science, Tsuda University
• Aram Karakhanyan, School of Mathematics, The University of Edinburgh
• Young-Heon Kim, Department of Mathematics, University of British Columbia
• Jun Kitagawa, Department of Mathematics, Michigan State University
• Nam Q. Le, Department of Mathematics, Indiana University
• Brendan Pass, Department of Mathematical and Statistical Sciences, University of Alberta
• Yi Wang, Department of Mathematics, Johns Hopkins University
• Shibing Chen, University of Science and Technology of China
• Hui Yu, Department of Mathematics, Columbia University
• Yu Yuan, Department of Mathematics, University of Washington
• Bin Zhou, School of Mathematical Sciences, Peking University
• Qian Zhang, The Australian National University
• Na Lei, DUT-RU International School of Information Science and Engineering Dalian University of Technology, P.R. China
• Paul Bryan, Macquarie University
• Min-Chun Hong, University of Queensland

### Australian speakers

• Ben Andrews, Mathematical Sciences Institute, ANU
• Gregoire Loeper, School of Mathematical Sciences, Monash University
• James McCoy, School of Mathematical and Physical Sciences, The University of Newcastle
• Valentina Wheeler, School of Mathematics and Applied Statistics, University of Wollongong
• Zhou Zhang, School of Mathematics and Statistics, The University of Sydney

## Sessions

19 August 2019
Time Session
8:50am
Opening Remarks
9am
Kahler-Einstein Metrics and Deformation of Fano Manifolds
Huai-Dong Cao, Lehigh University, Bethlehem
A theorem of N. Koiso in early 1980’s states that if a Fano Kahler-Einstein (KE) manifold X does not admit any nontrivial holomorphic vector field then each small deformation (of complex structure) of X also admits a Kahler-Einstein metric. In this talk, we shall present a new necessary and sufficient condition on the existence of Kahler-Einstein metrics on small deformations of a Fano KE manifold with nontrivialholomorphic vector fields. This is a joint work with Xiaofeng Sun, S.-T.
9:50am
Hessian estimates for semiconvex solutions to quadratic Hessian equation
Yu Yuan, University of Washington, Seattle
We present a priori interior Hessian estimates for semiconvex solutions to the quadratic Hessian equation. Previously, this result was known for almost convex solutions. This is joint work with Ravi Shankar.
10:35am
Morning Tea
11am
A proof of Weyl problem in isometric embedding via solving the Darboux equation
Bo Guan, Ohio State, Columbus
The classical Weyl problem asks whether every positively curved closed 2-dimensional Riemannian manifold can be isometrically embedded in Euclidean 3-space. This was solved affirmatively by Nirenberg and Pogorelov independently in early 1950’s. In this talk we report some preliminary work in attempt to give a proof by solving the Monge-Ampere type Darboux equation on given closed surface.
11.50am
Mean curvature flow supported on pinching cylinders
Valentina Wheeler, University of Wollongong
In this talk, we discuss recent results (joint with G. Wheeler) on the mean curvature flow with free boundary supported on a cylindrical hypersurface. Our focus is on describing the nature of singularities: Type 0, that result in a loss of domain, Type 1, where the second fundamental form blows up like 1/sqrt{T-t}, and Type 2, where the second fundamental form blows up faster than that. We show that all three kinds of singularities can occur, and that their occurrence is dependent on growth and decay properties of the support hypersurface. The proof relies on a new kind of pinching estimate that is special to this setting and of independent interest.
12:35pm
Lunch
2pm
A rigidity theorem for ideal surfaces with flat boundary
James McCoy, University of Newcastle
We are interested in surfaces with boundary satisfying a sixth order nonlinear elliptic partial differential equation associated with extremisers of the $L^2$-norm of the gradient of the mean curvature. We show that such surfaces satisfying so-called flat boundary conditions' and small $L^2$-norm of the second fundamental form are necessarily planar. This is joint work with Glen Wheeler.
2:50pm
The geometry of the free boundary near the fixed boundary generated by a fully nonlinear uniformly elliptic operator
Emanuel Indrei, Purdue, West Lafayette
The dynamics of how the free boundary intersects the fixed boundary has been the object of study in the classical dam problem, which is a mathematical model describing the filtration of water through a porous medium split into a wet and dry part. By localizing around a point at the intersection of free and fixed boundary, one is led to a PDE generated by a fully nonlinear uniformly elliptic operator. This talk focuses on the regularity problem of the free boundary.
3:35pm
Afternoon tea
4pm
Hyperbolic 3-manifolds, embeddings and an invitation to the Cross Curvature Flow
Paul Bryan, Macquarie University, Sydney
Hyperbolic three manifolds, particularly those of finite volume, are important in the study of three-manifold topology. Out of the eight geometries arising in Thurston's geometrisation program, only the hyperbolic ones are not explicitly. The cross curvature flow was introduced by Hamilton and Chow as a promising tool for negatively curved metrics to hyperbolic metrics. There is a natural integrability condition ensuring isometric embeddability in Minkowksi space as a spacelike co-compact hypersurface in which case the cross curvature flow is equivalent to the Gauss curvature flow. By Andrews et. al. the situation is completely understood with smooth convergence to a hyperbolic metric. The general case remains an open problem, yet some results are known in favour of the general case such as stability of the hyperbolic metric due to Knopf and Young as well as monotone quantities.
20 August 2019
Time Session
9am
Solving the optimal transport problem with finite difference approximations of the Monge-Ampere equation
Jean-David Benamou, Inria, Paris
I will show how a carefully designed monotone finite difference discretization of the Monge-Ampere equation can lead to a fast solver for optimal transport problems. This applies to continuous transport maps but can also be used to characterize discontinuous dual maps.
9:50am
Optimal transport with discrete long range mean field interactions
Gregoire Loeper, Monash University, Melbourne
We study an optimal transport problem where, at some intermediate time, the mass is accelerated by either an external force field, or self-interacting. We obtain regularity of the velocity potential, intermediate density, and optimal transport map, under conditions on the interaction potential that are related to the so-called Ma-Trudinger-Wang condition from optimal transport.
10:35am
Morning Tea
11am
Optimal transportation and interpretable deep learning
Xianfeng David Gu, Stony Brook
In this talk, we show optimal transportation theory can be applied to explain deep learning methods, especially the generative adversarial networks (GANs). By using the optimal transportation view of GAN model, we show that the discriminator computes the Kantorovich potential, the generator calculates the transportation map. For a large class of transportation costs, the Kantorovich potential can give the optimal transportation map by a close-form formula. This shows the adversarial competition can be avoided, and the computational architecture can be simplified. Furthermore, the regularity theory of optimal transportation map can explain the mode collapse issue in deep learning. Optimal transportation map can be reduced to solve Monge-Ampere equations, which can be approximated using convex geometric algorithms. Preliminary experimental results show the geometric method improves the transparency, efficiency and accuracy, and avoids mode collapsing. It outperforms conventional methods, for approximating probability measures with multiple clusters in low dimensional space.
11:50am
Discrete optimal transport and its applications in shape analysis
Na Lei, Dalian University of Technology
Finding the optimal mass transportation map is equivalent to solve the Monge-Ampere equation, which has intrinsic relations with Minkowski and Alexandrov problems in convex geometry. In this talk, we follow Gu, Luo, Sun and Yau's constructive proof for the classical Alexdrov theorem, introduce a practical algorithm which is a variational approach to solve the discrete optimal mass transportation problem. The method has been applied in engineering and medicine fields, including surface registration, human expression classification and brain cortical surface classification and so on.
12:35pm
Lunch
2pm
K-surfaces with free boudnaries
Aram Karakhanyan, University of Edinburgh
Consider a pair of parallel hyperplanes in R^{n+1} and a strictly convex closed submanifold of codimension 2 laying on one of the planes. Is there a convex hypersurface of constant Gauss curvature K such that it is trapped between the planes, the submanifold is on its boundary, and thehypersurface strikes the other plane at given constant angle? In this talk we will study this problem, which can also be interpreted as the Alt-Caffarelli problem for the Monge-Ampere equation. Moreover, it also relates to the problem of isometric embedding of a positive metric on the annulus with partially prescribed boundary and optimal transport with free mass.
2:50pm
Boundary regularity for Monge-Ampere equations with unbounded right hand side
Qian Zhang, Australian National University
We consider Monge-Ampere equations with right hand side $f$ that degenerate to $\infty$ near the boundary of a convex domain $\Omega$, which are of the type $\det\,D^2u=f$ in $\Omega$, $f\sim d_{\partial\Omega}^{-\alpha}$ near $\partial\Omega$, where $d_{\partial\Omega}$ represents the distance to the boundary of the domain $\Omega$ and $-\alpha$ is a negative power with $\alpha\in(0,2)$. We study the boundary regularity of the solutions and establish a localization theorem for boundary sections.
3:35pm
Afternoon tea
4pm
Volume estimates of singular set of Ricci limit space and harmonic functions
Wenshuai Jiang, University of Sydney & ZJU
In this talk, first we will discuss the quantitative volume estimate of the singular set of noncollapsed Ricci limit space, which is based on the joint work with Professor Jeff Cheeger and Professor Aaron Naber. As an application of the quantitative estimates, in the second part of this talk we will consider the volume estimates of singular set and nodal set of harmonic functions on manifolds with lower Ricci curvature.
5:30pm
Conference dinner
21 August 2019
Time Session
8:30am - 5pm
Conference excursion
22 August 2019
Time Session
9am
Weak and smooth solutions to the semigeostrophic system
Mikhail Feldman, University of Wisconsin, Madison
The semigeostrophic (SG) system is a model of large scale atmosphere/ocean flows. Solutions of this system are expected to contain singularities corresponding to the atmospheric fronts, and need to be understood in the appropriate weak sense. Most of known results were obtained for the SG system with constant Coriolis parameter, by rewriting the problem in the "dual variables" and using Monge-Kantorovich mass transport techniques. We will survey the results on existence of weak solutions, and describe recent results on weak-strong uniqueness. A more physically realistic SG model has variable Coriolis parameter. Dual space is not available in this case. We work directly in the original "physical" coordinates, and show existence of smooth solutions for short time on two-dimensional torus. The solution is obtained by a time-stepping procedure which involves solving Monge-Ampere type equations on each step. This talk is based on joint works with A. Tudorascu and with J. Cheng and M. Cullen.
9:50am
Brunn-Minkowski Theory and Minkowski problem
Yong Huang, Hunan University, Changsha
In this talk, we will recall that the history of Brunn-Minkowski theory, and how to solve Minkowski problem by using Aleksandrov’s variational method, continuity method, geometric flow. In particular, a recent joint work with using the anisotropic Gauss curvature flow, the regularity of Lp dual Minkowski problem with Chuanqiang Chen, Yiming Zhao will be particularly discussed.
10:35am
Morning tea
11am
Singular Abreu equations and minimizers of convex functionals with a convexity constraint
Nam Q. Le, Indiana University, Bloomington
Abreu type equations are fully nonlinear, fourth order, geometric PDEs which can be viewed as systems of two second order PDEs, one is a Monge-Ampere equation and the other one is a linearized Monge-Ampere equation. They first arise in the constant scalar curvature problem in differential geometry and in the affine maximal surface equation in affine geometry. This talk discusses the solvability and convergence properties of second boundary value problems of singular, fourth order equations of Abreu type arising from approximation of convex functionals whose Lagrangians depend on the gradient variable, subject to a convexity constraint. These functionals arise in different scientific disciplines such as Newton's problem of minimal resistance in physics and the Rochet-Choné model of monopolist's problem in economics. This talk explains how minimizers of the 2D Rochet-Choné model can be approximated by solutions of singular Abreu equations.
11:50am
Global regularity of optimal transport maps
Shibing Chen, USTC, Hefei
I will talk about the global smoothness of solutions to the Monge-Ampere equation with the second boundary condition. Besides its important connection to the optimal transport problem, it has many interesting applications in geometric problems such as prescribing Gauss curvature problem and minimal Lagrangian graphs. The talk is based on joint works with Jiakun Liu and Xu-Jia Wang.
12:35pm
Lunch
2pm
Optimal transport between unequal dimensions
Brendan Pass, University of Alberta, Edmonton
I will discuss joint work with Robert McCann on the optimal transport problem between densities supported on manifolds with different dimensions. We show that the problem is equivalent to a non-local analog of the Monge-Ampere equation. We also show that, under certain topological conditions, the solution is smooth if and only if a local variant of the equation admits a smooth, uniformly elliptic solution. We show that this local equation is elliptic, and $C^{2,α}$ solutions can therefore be bootstrapped to obtain higher regularity results, assuming smoothness of the corresponding differential operator, which we prove under simplifying assumptions. For one-dimensional targets, our sufficient criteria for regularity of solutions to the resulting ODE are considerably less restrictive than those required by earlier works.
2:50pm
Regularity for weak solutions of generated Jacobian equations
Jun Kitagawa, Michigan State, East Lansing
Generated Jacobian equations are a class of Monge-Ampere type equations that model the optimal transport problem, and many near-field optics problems. In this talk I will discuss local $C^{1, \alpha}$ regularity results for weak solutions of generated Jacobian equations of Aleksandrov type. A key ingredient is a quantitative geometric condition, related to the characterization of the Ma-Trudinger-Wang condition due to Loeper in the optimal transport case. This talk is based on joint work with N. Guillen.
3:35pm
Afternoon tea
4pm
A new phenomenon involving inverse curvature flows in hyperbolic space
Xianfeng Wang, ANU & Nankai University
Inverse curvature flows for hypersurfaces in hyperbolic space have been investigated intensively in recent years. In 2015, Hang and Wang constructed an example to show that the limiting shape of the inverse mean curvature flow in hyperbolic space is not necessarily round after scaling. This was extended by Li, Wang and Wei in 2016 to the inverse curvature flow in hyperbolic space by $H^{-\alpha}$ with power $\alpha\in(0,1)$. Recently, we discover a new phenomenon involving inverse curvature flows in hyperbolic space. We find that for a large class of symmetric and 1-homogeneous curvature functions $F$ of the shifted Weingarten matrix $\mathcal{W}-I$, the inverse curvature flow with initial horospherically convex hypersurface in hyperbolic space and driven by $F^{-\alpha}$ with $\alpha\in(0,1]$ will expand to infinity in finite time. The flow is asymptotically round smoothly and exponentially as the maximum time is approached, which means that circumradius minus inradius of the flow hypersurface decays to zero exponentially and that the flow becomes exponentially close to a flow of geodesic spheres. We also construct a counterexample to show that our results cannot be extended to the case with power $\alpha>1$. This is a joint work with Dr. Yong Wei and Dr. Tailong Zhou.
23 August 2019
Time Session
9am
TBA
Ben Andrews, Australian National University
9:50am
A priori estimates for the complex Monge-Ampere equation
Bin Zhou, Peking University, Beijing
We discuss a Moser-Trudinger type inequality for pluri-subharmonic functions, and use it to establish the a priori estimates, including the uniform estimate and the H\"older continuity, for solutions to the complex Monge-Amp\ere equation with the right-hand side in $L^p$ for some $p>1$. Our proof uses various PDE techniques but not the capacity theory (or other pluri-potential theory), and therefore answered a question raised by Blocki and Kolodziej.
10:35am
Morning tea
11am
The Monge problem in Brownian stopping optimal transport
Young-Heon Kim, UBC, Vancouver
We discuss a recent progress in an optimal Brownian stopping problem, called the optimal Skorokhod embedding problem, which is an active research area especially in relation to mathematical finance. Given two probability measures with appropriate order, the problem considers the stopping time under which the Brownian motion carries one probability measure to the other, while minimizing the transportation cost. We focus on the cost given by the distance between the initial and the final points. A strong duality result of this optimization problem is obtained, which enables us to prove that the optimal stopping time is given by the first hitting time to a barrier determined by the optimal dual solutions. The main part of this talk is based on joint work with Nassif Ghoussoub (UBC) and Aaron Palmer (UBC).
11:50am
The second boundary value problem for a discrete Monge-Ampere equation with symmetrization
Gerard Awanou, University of Illinois, Chicago
In this work we propose a natural discretization of the second boundary condition for Monge-Ampere type equations. For the discretization of the differential operator, we use a recently proposed scheme, which is between wide stencils and power diagrams. Existence, unicity and stability of the solutions to the discrete problem are established as well as the convergence of a damped Newton's method. Convergence results to the continuous problem are given.
12:35pm
Lunch
2pm
Complex Monge-Ampere equations with degenerate cohomology
Zhou Zhang, University of Sydney
the complex Monge-Ampere equation is a central topic in complex differential geometry, especially after the celebrated Calabi-Yau Theorem. We talk about the extension when the cohomology class is no longer Kahler, i.e. degenerate, which appears naturally when searching for the canonical metric which is inevitably singular in general. We soon encounter essential difficulties for regularity. The Kahler-Ricci flow provides an effective way for construction and also to illustrate the situation.
2:50pm
The Ericksen-Leslie System in nematic liquid crystals
Min-Chun Hong, University of Queensland, Brisbane
Liquid crystals are states of matter intermediate between solid crystals and normal isotropic liquids. Based on a generalization of the static liquid crystal theory, Ericksen and Leslie in the 1960s proposed some constitutive equations for nematic liquid crystals in the hydrodynamic theory of liquid crystals. The Ericksen-Leslie theory has been verified in physics as one of the successful theories for modelling the nematic liquid crystal flow. In this talk, we will survey some results on the existence of the solution of the Ericksen-Leslie system and present some new results on their Ginzburg-Landau approximations.
3:35pm
Afternoon tea

## Registration is now open

### Registration fees

• General registration: $300 • Student registration:$250

The Pavilion, Kendalls on the Beach Holiday Park, Kiama, NSW.

Brittany Joyce