**
Speaker: Liuwei Gong,
**
Rutgers University

**
Title:
**
A compactness theorem for the Yamabe problem, Part III

**
Abstract:
**
This is the third talk outlining the paper by Khuri, Marques and Schoen, 2009. Using the correction term in the second talk, we will define the
Pohozaev's quadratic form and reduce the Weyl vanishing theorem and the local sign restriction to the positive definiteness of the Pohozaev's quadratic form.

**
Speaker: Liuwei Gong,
**
Rutgers University

**
Title:
**
A compactness theorem for the Yamabe problem, Part IV

**
Abstract:
**
This is the last talk outlining the paper by Khuri, Marques and Schoen, 2009. We will outline the proof of the positive definiteness of the
Pohozaev's quadratic form when $n \leq 24$.

***********************************

**
Speaker: Liuwei Gong,
**
Rutgers University

**
Title:
**
A compactness theorem for the Yamabe problem, Part I

**
Abstract:
**
This is the first talk outlining a
paper by Khuri, Marques and Schoen, 2009. We will give the definitions of different blowup sequences and reduce the compactness theorem to the Weyl vanishing theorem and the local sign restriction.

**
Speaker: Liuwei Gong,
**
Rutgers University

**
Title:
**
A compactness theorem for the Yamabe problem, Part II

**
Abstract:
**
This is the second talk outlining the paper by Khuri, Marques and Schoen, 2009. We will prove a sharp pointwise estimate of an isolated simple blowup sequence at the blowup point. Here, a correction term is introduced.

**
Speaker: Fanghua Lin,
**
Courant Institute, New York University

**
Title:
**
Critical Point Sets of Solutions in Elliptic Homogenization

**
Abstract:
**
The quantitative uniqueness and the geometric measure estimates for the nodal and critical point sets of solutions of second order elliptic equations depend crucially on the bound of the associated Almgren's frequency function. The latter is possible (only) when the leading coefficients of equations are uniformly Lipschitz. One does not have this uniform Lipschitz continuity for coefficients of equations in elliptic homogenization. Instead, by using quantitative homogenization, successive harmonic approximation and suitable L^2-renormalization, we shall see how one can get a uniform estimate (independent of a small parameter characterizing the nature of homogenization) of co-dimension two Hausdorff measure as well as the Minkowski content of the critical point sets. A key element is an estimate of "turning" for the projection of a non-constant solution onto the subspace of spherical harmonics of order N, when the doubling index of solution on annular regions is trapped near N.

**
Speaker: Ruobing Zhang,
**
Princeton University

**
Title:
**
Singular sets of solutions of elliptic equations

**
Abstract:
**
In this talk, we will discuss some basic techniques in understanding the structure of the singular sets of solutions of elliptic equations. This is a very classical and important topic in geometric measure theory. There are analogues in various areas of geometric analysis.

**
Speaker: Inigo Urtiaga Erneta,
**
Rutgers University

**
Title:
**
Estimates controlling a function by its radial derivative and applications to stable solutions

**
Abstract:
**
Functional inequalities are among the most fundamental tools in analysis. One important example is the Poincaré inequality, where a function is controlled by its gradient. In this talk, we will discuss some new estimates of X. Cabré controlling a function by only its radial derivative. We will also give applications to the regularity of stable solutions to elliptic equations.

**
Speaker: Shaoming Guo,
**
University of Wisconsin

**
Title:
**
Oscillatory integral operators on manifolds and related Kakeya and Nikodym problems

**
Abstract:
**
The talk is about oscillatory integral operators on manifolds. Manifolds of constant sectional curvatures are particularly interesting, and we will see that very good estimates on these manifolds can be expected. We will also discuss Kakeya and Nikodym problems on general manifolds, in particular, manifolds satisfying Sogge’s chaotic curvatures.

**
Speaker: Maximilien Hallgren,
**
Rutgers University

**
Title:
**
Quantitative estimates for singular sets of elliptic equations (Part I)

**
Abstract:
**
This is the first of a series of talks outlining Naber-Valtorta's paper "Volume estimates on the critical sets of solutions to elliptic PDEs". We will review orthogonal polynomial expansions of solutions to the Laplace and heat equations, and how these expansions help us study the elliptic and parabolic frequency functions. We will also discuss sharp almost-rigidity and almost-splitting theorems for the frequency functions and use these to study the set of frequency-pinched points.

**
Speaker: Maximilien Hallgren,
**
Rutgers University

**
Title:
**
Quantitative estimates for singular sets of elliptic equations (Part II)

**
Abstract:
**
This is the second of a series of talks outlining Naber-Valtorta's paper "Volume estimates on the critical sets of solutions to elliptic PDEs". We will see sharp almost-rigidity and almost-splitting theorems for the frequency functions, which show that the frequency pinched points are almost-contained in a codimension two plane. We then define the critical radius and study this quantity in the case of a codimension two almost-symmetry.

**
Speaker: Maximilien Hallgren,
**
Rutgers University

**
Title:
**
Quantitative estimates for singular sets of elliptic equations (Part III)

**
Abstract:
**
This is the third of a series of talks outlining Naber-Valtorta's paper "Volume estimates on the critical sets of solutions to elliptic PDEs". We first show lower bounds for the critical radius of approximate spherical harmonics with a codimension two symmetry. We then combine the tools developed so far with a covering argument to prove bounds on the quantitative nodal and singular sets of harmonic functions.

**
Speaker: Qing Han,
**
University of Notre Dame

**
Title:
**
Asymptotic Analysis of Gradient Systems

**
Abstract:
**
In this lecture, we study asymptotic behaviors of global and bounded solutions of gradient differential systems and discuss how to derive the long-time convergence of solutions by “gradient inequalities”. The idea that gradient inequalities force convergence of solutions of gradient systems was known to Lojasiewicz, who derived his celebrated gradient inequality for analytic gradient maps on finite-dimensional spaces. Simon proved infinite-dimensional gradient inequalities for analytic gradient maps associated with variational problems. The techniques developed by Lojasiewicz and Simon have been improved and applied to many problems.

**
Speaker: Han Lu,
**
University of Notre Dame

**
Title:
**
A constructive proof for the existence of $\sigma_2$-Yamabe problem for $n=8$

**
Abstract:
**
Consider the compact Riemannian manifold $(M,g)$ of dimension $n\geq 5$. The $\sigma_2$ Yamabe problem seeks to find a conformal metric to $g$ that has a constant $\sigma_2$ curvature. Sheng-Trudinger-Wang have previously proved the existence of such a metric when $n\geq 5$, while Ge-Wang have provided a constructive proof for $n\geq 9$. In this talk, we will present a constructive proo
f for the case $n=8$ and discuss the obstruction for such construction when $5\l
eq n\leq 7$. The proof is a joint work with Bin Deng and Juncheng Wei.

**
Speaker: Zihui Zhao,
**
University of Chicago

**
Title:
**
Quantitative boundary unique continuation II

**
Abstract:
**
In the first part, I have talked about recent progress in quantitative boundary unique continuation, and in particular, the work of X. Tolsa and my joint work with C. Kenig estimating the size of the singular set for harmonic functions near the boundary. In the second part, I will discuss some key ideas in their proofs. Both proofs rely on the monotonicity of Almgren's frequency function (which is a quantifier of the growth rate of a harmonic function), to different extents. However, the former proof is a harmonic analysis approach inspired by the work of Logunov on Yau's conjecture for Laplace eigenfunctions, and the latter uses tools from geometric measure theory inspired by the work of Naber-Valtorta on geometric variational problems.

**
Speaker: Jian Song,
**
Rutgers University

**
Title:
**
New phenomena in global complex hessian equations (I)

**
Abstract:
**
We will explain the relations among subsolution, algebraic stability conditions and solvability for global complex hessian equations such as the J-equation and the deformed Yang-Mills equation. We will investigate canonical solutions of minimal singularities when such equations do not admit smooth solutions. These new phenomena occur in explicit examples in complex dimension 2.

**
Speaker: Jian Song,
**
Rutgers University

**
Title:
**
New phenomena in global complex hessian equations (II)

**
Abstract:
**
We will explain the relations among subsolution, algebraic stability conditions and solvability for global complex hessian equations such as the J-equation and the deformed Yang-Mills equation. We will investigate canonical solutions of minimal singularities when such equations do not admit smooth solutions. These new phenomena occur in explicit examples in complex dimension 2.

**
Speaker: Jian Song,
**
Rutgers University

**
Title:
**
New phenomena in global complex hessian equations (III)

**
Abstract:
**
We will explain the relations among subsolution, algebraic stability conditions and solvability for global complex hessian equations such as the J-equation and the deformed Yang-Mills equation. We will investigate canonical solutions of minimal singularities when such equations do not admit smooth solutions. These new phenomena occur in explicit examples in complex dimension 2.

**
Speaker: Bin Guo,
**
Rutgers University -- Newark

**
Title:
**
Uniform estimates for fully nonlinear PDEs on complex manifolds (I)

**
Abstract:
**
We will discuss recent joint works with Phong-Tong on the method of comparison using auxiliary complex Monge-Ampere equations. As an application, we will derive uniform estimates for a class of fully nonlinear PDEs on Kahler manifolds.

**
Speaker: Bin Guo,
**
Rutgers University -- Newark

**
Title:
**
Uniform estimates for fully nonlinear PDEs on complex manifolds (II)

**
Abstract:
**
We will discuss recent joint works with Phong-Tong on the method of comparison using auxiliary complex Monge-Ampere equations. As an application, we will derive uniform estimates for a class of fully nonlinear PDEs on Kahler manifolds.

**
Speaker: Xuanlin Shu,
**
Rutgers University

**
**

**
**

**
Speaker: Xuanlin Shu,
**
Rutgers University

**
**

**
**

**
Speaker: Xuanlin Shu,
**
Rutgers University

**
**

**
**

**
Speaker: Xuanlin Shu,
**
Rutgers University

**
**

**
**

**
Speaker: Jeaheang Bang,
**
University of Texas at San Antonio

**
Title:
**
Liouville-type Theorems for Steady Solutions to the Navier-Stokes System in a Slab

**
Abstract:
**
I will present on my recent work with Changfeng Gui, Yun Wang, and Chunjing Xie. In this work, Liouville-type theorems for the steady incompressible Navier-Stokes system are investigated for solutions in a three-dimensional slab with either no-slip boundary conditions or periodic boundary conditions. For the no-slip boundary conditions, we proved that any bounded solution is trivial if it is axisymmetric or ru^r is bounded, and that general three-dimensional solutions must be Poiseuille flows when the velocity is not big in L^\infty space. For the periodic boundary conditions, we proved that the bounded solutions must be constant vectors if either the swirl or radial velocity is independent of the angular variable, or ru^r decays to zero as r tends to infinity. The proofs are based on energy estimates, and the key technique is to establish a Saint-Venant type estimate that characterizes the growth of Dirichlet integral of nontrivial solutions. During the talk, I will present on the proofs in detail as much as time permits.

**
The talks will be held virtually:
**

Zoom Link
**
Meeting ID:
**
924 8937 5526
**
Passcode:
**
565238

**
Speaker: Freid Tong,
**
Columbia University

**
Title:
**
Moser-Trudinger inequality for complex Monge-Ampere and applications. I

**
Abstract:
**
We will discuss some recent works of Wang-Wang-Zhou on a proof of Moser-Trudinger inequalities for complex Monge ampere equation. We will then discuss its applications to establish apriori estimates for the complex Monge-Ampere equation which previously can only be obtained using the methods of pluripotential theory.

**
Speaker: Freid Tong,
**
Columbia University

**
Title:
**
Moser-Trudinger inequality for complex Monge-Ampere and applications. II

**
Abstract:
**
We will discuss some recent works of Wang-Wang-Zhou on a proof of Moser-Trudinger inequalities for complex Monge ampere equation. We will then discuss its applications to establish apriori estimates for the complex Monge-Ampere equation which previously can only be obtained using the methods of pluripotential theory.

**
Speaker: Jean Van Schaftingen,
**
UCLouvain, Belgium

**
Title:
**
Marcinkiewicz-Gagliardo weak-type formulas for Sobolev norms

**
Abstract:
**
First-order Sobolev spaces have their norm defined in terms of weak derivative whereas fractional Sobolev spaces are endowed with the Gagliardo semi-norm controlling some integral of a differential quotient; this results in a qualitative jump at the endpoint. These two perspectives were reconciled through the introduction of a suitable corrective factor by Jean Bourgain, Haïm Brezis and Petru Mironescu have reconciled. In a recent work in collaboration with Haïm Brezis (Rutgers, Technion Haifa and Sorbonne) and Po-Lam Yung (Chinese University of Hong Kong and Australian National University), we are proving that Sobolev norms are equivalent to a Marcinkiewicz weak-type quantity associated to the Gagliardo seminorm. I will present how these estimates can be obtained through various tools of real analysis.

**
Speaker: Wanxing Liu,
**
Northwestern University

**
Title:
**
Convergence of cscK metrics on smooth minimal models of general type

**
Abstract:
**
On a smooth minimal model of general type, it is known that there exists a unique sequence of cscK metrics in the neighborhood of the canonical class, which is the perturbation of the canonical class by a
fixed Kahler
metric. In this talk, we will consider the local smooth convergence behavior of such a
sequence. In particular, we will discuss how to establish uniform zeroth order estimate and uniform upper entropy bound for it.

**
Speaker: Jian Song,
**
Rutgers University

**
Title:
**
Constant scalar curvature Kahler metrics

**
Abstract:
**
We will review some a priori estimates in the important
work of Chen-Cheng on the existence of constant scalar curvature Kahler metrics.

**
Speaker: Hui Yu,
**
Columbia University

**
Title:
**
Improvement of flatness in elliptic PDEs and free boundary problems

**
Abstract:
**
In this talk, we discuss the classic technique of improvement of flatness and several of its recent applications to free boundary problems.

**
Speaker: Jian Song,
**
Rutgers University

**
Title:
**
Constant scalar curvature Kahler metrics. II

**
Abstract:
**
We will continue to discuss a priori estimates in the work of Chen-Cheng on constant scalar curvature Kahler metrics. We will also apply their techniques for the Kahler-Ricci flow on Fano manifolds.

**
Speaker: Bin Guo
**
Rutgers University -- Newark

**
Title:
**
A new proof of Evans-Krylov estimate for Monge-Ampere equations

**
Abstract:
**
We'll discuss a recent proof of Wang and Wu on the C^{2,\alpha} estimate for solutions of real MA equations. The idea is to employ the Green function for the linearized operator and a contradiction argument.

**
Speaker: Zongyuan Li
**
Rutgers University

**
Title:
**
Unique continuation for elliptic and parabolic equations. I

**
Abstract:
**
In these two talks, we introduce the unique continuation for elliptic and parabolic equations.
In the first talk, we will discuss the strong and weak unique continuation for elliptic equations. We will mention some open problems, exciting progress made by Logunov, et. al., and a proof of a simple unique continuation result using the Carleman estimate. In the second talk, we will discuss the application of a parabolic unique continuation result in the regularity criteria for 3D Navier-Stokes equations. The second talk will be based on Escauriaza-Seregin-Sverak's 2003 ARMA paper.

**
Speaker: Zongyuan Li
**
Rutgers University

**
Title:
**
Unique continuation for elliptic and parabolic equations. II

**
Abstract:
**
In these two talks, we introduce the unique continuation for elliptic and parabolic equations.
In the first talk, we will discuss the strong and weak unique continuation for elliptic equations. We will mention some open problems, exciting progress made by Logunov, et. al., and a proof of a simple unique continuation result using the Carleman estimate. In the second talk, we will discuss the application of a parabolic unique continuation result in the regularity criteria for 3D Navier-Stokes equations. The second talk will be based on Escauriaza-Seregin-Sverak's 2003 ARMA paper.

**
Speaker: Siyuan Lu,
**
McMaster University

**
Title:
**
Capacity in General Relativity

**
Abstract:
**
The capacity estimate plays an important role in Bray's proof of Riemannian Penrose inequality. In this talk, we will review some capacity estimates in general relativity due to Bray, Bray-Miao and Mantoulidis-Miao-Tam.

**
Speaker: Dennis Kriventsov,
**
Rutgers University

**
Title:
**
Boundary Harnack Principles

**
Abstract:
**
Boundary Harnack inequalities are common tools in free boundary problems, potential theory, and harmonic analysis on domains. We will look at the proof of the boundary Harnack inequality on Lipschitz domains, discuss what other kinds of domains they are available for, and then consider some recent results on equations with right-hand side (based on joint work with Mark Allen and Henrik Shahgholian).

**
Speaker: Jacob Sturm
**
Rutgers University

**
Title:
**
Singular Complex Monge Ampere equations

**
Abstract:
**
The Complex Monge-Ampere equation has the form MA(phi) = mu, where mu is a given Borel measure on a domain in C^n (resp. complex manifold) X, phi is an unknown plurisubharmonic (resp. quasi-plurisubharmonic) function on X, and MA is the Monge-Ampere operator. The existence, uniqueness and regularity questions have been widely studied in the case where mu is absolutely continuous, but much less is known is known if mu is singular (e.g. if mu charges a point). I will discuss some partial results in the singular case and will also describe some examples.

**
Speaker: Qing Han,
**
Notre Dame University

**
Title:
**
The isometric embedding of abstract surfaces in the 3-dim Euclidean space

**
Abstract:
**
A surface in the 3-dim Euclidean space can be viewed as the image of a map from a planar domain to the 3-dim Euclidean space, at least locally. The standard metric in the Euclidean space induces a metric on the surface. The induced metric on the surface can be transformed to an abstract metric by the abovementioned map. Now, we consider the converse question. Given an abstract metric on a planar domain, can we find a surface in the 3-dim Euclidean space whose induced metric is the given abstract metric? This is the isometric embedding problem we will discuss. It started with a conjecture by Schlaefli in 1873 that this can always be achieved near any given point. This conjecture is widely open and there are only a few results under various conditions. The question can be reformulated in terms of partial differential equations and is essentially a PDE problem. Despite the technical description, the underlying equation has a simple form.

**
The talks will be held virtually:
**

Zoom Link
**
Meeting ID:
**
948 4160 4914
**
Passcode:
**
654986

**
Speaker: Jian Song, Rutgers University
**

**
Title:
**
Volume comparison and its applications for the Ricci flow. I

**
Abstract:
**
We will review the recent work of Bamler on volume estimates for the Ricci flow. We also show such estimates can be applied to establish and improve the relative volume comparison of Tian-Zhang for the Ricci flow. Finally, we prove a uniform diameter bound for long time solutions of the Kahler-Ricci flow, which is a natural extension of Perelman's diameter estimate for the Fano Ricci flow.

**
Speaker: Jian Song, Rutgers University
**

**
Title:
**
Volume comparison and its applications for the Ricci flow. II

**
Abstract:
**
We will review the recent work of Bamler on volume estimates for the Ricci flow. We also show such estimates can be applied to establish and improve the relative volume comparison of Tian-Zhang for the Ricci flow. Finally, we prove a uniform diameter bound for long time solutions of the Kahler-Ricci flow, which is a natural extension of Perelman's diameter estimate for the Fano Ricci flow.

**
Speaker: Weiyong He, University of Oregon
**

**
Title:
**
The Gursky-Streets equations

**
Abstract:
**
Gursky and Streets introduced a family of fully nonlinear elliptic equations, as geodesic equations of the space of metrics in a fixed conformal class.
The set-up of geometric structure was motivated by the Kahler geometry, and can be used to prove the uniqueness of sigma_2 Yamabe problem in dimension four.
We discuss the Gursky-Streets equations, the regularity and the applications.

**
Speaker: Chao Li, Princeton University
**

**
Title:
**
Generalized soap bubbles and the topology of manifolds with positive scalar curvature

**
Abstract:
**
It has been a classical question which manifolds admit Riemannian metrics with positive scalar curvature. I will first review some history of this question, and present some recent progress, ruling out positive scalar curvature on closed aspherical manifolds of dimensions 4 and 5 (as conjectured by Schoen-Yau and by Gromov), as well as complete metrics of positive scalar curvature on an arbitrary manifold connect sum with a torus. Applications include a Schoen-Yau Liouville theorem for all locally conformally flat manifolds. The proofs of these results are based on analyzing generalized soap bubbles - surfaces that are stable solutions to the prescribed mean curvature problem. This talk is based on joint work with O. Chodosh.

**
Speaker: Jiahong Wu, Oklahoma State University
**

**
Title:
**
Global solutions of 3D incompressible MHD system with mixed partial dissipation
and magnetic diffusion near an equilibrium

**
Abstract:
**
This talk presents the global existence and stability result on solutions near a background magnetic
field to the 3D incompressible magnetohydrodynamic (MHD) system with horizontal dissipation and
vertical magnetic diffusion. Due to the lack of the vertical dissipation and the horizontal magnetic
diffusion, the stability problem concerned here is difficult and classical approaches no longer apply.
This work exploits the extra smoothing and stabilizing effect in the wave structure resulting from
the coupling and interaction of the velocity and the magnetic fields. By constructing a suitable
energy functional to incorporate this stabilizing effect and employing the bootstrapping argument,
we are able to show the desired existence and stability result.

**
Speaker: Yannick Sire, Johns Hopkins University
**

**
Title:
**
Rigidity in stationary Euler equation with free boundary

**
Abstract:
**
We investigate a free boundary version of the results of Hamel and Nadirashvili on Liouville theorems for stationary Euler equations in 2 dimensions. I will first introduce stationary Euler with free boundary. Then, I will explain how to reduce the equations to an overdetermined semi-linear problem, for which a Serrin’s type result can be shown.

**
This talk will be held virtually on WebEx, copy and paste the following web
address:
**

**
https://rutgers.webex.com/rutgers/j.php?MTID=m3cd0c939b5693503d0bdf1aa9410b8a6
**

**
Speaker: Zongyuan Li, Rutgers University
**

**
Title:
**
Elliptic boundary value problems on irregular domains

**
Abstract:
**
In this talk, we will discuss the elliptic boundary value problems on irregular domains, with non-smooth boundary data. We will first setup the problems with $L_p$ boundary data, where the boundary conditions are understood in the sense of the ``non-tangential limit''. Some interesting ``good and bad''
domains will be given in the process. For the second part, let us focus on problems on Lipschitz domains. We will introduce an interpolation method which is by proving the boundary reverse H\"older inequality via the theory of weak solutions. As a concrete example, we will present a recent joint work with H. Dong on the mixed Dirichlet-conormal problems.

**
This talk will be held virtually on WebEx, copy and paste the following web
address:
**

**
https://rutgers.webex.com/rutgers/j.php?MTID=m3cd0c939b5693503d0bdf1aa9410b8a6
**

**
Speaker: Jian Song, Rutgers University
**

**
Title:
**
Complex hessian quotient equations. II.

**
Abstract:
**
The J-flow is proposed by Donaldson to study symplectomorphisms of a Kaher manifold. The J-equation, critical equation of the J-flow, corresponds to a standard complex hessian quotient equation. The global J-equation does not always admit a smooth solution due to topological obstructions. We review both the analytic and topological conditions for solving the J-equation. These equivalent conditions are natural analogues of the Nakai criterion in algebraic geometry.

**
This talk will be held virtually on WebEx, copy and paste the following web
address:
**

**
https://rutgers.webex.com/rutgers/j.php?MTID=m3cd0c939b5693503d0bdf1aa9410b8a6
**

**
Speaker: Jian Song, Rutgers University
**

**
Title:
**
Complex hessian quotient equations

**
Abstract:
**
The J-flow is proposed by Donaldson to study symplectomorphisms of a Kaher manifold. The J-equation, critical equation of the J-flow, corresponds to a standard complex hessian quotient equation. The global J-equation does not always admit a smooth solution due to topological obstructions. We review both the analytic and topological conditions for solving the J-equation. These equivalent conditions are natural analogues of the Nakai criterion in algebraic geometry.

**
This talk will be held virtually on WebEx, copy and paste the following web
address:
**

**
https://rutgers.webex.com/rutgers/j.php?MTID=m3cd0c939b5693503d0bdf1aa9410b8a6
**

**
Speaker: Jeaheang Bang, Rutgers University
**

**
Title:
**
A Liouville Theorem for the Euler Equations in the Plane. II

**
Abstract:
**
In these two expository talks, I will talk about the paper by Hamel and Nadirashvili in 2019. They proved that any bounded solution to the stationary Euler Equations in the plane is necessarily a shear flow provided that the solution does not have stagnation points (not even at infinity). The proof is twofold. First, they studied the geometrical properties of the streamlines and of the gradient flows—first talk. Second, they derived some logarithmic estimates on the argument of the flow in large balls—second talk.

**
This talk will be held virtually on WebEx, copy and paste the following web
address:
**

**
https://rutgers.webex.com/rutgers/j.php?MTID=m3cd0c939b5693503d0bdf1aa9410b8a6
**

**
Speaker: Jeaheang Bang, Rutgers University
**

**
Title:
**
A Liouville Theorem for the Euler Equations in the Plane. I

**
Abstract:
**
In these two expository talks, I will talk about the paper by Hamel and Nadirashvili in 2019. They proved that any bounded solution to the stationary Euler Equations in the plane is necessarily a shear flow provided that the solution does not have stagnation points (not even at infinity). The proof is twofold. First, they studied the geometrical properties of the streamlines and of the gradient flows—first talk. Second, they derived some logarithmic estimates on the argument of the flow in large balls—second talk.

**
This talk will be held virtually on WebEx, copy and paste the following web
address:
**

**
https://rutgers.webex.com/rutgers/j.php?MTID=m3cd0c939b5693503d0bdf1aa9410b8a6
**

**
Speaker: Weisong Dong, Tianjing University
**

**
Title:
**
C-subsolution and its applications in a priori estimates for
solutions of fully nonlinear equations

**
Abstract:
**
In this expository talk, I will introduce the definition
of C-subsolution given by
G. Szekelyhidi in 2018 JDG. This is more powerful in a priori estimates
than the usual subsolution. I will show the role of C-subsolution in
the C^0 and C^2 estimates for solutions.

**
This talk will be held virtually on WebEx, copy and paste the following web
address:
**

**
https://rutgers.webex.com/rutgers/j.php?MTID=m3cd0c939b5693503d0bdf1aa9410b8a6
**

**
Speaker: Jingang Xiong, Beijing Normal University
**

**
Title:
**
Isolated singularities of solutions to the Yamabe equation. II.

**
Abstract:
**
I will talk about the asymptotic behavior of local solutions to the Yamabe equation near an isolated singularity, when the metric is not locally
conformally flat.
I will show that when the dimension is less than or equal to 6, any solution is asymptotic to a radially symmetric Fowler solution. This is based on two papers: F. Marques 2008 for dimensions less than 6 and a recent preprint of L. Zhang and myself for dimension 6.
When the metric is locally
conformally flat, the above mentioned asymptotic behavior holds in all dimensions, which was proved in the classical paper of Caffarelli-Gidas and Spruck 1989. See also an important refinement due to Korevaar, Mazzeo, Pacard and Schoen 1999.

**
This talk will be held virtually on WebEx, copy and paste the following web
address:
**

**
https://rutgers.webex.com/rutgers/j.php?MTID=m3cd0c939b5693503d0bdf1aa9410b8a6
**

**
Speaker: Jingang Xiong, Beijing Normal University
**

**
Title:
**
Isolated singularities of solutions to the Yamabe equation. I.

**
Abstract:
**
I will talk about the asymptotic behavior of local solutions to the Yamabe equation near an isolated singularity, when the metric is not locally
conformally flat.
I will show that when the dimension is less than or equal to 6, any solution is asymptotic to a radially symmetric Fowler solution. This is based on two papers: F. Marques 2008 for dimensions less than 6 and a recent preprint of L. Zhang and myself for dimension 6.
When the metric is locally
conformally flat, the above mentioned asymptotic behavior holds in all dimensions, which was proved in the classical paper of Caffarelli-Gidas and Spruck 1989. See also an important refinement due to Korevaar, Mazzeo, Pacard and Schoen 1999.
`

**
This talk will be held virtually on WebEx, copy and paste the following web
address:
**

**
https://rutgers.webex.com/rutgers/j.php?MTID=m3cd0c939b5693503d0bdf1aa9410b8a6
**

**
Speaker: Jeaheang Bang, Rutgers University
**

**
Title:
**
Construction of a Smooth, Compactly-Supported Solution to the Three-Dimensional Stationary Euler Equations

**
Abstract:
**
It is an expository talk about the work of Constantin, La, and Vicol in 2019. In this paper, they constructed a smooth, compactly-supported
solution to the stationary Euler equations in the three-dimensional Euclidean space. To do so, they seek for axisymmetric solutions and use some specific ansatz, which leads to Hicks equation (equivalent to Grad-Shafranov equation). A solution to Hicks equation have meaning as a stream function of a solution to the Euler equations. They try to construct a solution to Hicks equation, satisfying some additional condition called localization. To construct a solution to Hicks equation with the localization condition, they use some transformation called hodograph. Majority of it boils down to existence of a solution to some ODEs. This leads to a smooth solution to the Euler equation (with some nice properties deduced from the localization condition) but it does not necessarily have compact support. Thanks to the localization condition, finally they can
'localize' the solution that they have found. In other words, they can immediately generate another solution with compact support. (This paper is inspired by the work of A. V. Gavrilov in 2019 about the Euler equations.)

**
This talk will be held virtually on WebEx, copy and paste the following web
address:
**

**
https://rutgers.webex.com/rutgers/j.php?MTID=m3cd0c939b5693503d0bdf1aa9410b8a6
**

**
Speaker: Jian Song, Rutgers University
**

**
Title:
**
Construction of cut-off functions

**
Abstract:
**
We will talk about the construction of cut-off functions constructed by Cheeger-Colding on Riemannian manifolds with Ricci curvature bounded below.

**
This talk will be held virtually on WebEx, copy and paste the following web
address:
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**
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**

**
Speaker: Jeaheang Bang, Rutgers University
**

**
Title:
**
Shear Flows of Euler Equations in a Two-dimensional Strip.

**
Abstract:
**
It is an expository talk about the work of François Hamel and Nikolai Nadirashvili in 2017. They proved that, in a two-dimensional strip, a steady flow of Euler equations with tangential boundary conditions and with no stationary point is a shear flow. The proof is based on the study of geometric properties of the streamlines of the flow and on one-dimensional symmetry results for solutions of some semi-linear elliptic equations.

**
This talk will be held virtually on WebEx, copy and paste the following web
address:
**

**
https://rutgers.webex.com/rutgers/j.php?MTID=m3cd0c939b5693503d0bdf1aa9410b8a6
**

**
Speaker: Dennis Kriventsov, Rutgers University
**

**
Title:
**
The Alt-Caffarelli functional with volume constraint.

**
Abstract:
**
I will discuss an old, classic example from free boundary theory: minimizing the Alt-Caffarelli functional with volume constraint. The paper is based on an obvious idea: finding it difficult to deal with a constraint, we replace it with a penalization. The catch is that the penalized functional is not differentiable with respect to domain variations, while the estimates available for it all seem to depend on the penalization parameter. This example explains how sometimes, on good days, one may still obtain Euler-Lagrange equations for non-differentiable functionals, and sometimes if you penalize correctly, the penalized and constrained problems turn out to have identical minimizers. While the example is free boundary based, I think the insights are applicable to many calculus of variations problems.

**
This talk will be held virtually on WebEx, copy and paste the following web
address:
**

**
https://rutgers.webex.com/rutgers/j.php?MTID=m3cd0c939b5693503d0bdf1aa9410b8a6
**

**
Speaker: Jian Song, Rutgers University
**

**
Title:
**
Hessian equations in complex geometry. II.

**
Abstract:
**
We will review results on both smooth and weak solutions to complex hessian equations. There exists obstruction to the existence of solutions to global complex hessian equations. We will establish relations between the solvability of such equations and positivity in intersection theory of
topological cycles and geometric structures of holomorphic fibration.

**
This talk will be held virtually on WebEx, copy and paste the following web
address:
**

**
https://rutgers.webex.com/rutgers/j.php?MTID=m3cd0c939b5693503d0bdf1aa9410b8a6
**

**
Speaker: Jian Song, Rutgers University
**

**
Title:
**
Hessian equations in complex geometry. I.

**
Abstract:
**
We will review results on both smooth and weak solutions to complex hessian equations. There exists obstruction to the existence of solutions to global complex hessian equations. We will establish relations between the solvability of such equations and positivity in intersection theory of topological cycles and geometric structures of holomorphic fibration.

**
This talk will be held virtually on WebEx, copy and paste the following web
address:
**

**
https://rutgers.webex.com/rutgers/j.php?MTID=m3cd0c939b5693503d0bdf1aa9410b8a6
**

**
Speaker: Jingang Xiong, Beijing Normal University
**

**
Title:
**
Existence of solutions of the steady Navier-Stokes equations in unbounded domains with pipe ends

**
Abstract:
**
I will present a paper of Amick 1977,
where he proved an existence result of the Navier-Stokes flow in unbounded domains with pipe ends.
Those flows converge to Poiseuille flows at the ends.
Amick assumed that the viscosity coefficient is greater than some positive constant
$\nu_0$. Without this assumption, the existence is still OPEN.

**
This talk will be held virtually on WebEx, copy and paste the following web
address:
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**
https://rutgers.webex.com/rutgers/j.php?MTID=m3cd0c939b5693503d0bdf1aa9410b8a6
**

**
Speaker: Jingang Xiong, Beijing Normal University
**

**
Title:
**
Energy decay estimates for steady Navier-Stokes equations in pipes

**
Abstract:
**
I will present a paper by Horgan-Wheeler in 1978,
where they established an exponential decay estimates
of the Dirichlet energy when the viscosity coefficient is large.
Their method is to derive an integro-differential inequality of the energy.

**
This talk will be held virtually on WebEx, copy and paste the following web
address:
**

**
https://rutgers.webex.com/rutgers/j.php?MTID=m3cd0c939b5693503d0bdf1aa9410b8a6
**

**
Speaker: Jeaheang Bang, Rutgers University
**

**
Title:
**
A Liouville Theorem for Weak Beltrami Flows

**
Abstract:
**
Abstract: This is an expository talk about the work of Dongho Chae and Jörg Wolf in 2016. They established a Liouville theorem for weak Beltrami flows in three dimensional Euclidean space. The sufficient condition that they impose in order to ensure the triviality of such flows is that some sort of tangential part of the flow decays at infinity in some weak sense (with no assumption on the normal part). This theorem is a generalization of all the known results of this problem. The main idea is to test the Euler equations against some simple vector fields over a ball and over an annulus.

**
Speaker: Jingang Xiong, Beijing Normal University
**

**
Title:
**
Steady
solutions of
the Navier-Stokes equations in unbounded channels and pipes. V

**
Abstract:
**
I will present two papers of Charles J. Amick: Ann. Scu. Norm. Sup. di Pisa 1977 and
Nonlinear Analysis 1978, where the existence of weak solutions approaching to Poiseuille flows
was proved under certain assumptions.

**
Speaker: Jingang Xiong, Beijing Normal University
**

**
Title:
**
Steady
solutions of
the Navier-Stokes equations in unbounded channels and pipes. IV

**
Abstract:
**
I will present two papers of Charles J. Amick: Ann. Scu. Norm. Sup. di Pisa 1977 and
Nonlinear Analysis 1978, where the existence of weak solutions approaching to Poiseuille flows
was proved under certain assumptions.

**
Speaker: Jingang Xiong, Beijing Normal University
**

**
Title:
**
Steady
solutions of
the Navier-Stokes equations in unbounded channels and pipes.III

**
Abstract:
**
I will present two papers of Charles J. Amick: Ann. Scu. Norm. Sup. di Pisa 1977 and
Nonlinear Analysis 1978, where the existence of weak solutions approaching to Poiseuille flows
was proved under certain assumptions.

**
Speaker: Jingang Xiong, Beijing Normal University
**

**
Title:
**
Steady
solutions of
the Navier-Stokes equations in unbounded channels and pipes. II

**
Abstract:
**
I will present two papers of Charles J. Amick: Ann. Scu. Norm. Sup. di Pisa 1977 and
Nonlinear Analysis 1978, where the existence of weak solutions approaching to Poiseuille flows
was proved under certain assumptions.

**
Speaker: Jingang Xiong, Beijing Normal University
**

**
Title:
**
Steady
solutions of
the Navier-Stokes equations in unbounded channels and pipes. I

**
Abstract:
**
I will present two papers of Charles J. Amick: Ann. Scu. Norm. Sup. di Pisa 1977 and
Nonlinear Analysis 1978, where the existence of weak solutions approaching to Poiseuille flows
was proved under certain assumptions.

**
Speaker: Ting Zhang, Zhejiang University
**

**
Title:
Introduction to the Incompressible Navier-Stokes equations. II.
**

**
Abstract:
**

**
Speaker: Ting Zhang, Zhejiang University
**

**
Title:
Introduction to the Incompressible Navier-Stokes equations. I.
**

**
Abstract:
**

**
Speaker: Jeaheang Bang, Rutgers University
**

**
Title:
**
A Liouville-Type Theorem for Beltrami Flows

**
Abstract:
**
This is an expository talk on the work of Dongho Chae and Peter Constantin in 2015. They presented a simple, short, and elementary proof that any three-dimensional Beltrami flow with a finite energy has to be trivial. The main idea of the proof is to use the continuity of the Fourier transform of functions belonging to L^1.

**
Speaker: Hongjie Dong, Brown University
**

**
Title:
**
Some partial regularity and regularity criteria results for the Navier-Stokes equations. I.

**
Abstract:
**
I
will first review some previous results on the conditional regularity of solutions to the incompressible Navier-Stokes equations (both stationary and time-dependent) in the critical Lebesgue spaces, and then discuss a recent work which mainly addressed the boundary regularity issue.

**
Speaker: Hongjie Dong, Brown University
**

**
Title:
**
Some partial regularity and regularity criteria results for the Navier-Stokes equations. II.

**Abstract:
**
I will first review some previous results on the conditional regularity of solutions to the incompressible Navier-Stokes equations (both stationary and time-dependent) in the critical Lebesgue spaces, and then discuss a recent work which mainly addressed the boundary regularity issue.

**
Speaker: Jiaping Wang, University of Minnesota
**

**
Title:
**
Structure at infinity for four dimensional shrinking Ricci solitons

**Abstract:
**
Ricci solitons, as self-similar solutions to the Ricci flows, play a prominent role in the study
of singularity formations of Ricci flows. The primary concern of the talk is four dimensional shrinking
Ricci solitons. We will discuss some joint work with Ovidiu Munteanu concerning their geometric
structure at infinity.

**
Speaker: Hao Jia, University of Chicago
**

**
Title:
**
Reading report of Kato's pioneering paper on $L^p$ solutions to
Navier Stokes equations

**
Abstract:
**
In a paper published in 1984, Kato made an important observation
that one can obtain global existence of solution for Navier Stokes if the
initial data is small in $L^3(R^3)$, which is scale invariant for Navier
Stokes. Various applications to long time decay of solutions were also
given. I will explain these results and if time permits, review some recent
progress on the Cauchy problem with initial data in scale invariant spaces.
(Refence: Tosio Kato ``Strong L^p-solutions of the Navier-Stokes equation
in R^m , with applications to weak solutions. Math. Z. 187 (1984), no.
4, 471-480.'')

**
Speaker: Bo Yang, Rutgers University
**

**
Title:
**
Kaehler-Ricci solitons

**
Abstract:
**
We introduce Kaehler-Ricci solitons and discuss applications
in the uniformization problems on Kaehler manifolds with nonnegative
curvature. This is an expository talk on the work of Chau-Tam. The
reference is MR2488949.

**
Speaker: Natasa Sesum, Rutgers University
**

**
Title:
**
Mean curvature flow of entire graphs. II

**
Abstract:
**
We discuss the joint work of Huisken and Ecker on the mean curvature flow of entire
graphs. We discuss the longtime existence and convergence of the class of entire g
raphs that grow at most linearly. We also tackle the question on how to handle the
previous problem once we drop the condition on a linear growth. We discuss the join
t work of Huisken and Ecker on the mean curvature flow of entire graphs. We discuss
the longtime existence and convergence of the class of entire graphs that grow at
most linearly. We also tackle the question on how to handle the previous problem on
ce we drop the condition on a linear growth.

**
Speaker: Natasa Sesum, Rutgers University
**

**
Title:
**
Mean curvature flow of entire graphs

**
Abstract:
**
We discuss the joint work of Huisken and Ecker on the mean curvature flow of entire graphs. We discuss the longtime existence and convergence of the class of entire graphs that grow at most linearly. We also tackle the question on how to handle the previous problem once we drop the condition on a linear growth. We discuss the joint work of Huisken and Ecker on the mean curvature flow of entire graphs. We discuss the longtime existence and convergence of the class of entire graphs that grow at most linearly. We also tackle the question on how to handle the previous problem once we drop the condition on a linear growth.

**
Speaker: Natasa Sesum, Rutgers University
**

**
Title:
**
Generic singularities of the mean curvature flow. II

**
Abstract:
**
It has long been conjectured that starting at a generic smooth closed embedded
surface in R^3 the mean curvature flow remains smooth until it arrives at a sing
ularity in a neighborhood of which the flow looks like concentric spheres or cyl
inders. That is, the only singularities of a generic flow are spherical or cylin
drical. The key in showing this conjecture is to show that shrinking spheres, cy
linders and planes are the only stable self-shrinkers under the mean curvature f
low. That was proved by Colding and Minicozzi and we will discuss parts of their
paper.

**
Speaker: Natasa Sesum, Rutgers University
**

**
Title:
**
Generic singularities of the mean curvature flow

**
Abstract:
**
It has long been conjectured that starting at a generic smooth closed embedded surface in R^3 the mean curvature flow remains smooth until it arrives at a singularity in a neighborhood of which the flow looks like concentric spheres or cylinders. That is, the only singularities of a generic flow are spherical or cylindrical. The key in showing this conjecture is to show that shrinking spheres, cylinders and planes are the only stable self-shrinkers under the mean curvature flow. That was proved by Colding and Minicozzi and we will discuss parts of their paper.

**
Speaker: Jian Song, Rutgers University
**

**
Title:
**
L^2 estimates and its applications in Riemannian and
algebraic geometry. III

**
Abstract:
**
We will discuss the partial C^0 estimates in the recent
paper of Donaldson and Sun as a solution to a conjecture of Tian. This
gives a compactness result for Kahler-Einstein manifolds with positive
scalar curvature.

**
Speaker: Jian Song, Rutgers University
**

**
Title:
**
L^2 estimates and its applications in Riemannian and
algebraic geometry. II

**
Abstract:
**
We will discuss the partial C^0 estimates in the recent
paper of Donaldson and Sun as a solution to a conjecture of Tian. This
gives a compactness result for Kahler-Einstein manifolds with positive
scalar curvature.

**
Speaker: Jian Song, Rutgers University
**

**
Title:
**
L^2 estimates and its applications in Riemannian and
algebraic geometry

**
Abstract:
**
Abstract: We will discuss the partial C^0 estimates in the recent
paper of Donaldson and Sun as a solution to a conjecture of Tian. This
gives a compactness result for Kahler-Einstein manifolds with positive
scalar curvature.

**
Speaker: Yanyan Li, Rutgers University
**

**
Title:
**
A theorem of Leray on Navier-Stokes equations and an open problem, I

**
Abstract:
**
This is the first of two expository talks. We will present a proof of a classic theorem of Jean Leray on the nonhomogeneous steady incompressible Navier-Stokes equations in a two dimensional domain. The theorem establishes the existence of a solution when the prescribed boundary velocity has zero flux through each boundary component. It remains up-to-date a challenging open problem whether the existence result holds under the (weaker) compatibility condition that the total velocity flux through the boundary is equal to zero. We will also give a brief survey on partial results on the open problem.
Both talks are meant to be accessible to first year graduate students.

**
Speaker: Yanyan Li, Rutgers University
**

**
Title:
**
A theorem of Leray on Navier-Stokes equations and an open problem, II

**
Abstract:
**
This is the first of two expository talks. We will present a proof of a classic theorem of Jean Lera
y on the nonhomogeneous steady incompressible Navier-Stokes equations in a two dimensional domain. Th
e theorem establishes the existence of a solution when the prescribed boundary velocity has zero flux
through each boundary component. It remains up-to-date a challenging open problem whether the existe
nce result holds under the (weaker) compatibility condition that the total velocity flux through the
boundary is equal to zero. We will also give a brief survey on partial results on the open problem.
Both talks are meant to be accessible to first year graduate students.