**
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**

**
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**

**
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.

**
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**

**
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**

**
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:
**

**
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**

**
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
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**

**
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**

**
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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**

**
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:
**

**
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.