**
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.