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− 
 
−  = Abstracts =
 
− 
 
−  ===Greg Drugan (U. of Washington)===
 
−  ''Construction of immersed selfshrinkers''
 
− 
 
−  Abstract: We describe a procedure for constructing immersed
 
−  selfshrinking solutions to mean curvature flow.
 
−  The selfshrinkers we construct have a rotational symmetry, and
 
−  the construction involves a detailed study of geodesics in the
 
−  upperhalf plane with a conformal metric.
 
−  This is a joint work with Stephen Kleene.
 
− 
 
−  ===Guo Luo (Caltech)===
 
−  ''Potentially Singular Solutions of the 3D Incompressible Euler Equations''
 
− 
 
−  Abstract:
 
−  Whether the 3D incompressible Euler equations can develop a singularity in
 
−  finite time from smooth initial data is one of the most challenging problems in
 
−  mathematical fluid dynamics. This work attempts to provide an affirmative answer to this
 
−  longstanding open question from a numerical point of view, by presenting a class of
 
−  potentially singular solutions to the Euler equations computed in axisymmetric
 
−  geometries. The solutions satisfy a periodic boundary condition along the axial direction
 
−  and noflow boundary condition on the solid wall. The equations are discretized in space
 
−  using a hybrid 6thorder Galerkin and 6thorder finite difference method, on specially
 
−  designed adaptive (moving) meshes that are dynamically adjusted to the evolving
 
−  solutions. With a maximum effective resolution of over $(3 \times 10^{12})^{2}$ near the
 
−  point of singularity, we are able to advance the solution up to $\tau_{2} = 0.003505$ and
 
−  predict a singularity time of $t_{s} \approx 0.0035056$, while achieving a
 
−  \emph{pointwise} relative error of $O(10^{4})$ in the vorticity vector $\omega$ and
 
−  observing a $(3 \times 10^{8})$fold increase in the maximum vorticity
 
−  $\norm{\omega}_{\infty}$. The numerical data is checked against all major blowup
 
−  (nonblowup) criteria, including BealeKatoMajda, ConstantinFeffermanMajda, and
 
−  DengHouYu, to confirm the validity of the singularity. A careful local analysis also
 
−  suggests that the blowingup solution develops a selfsimilar structure near the point of
 
−  the singularity, as the singularity time is approached.
 
− 
 
−  ===Xiaojie Wang(Stony Brook)===
 
−  ''Uniqueness of Ricci flow solutions on noncompact manifolds''
 
− 
 
−  Abstract:
 
−  Ricci flow is an important evolution equation of Riemannian metrics.
 
−  Since it was introduced by R. Hamilton in 1982, it has greatly changed the landscape of riemannian geometry. One of the fundamental question about ricci flow is when is its solution to initial value problem unique. On compact manifold, with arbitrary initial metric, it was confirmed by Hamilton. On noncompact manifold, we only know this is true when further restrictions are imposed to the solution. In this talk, we will discuss various conditions that guarantee the uniqueness. In particular, we will discuss in details with the following uniqueness result. Let $(M,g)$ be a complete noncompact noncollapsing $n$dimensional riemannian manifold, whose complex sectional curvature is bounded from below and scalar curvature is bounded from above. Then ricci flow with above as its initial data, on $M\times [0,\epsilon]$ for some $\epsilon>0$, has at most one solution in the class of complete riemannian metric with complex sectional curvature bounded from below.
 
− 
 
−  ===Roman Shterenberg(UAB)===
 
−  ''Recent progress in multidimensional periodic and almostperiodic spectral
 
−  problems''
 
− 
 
−  Abstract: We present a review of the results in multidimensional periodic
 
−  and almostperiodic spectral problems. We discuss some recent progress and
 
−  old/new ideas used in the constructions. The talk is mostly based on the
 
−  joint works with Yu. Karpeshina and L. Parnovski.
 
− 
 
−  ===Antonio Ache(Princeton)===
 
−  ''Ricci Curvature and the manifold learning problem''
 
− 
 
−  Abstract: In the first half of this talk we will review several notions of coarse or weak
 
−  Ricci Curvature on metric measure spaces which include the works of LottVillani, Sturm
 
−  and Ollivier. The discussion of the notion of coarse Ricci curvature will serve as
 
−  motivation for developing a method to estimate the Ricci curvature of a an embedded
 
−  submaifold of Euclidean space from a point cloud which has applications to the Manifold
 
−  Learning Problem. Our method is based on combining the notion of ``Carre du Champ"
 
−  introduced by BakryEmery with a result of Belkin and Niyogi which shows that it is
 
−  possible to recover the rough laplacian of embedded submanifolds of the Euclidean space
 
−  from point clouds. This is joint work with Micah Warren.
 
− 
 
−  ===JeanMichel Roquejoffre (Toulouse)===
 
−  ''Front propagation in the presence of integral diffusion''
 
− 
 
−  Abstract: In many reactiondiffusion equations, where diffusion is
 
−  given by a second order elliptic operator, the solutions
 
−  will exhibit spatial transitions whose velocity is asymptotically
 
−  linear in time. The situation can be different when the diffusion is of the
 
−  integral type, the most basic example being the fractional Laplacian:
 
−  the velocity can be timeexponential. We will explain why, and
 
−  discuss several situations where this type of fast propagation
 
−  occurs.
 
− 
 
−  ===Myoungjean Bae (POSTECH)===
 
−  ''Free Boundary Problem related to EulerPoisson system''
 
− 
 
−  One dimensional analysis of EulerPoisson system shows that when incoming
 
−  supersonic flow is fixed, transonic shock can be represented as a monotone
 
−  function of exit pressure. From this observation, we expect wellposedness
 
−  of transonic shock problem for EulerPoisson system when exit pressure is
 
−  prescribed in a proper range. In this talk, I will present recent progress
 
−  on transonic shock problem for EulerPoisson system, which is formulated
 
−  as a free boundary problem with mixed type PDE system.
 
−  This talk is based on collaboration with Ben Duan(POSTECH), Chujing Xie(SJTU)
 
−  and Jingjing Xiao(CUHK).
 
− 
 
−  ===Changhui Tan (University of Maryland)===
 
−  ''Global classical solution and long time behavior of macroscopic flocking models''
 
− 
 
−  Abstract: Selforganized behaviors are very common in nature and human societies.
 
−  One widely discussed example is the flocking phenomenon which describes
 
−  animal groups emerging towards the same direction. Several models such
 
−  as CuckerSmale and MotschTadmor are very successful in characterizing
 
−  flocking behaviors. In this talk, we will discuss macroscopic representation
 
−  of flocking models. These systems can be interpreted as compressible Eulerian
 
−  dynamics with nonlocal alignment forcing. We show global existence of classical solutions and long time
 
−  flocking behavior of the system, when initial profile satisfies a threshold condition. On the other hand, another set
 
−  of initial conditions will lead to a finite time break down of the system. This
 
−  is a joint work with Eitan Tadmor.
 
− 
 
−  ===Hongjie Dong (Brown University)===
 
−  ''Parabolic equations in timevarying domains''
 
− 
 
−  Abstract: I will present a recent result on the Dirichlet boundary value
 
−  problem for parabolic equations in timevarying domains. The equations are
 
−  in either divergence or nondivergence form with boundary blowup loworder
 
−  coefficients. The domains satisfy an exterior measure condition.
 
− 
 
−  ===Hao Jia (University of Chicago)===
 
−  ''Long time dynamics of energy critical defocusing wave equation with
 
−  radial potential in 3+1 dimensions.''
 
− 
 
−  Abstract: We consider the long term dynamics of radial solution to the
 
−  above mentioned equation. For general potential, the equation can have a
 
−  unique positive ground state and a number of excited states. One can expect
 
−  that some solutions might stay for very long time near excited states
 
−  before settling down to an excited state of lower energy or the ground
 
−  state. Thus the detailed dynamics can be extremely complicated. However
 
−  using the ``channel of energy" inequality discovered by T.Duyckaerts,
 
−  C.Kenig and F.Merle, we can show for generic potential, any radial solution
 
−  is asymptotically the sum of a free radiation and a steady state as time
 
−  goes to infinity. This provides another example of the power of ``channel
 
−  of energy" inequality and the method of profile decompositions. I will
 
−  explain the basic tools in some detail. Joint work with Baoping Liu and
 
−  Guixiang Xu.
 
− 
 
−  ===Alexander Pushnitski (King's College)===
 
−  ''An inverse spectral problem for Hankel operators''
 
− 
 
−  Abstract:
 
−  I will discuss an inverse spectral problem for a certain class of Hankel
 
−  operators. The problem appeared in the recent work by P.Gerard and S.Grellier as a
 
−  step towards description of evolution in a model integrable nondispersive
 
−  equation. Several features of this inverse problem make it strikingly (and somewhat
 
−  mysteriously) similar to an inverse problem for SturmLiouville operators. I will
 
−  describe the available results for Hankel operators, emphasizing this similarity.
 
−  This is joint work with Patrick Gerard (Orsay).
 
− 
 
−  ===Ronghua Pan (Georgia Tech)===
 
−  ''Compressible NavierStokesFourier system with temperature dependent dissipation''
 
− 
 
−  Abstract: From its physical origin such as ChapmanEnskog or Sutherland, the viscosity and
 
−  heat conductivity coefficients in compressible fluids depend on absolute temperature
 
−  through power laws. The mathematical theory on the wellposedness and regularity on this
 
−  setting is widely open. I will report some recent progress on this direction,
 
−  with emphasis on the lower bound of temperature, and global existence of
 
−  solutions in one or multiple dimensions. The relation between thermodynamics laws
 
−  and NavierStokesFourier system will also be discussed. This talk is based on joint works
 
−  with Junxiong Jia and Weizhe Zhang.
 