Configuration and the RankineHugoniot type condition

1
Minimal surfaces in the Heisenberg group

• Configuration and the Rankine-Hugoniot type
  condition

2

• The research team consisting of Dr. Jih-Hsin
  Cheng and Dr. Jenn-Fang Hwang at IMAS has
  collaborated with Professor Paul Yang of
  Princeton University and Professor Andrea
  Malchiodi of SISSA on the study of minimal
  surfaces in the Heisenberg group. This is a
  degenerate elliptic and degenerate hyperbolic
  equation. They proved a Bernstein-type theorem.
  They also proved the existence of minimizers and
  pointed out a condition for characteristic lines
  to meet along a singular curve. It is a
  Rankine-Hugoniot type jump condition. As a
  consequence, a C2-smooth solution from the PDE
  viewpoint may not be a minimizer.

3
Quantitative Analysis of Boltzmann Equation
4
Quantitative Analysis of Boltzmann Equation

• Tai-Ping Liu and his collaborators have made
  progresses on the quantitative theory for
  Boltzmann equation in kinetic theory. In the
  first paper, three dimensional wave structure,
  including the weak Huygens' waves, see Figure 1,
  is obtained. Besides this close link with fluid
  dynamics, it is also shown that Boltzmann
  equation carries particle-like waves. The second
  paper offers the first analytical understanding
  of the thermal transpiration, a phenomena whose
  physical explanation consumed the last three
  years of the great physicist Maxwell. The third
  paper studies the refleciton of waves off the
  boundary, Figure 2. This is the first general
  study of the initial-boundary phenomena for
  Boltzmann equation. The Boltzmann equation allows
  for much richer boundary phenomena
• than the fluid equations such as the Euler
  and Navier-Stokes equations. We are continue to
  explore this important research direction.

5

• Connections between symmetry and super-symmetry
• In mathematics symmetry may be regarded as
  certain objects that are either groups or
  algebras. In physics the notion of symmetry was
  soon realized to be insufficient to describe all
  known physical phenomena. In order to resolve
  these restrictions the notion of super-symmetry
  was introduced.
• In 2004 Shun-Jen Cheng in joint works with Ruibin
  Zhang, in their investigations of representations
  of certain simple Lie superalgebras, noticed
  remarkable connections between the representation
  theories of Lie algebras and Lie superalgebras,
  i.e. symmetries and super-symmetries. In 2006 in
  a joint work with Weiqiang Wang they established,
  for the first time, such a direct connection.
  They proved that the general linear Lie algebras
  and the general linear Lie superalgebras share
  certain common important representation-theoretica
  l invariants, called Kazhdan-Lusztig polynomials.
  Subsequently related results for
  infinite-dimensional Lie superalgebras have been
  established by Shun-Jen Cheng and Jae-Hoon Kwon.