The 6th HU and SNU Symposium on Mathematics
- Geometry and Topology -
as a part of The 11th HU-SNU Joint Symposium
アブストラクト
Jongil Park
- Symplectic $4$-manifolds with $b_2^+=1$ versus complex surfaces
with $p_g=0$
(Please note that a title is a little bit changed)
- One of the fundamental problems in the study of $4$-manifolds is
to find a new family of simply connected smooth (symplectic, complex)
$4$-manifolds. Though many interesting $4$-manifolds have been constructed
using techniques such as fiber sum, rational blow-down, knot surgery,
Luttinger surgery and so on, it is still very hard to find a new family of
$4$-manifolds with small Euler characteristic.
Since I discovered a new simply connected symplectic $4$-manifold with
$b_2^+ =1$ and $c_1^2=2$ in 2004 using a rational blow-down surgery, many
new simply connected $4$-manifolds with small Euler characteristic have
been constructed and now it is one of most active research areas in
$4$-manifolds theory to find a new family of $4$-manifolds with $b_2^+ =1$
(equivalently $p_g=0$ in complex category).
The aim of this talk is to review recent development in the construction
of $4$-manifolds with small Euler characteristic. In particular, I'd like
to survey the existence problems of simply connected symplectic
$4$-manifolds with $b_2^+=1$ and complex surfaces of general type with
$p_g=0$.
Jaehyun Hong
- Analytic continuation of holomorphic maps respecting varieties of
minimal rational tangents
- In this talk, we will discuss how to extend a local holomorphic
embedding respecting the variety of minimal rational tangents. As an
application, we will show that, given an equivariant embedding of a
homogeneous variety X into a homogeneous variety Y, a local
biholomorphism f from X into Y becomes the given embedding up to the group
action, provided that f respects the varieties of minimal rational
tangents. This is a joint work with N. Mok.
Cheol-hyun Cho
- On Floer cohomology in toric manifolds
- We review developments of Floer homology theory due to Fukaya
Oh, Ohta and Ono and
explain recent advances in the computations of toric manifold cases.
Sangwook Lee
- Potentials of homotopic cyclic A-infinity algebras
- For an A-infinity algebra with cyclic inner product, a potential
recording the structure constants can be defined. We explain how to
construct a potential directly in the case that cyclic inner product is
given in an A-infinite quasi-isomorphic algebra. (joint work with
Cheol-Hyun Cho).
Heesang Park
- A simply connected surface of general type with $p_g=0$ and $K^2=4$
- We construct a simply connected minimal complex surface of
general type
with $p_g=0$ and $K^2=4$ by using a rational blow-down surgery and
$\mathbb{Q}$-Gorenstein smoothing theory.
Takuro Abe
- A logarithmic module of a generalized multiarrangement
- A multiarrangement is a pair of an arrangement of hyperplanes and
positive integer-valued multiplicities. We can associate the logarithmic
vector field to a multiarrangement, which has been playing an important
role in the arrangement theory. Recently, several researches on it
make us feel that negative multiplicities on hyperplanes and
the associated logarithmic module should be also important if we
could formulate them correctly.
In this talk, we will give a definition of a generalized
multiarrangement including negative multiplicities, and the
associated logarithmic module. As an application, a duality of
Coxeter multiarrangements will be given.
Mayuko Kon
- On a Hopf hypersurface of a complex space form
-
A Hopf hypersurface is defined to be a real hypersurface
of a complex space form whose structure vector field is a
principal curvature vector field. We study some conditions
on the holomorphic distribution on real hypersurfaces which
contain the definition of a Hopf hypersurface. On the other
hand, we study real hypersurfaces whose structure vector field
is an eigenvector field of the Ricci operator.
Ryohei Hattori
-
Analogies to Gauss' results on the arithmetic-geometric mean
and the hypergeometric function
- Let m(a, b) be an arithmetic-geometric mean of two
positive numbers a and b. That is, it is a common
limit of sequences {a_n} and {b_n} defined by
a_1 = a,
b_1 = b,
a_{n+1} = (a_n+b_n)/2,
b_{n+1} = (a_n*b_n)^(1/2).
Gauss showed the equation
a/m(a,b) = F( 1/2, 1/2, 1 ;1-(b/a)^2)
in 1799, where F(p, q, r; z) denotes the hypergeometric
function with parameters p, q, and r.
We can prove this equation simply. The key of the proof
is, to use transformation formulas for the
hypergeometric function skillfully.
It is well known that there are several types of
transformation formulas of the hypergeometric function.
In this talk, we use those in the Goursat's paper in 1881,
and give extensions of arithmetic-geometric mean and
introduce analogies to Gauss' theorem.
Daisuke Hirose
-
A positive characteristic analogue of the log canonical singularity
- The log canonical singularity is a class of singularities,
which appeared in birational geometry.
Since this singularity is defined via resolution of singularities,
the base field is always the complex number field.
On the other hand,
for a field of positive characteristic,
an another type of singularity is defined via the Frobenius morphism.
It is called the F-pure singularity.
After reduction to mod p >0,
F-pure singularities imply log canonical singularities.
However these two type of singularities have different behavior.
In this talk,
the speaker introduces the notion of two type of singularities
and gives examples which illustrate their differences.
Jiro Adachi
- A certain handle decomposition and Engel structure
- An Engel structure is a distribution of tangent subspace of dimension $2$
on a $4$-dimensional manifold which is maximally non-integrable.
Engel structure has no local invariant, like contact structure.
Contact structures on $3$-dimensional manifolds
have an important relation to $3$-dimensional topology.
However, we know little about parallelizable $4$-dimensional manifolds,
only where Engel structures exist.
I will talk on some geometry
of closed parallelizable $4$-dimensional manifolds
from the view point of Engel geometry.