With S. Marotta and R. Stankewitz we study, for each nonzero complex
parameter β, the Caruso semigroup Sβ of Mobius transformations of the
Riemann sphere generated by β+1/z and -β+1/z. In particular we
investigate
the dynamics of Sβ on its dual Julia set J'β, the closure of the set
of attracting
fixed points of elements of Sβ. Our approach is based on the
well-studied Riley
group Pβ generated by z + β and
z/(βz+1) and its limit set Λβ.
One has
J'β ⊂ Λβ,
but this inclusion may be strict. We study cases when J'β is an
attractor and
certain limiting cases as well, when we are able to find symbolic
dynamics for the
semigroup. Our work employs Riley group results of Keen/Series and
Wright.
In this talk I will explain the geometric quantization from the
viewpoint of the spectral convergence.
We take a Lagrangian fibration on a symplectic manifold and a family
of compatible complex structures tending to the real polarization
given by the fibration,
and show a spectral convergence of the d-bar Laplacian on the
prequantum line bundle to the spectral structure related to the set of
Bohr-Sommerfeld fibers.
This talk is based on the joint work with Mayuko Yamashita.
In this talk, we will give a sufficient condition for (strong)
stability of non-proper functions (with respect to the Whitney
topology). As an application, we will give a strongly stable but not
infinitesimally stable function. We will further show that any Nash
function on the Euclidean space becomes stable after a generic linear perturbation.
A stable map is a smooth map between manifolds which is right-left
equivalent to any map in a small neighborhood in the space of smooth
maps. A typical example is a Morse function. In the talk, we focus on
stable maps from compact orientable 3-manifolds bounded by (possibly
empty) tori to the real plane. We introduce the notion of stable map
complexity for a compact orientable 3-manifold counting, with some
weights, the minimal number of singular fibers of codimension 2 of
stable maps, and prove that this number equals the minimal number of
vertices of certain simple polyhedrons, called branched shadows. It is
known by F. Costantino and D. Thurston in 2008 that the minimal number
of vertices of shadows of a 3-manifold estimates its hyperbolic
volume. Combining this with our result, we may conclude that the
stable map complexity estimates the hyperbolic volume. This is a joint
work with Yuya Koda in Hiroshima University.
Poisson boundary is defined to be a probablity space
describing the distribution of a given random walk at time infinity.
We show that if a finitely generated group G acts on locally compact
CAT(0) space Y, then either G leaves a flat subspace in Y invariant,
or there exists a naturally induced equivariant map from the Poisson
boundary of G into the boundary of Y. This result is a refinement of
a theorem due to Uri Bader, Bruno Duchesne, and Jean Lécureux.
Unlike their approach, our proof is based on the use of discrete
harmonic map from G into Y.
The disk potential function of a Lagrangian submanifold plays an
important role in Lagrangian Floer theory and Mirror symmetry. In this
talk, I will review the construction of disk potential functions and
explain how to compute the disk potential functions for quadric
hypersurfaces.
Neural network in machine learning shares a common starting point as
quiver representation theory. In this talk, I will build an
algebro-geometric formulation of a `computing machine' which is
well-defined over the moduli space. The main algebraic ingredient is
to extend noncommutative geometry of Connes, Cuntz-Quillen, Ginzburg
to near-rings, which capture the non-linearity that is not present in
usual representation theory. Metrics over the moduli spaces are
crucial. I will explain a uniformization between spherical, Euclidean
and hyperbolic moduli of framed quiver representations.
We present an effective BV quantization theory for chiral deformation
of two dimensional conformal field theories. We explain a connection
between the quantum master equation and the chiral homology for vertex
operator algebras. As an application, we construct correlation
functions of the curved beta-gamma/b-c system and establish a coupled
equation relating to chiral homology groups of chiral differential
operators. This can be viewed as the vertex algebra analogue of the
trace map in algebraic index theory.
Strominger-Yau-Zaslow conjectured that Calabi-Yau manifolds admit
fibration structure with fibres being minimal tori. This gives a
strong geometric description of Calabi-Yau manifolds. Moreover, the
mirror of the Calabi-Yau manifolds can be constructed via the dual
torus fibration. The conjecture is a guiding principle for the
development of mirror symmetry, while the original conjecture largely
remains open. In this talk, I will have a general discussion about
the conjecture and then discuss the recent developments on some log
Calabi-Yau surfaces. This is the first time a full SYZ conjecture is
verified. The talk is based on joint works with T. Collins and A. Jacob.
A Hurwitz group is a conformal automorphism group of a compact Riemann
surface with precisely 84(g-1) automorphisms, where g is the genus of
the surface. Our starting point is a result on the smallest Hurwitz
group PSL(2,7) which is the automorphism group of the Klein
surface. In this talk, we generalize it to various classes of simple
Hurwitz groups and discuss a relationship between the surface symmetry
and spectral asymmetry for compact Riemann surfaces. To be more
precise, we show that the reducibility of an element of a simple
Hurwitz group is equivalent to the vanishing of the eta invariant of
the corresponding mapping torus.
The Wodzicki residue can be used to build Chern classes and
Chern-Simons classes for infinite dimensional bundles with
pseudodifferential operator-valued connections. This setup occurs
naturally in the Riemannian geometry of loop spaces. The Chern-Simons
classes give information on the topology of the diffeomorphism group
of the line bundles in geometric quantization of symplectic
manifolds. This has intriguing (i.e. vague) connections to the
geometric quantization Fock space and to anomalies in abelian QFT.
We construct a rigorous BV quantization of the half-twisted heterotic
sigma model with target a complex manifold X equipped with a
holomorphic vector bundle E, study its local anomalies, and relate the
algebra of observables to chiral differential operators acting on
E. This is joint work with Owen Gwilliam, James Ladouce, and Brian
Williams.
We first review the works of Bridgeland-Smith and
Haiden-Katzarkov-Kontsevich on stability conditions via quadratic
differentials. Then we introduce the q-deformation of categories,
stability conditions and quadratic differentials that link/unify the
works of BS and HKK. The talk is based on joint work with Akishi Ikeda
and Yu Zhou.
Integrals over configuration spaces have made their frequent
appearances through Feynman graph integrals (such as in 2d chiral
CFTs) and their various mathematical disguises in recent studies of
Gromov-Witten theory and mirror symmetry. These integrals are
constructed out of Green's functions and usually have serious
divergences. My talk will be focused on explaining how to
intrinsically regularized these divergent integrals for the case of
Riemann surfaces without having to deal with (in particular, multiply)
distributions by using the complex-analytic structure of Riemann
surfaces, and discussing properties of these integrals. I will first
outline the definition of our notion of regularized integrals, then
some of their main properties with examples, and finally applications
in mirror symmetry for elliptic curves. The talk will be based on
recent joint works with Si Li.
The Crepant Transformation Conjecture, proposed by Ruan, asserts
certain equivalence between Gromov-Witten theory of two
manifolds/orbifolds which are related by a crepant transformation. In
general, the higher genus Crepant Transformation Conjecture is
quite hard to study. Even the formulation of the Crepant
Transformation Conjecture in the higher genus case is subtle. In this
talk, I will
explain the proof of the all genus Crepant Transformation Conjecture
for general toric Calabi-Yau 3-orbifolds. We will consider
the higher genus Gromov-Witten theory of toric Calabi-Yau 3-orbifolds
in both open-string sector and closed-string sector. This talk
is based on a project joint with Bohan Fang, Chiu-Chu Melissa Liu, and
Song Yu.
