 Past seminars:
Fall 2006,
Spring 2007,
Fall 2007
Spring 2008,
Fall 2008,
Spring 2009,
and
Fall 2009
|
Spring 2010:
|
Jan. 29: Ara Basmajian (Hunter College and Graduate Center of CUNY)
Universal Length Bounds for Non-Simple Closed Geodesics
We consider the relationship between the length of a closed geodesic on a hyperbolic
surface and its self-intersection number. In the 1993 paper The stable neighborhood theorem and lengths of closed geodesics
it was shown that a closed geodesic with intersection number k has length bounded from below by a constant Mk
that goes to infinity with k. The Mk are universal constants in that they only depend on k and not
on any particular hyperbolic structure. In this talk we show that the Mk grow like Log k. Our techniques
are elementary and involve using a quantitative version of the Margulis lemma.
|
Feb. 5: Sarah Koch (Harvard University)
Böttcher Coordinates in Cm
A well-known theorem of Böttcher asserts that an analytic germ
f:(C,0) → (C,0) which has a superattracting fixed
point at 0, more precisely of the form f(z) = a zk + o(zk) for some non-zero
a, is analytically conjugate to the power map z ↦ a zk by an
analytic germ (C,0) → (C,0) which is tangent
to the identity at 0. In this talk, we generalize this result and
give a Böttcher criterion for analytic maps in several complex
variables.
|
Feb. 12: No meeting
|
Feb. 19: Trevor Clark (Toronto/Stony Brook University)
Regular or Stochastic Dynamics in Higher Degree Families of Unimodal Maps
About twenty years ago, Palis conjectured that typical dynamical systems should possess good statistical properties.
Through the work of Avila, Lyubich, de Melo and Moreira, this has been proven for unimodal maps with a non-degenerate
critical point. I will show how to remove the condition on the critical point in analytic families of unimodal maps;
along the way proving that the hybrid classes in the space of unimodal maps yield a lamination near all but
countably many maps in the family. The essential difference in the higher
degree case is the presence of non-renormalizable maps without "decay of geometry." The key to their study is
the use of a generalized renormalization operator, which has much in common with the usual renormalization operator.
|
Feb. 26: Seminar cancelled due to heavy snow
|
Mar. 5: Reza Chamanara (Brooklyn College of CUNY)
Rigidity of Inversive Distance Disk Patterns
The study of configurations of disks with prescribed combinatorial
and geometric patterns is an important part of combinatorial geometry.
The most well-known example is the theory of circle packing with its numerous
applications in discrete analytic function theory. The basis of the theory of
circle packing is the so called Koebe's theorem which states that given any
triangulation K of the sphere S2, there is a circle packing
PK in S2 with the tangency pattern prescribed by K.
Moreover, the disks in PK have mutually disjoint interiors and PK
is unique up to application of a Mobius map. Theorems by Andreev and Rivin on the existence and uniqueness
of convex compact or convex ideal hyperbolic polyhedra can be interpreted as theorems about
configurations of disks with prescribed patterns of intersection.
I will discuss the possible extensions of such results to patterns of pairwise disjoint disks on the sphere.
In particular, I will show how such patterns define a weighted planar graph. Moreover, I will explain why
the weighted graph determines the disk pattern up to application of a Mobius map.
|
Mar. 12: Robert Devaney (Boston University)
Dynamics of Family of Maps zn + C/zn: Why the Case n = 2 is Crazy
In this talk we describe some of the interesting dynamics associated to the family of complex maps
zn + C/zn, where C is a complex parameter and n > 1.
It turns out that the case n = 2 is very different from the case n > 2. For example,
when n > 2 there is a "McMullen domain" in the parameter space which is surrounded by infinitely
many "Mandelpinski" necklaces, but there is no such structure when n = 2. Also, when n = 2,
as C tends to 0, the Julia sets of these maps converge to the closed unit disk, but this never
happens when n > 2. So the dynamical behavior of these maps near the parameter C = 0 is much more complicated
when n = 2.
|
Mar. 19: Gabino Gonzalez-Diez (UAM, Spain)
On Beauville Surfaces and the Genus of the Curves Arising in Their Construction
A Beauville surface is a 2-dimensional compact complex manifold of the form
S = (C1 x C2) / G, where C1 and C2
are compact Riemann surfaces of hyperbolic type and G is a finite group acting freely on the product
C1 x C2 in such a way that each of the factors is preserved by the
action and, moreover, the quotient Ci / G is an orbifold of genus zero with
three cone points. Beauville surfaces were introduced by Catanese following an
initial example of Beauville in which C1 = C2 is the Fermat curve of genus
6 with the equation X05+X15+X25 = 0
and G is isomorphic to Z5 x Z5.
In this talk, I shall discuss questions such as which groups G and which
genera g1, g2 of C1, C2 can arise in the construction of Beauville
surfaces. In particular, I will show that g1 and g2 have to be at least 6
and that if g1 = g2 = 6 then S agrees with (one of the two) Beauville
examples. The proof of this fact will rely on methods belonging to the theory
of Fuchsian groups and Riemann surfaces.
|
Mar. 26: Xiao Jun Huang (Rutgers University)
TBA
|
Apr. 2: No Meeting
|
Apr. 9: Jian Song (Rutgers University)
TBA
|
Apr. 16: Moira Chas (Stony Brook University)
TBA
|
Apr. 23: Melkana Brakalova (Fordham University)
TBA
|
Apr. 30: Shou-Wu Zhang (Columbia University)
Calabi Theorem and Algebraic Dynamics
In this talk, I will first state a theorem of Calabi for
both complex and p-adic manifolds and show its applications
to dynamical systems on complex varieties in terms of preperiodic
points. Then I will discuss dynamical Manin-Mumford conjectures
with both supporting examples and counterexamples.
This is a report on joint works with Xinyi Yuan, and with Ghioca and Tucker
respectively.
|
May 7: Clifford Earle (Cornell University)
TBA
|
May 14: Sudeb Mitra (Queens College of CUNY)
TBA
|
