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Residues and Dynamics

12-13 September 2016
Place: Aula Magna, Department of Mathematics, University of Pisa

List of Speakers:

Paolo Arcangeli (University of Roma "Sapienza")
Anna Miriam Benini (University of Roma "Tor Vergata")
Fabrizio Bianchi (University "Paul Sabatier", Toulouse & University of Pisa)
Cinzia Bisi (University of Ferrara)
Ko Fujisawa (Hokkaido University)
Takeshi Izawa (Hokkaido University of Science)
Toru Ohmoto (Hokkaido University)
Tatsuo Suwa (Hokkaido University)

Titles and Abstracts:

-Paolo Arcangeli (University of  Roma "Sapienza")
Title: Index theorems for couples of holomorphic self-maps.
Abstract: Let M be a n-dimensional complex manifold and f and g two distinct holomorphic self-maps of M. Suppose that f and g coincide on a globally irreducible compact hyper surface S of M. We show that if one of the two maps is a local biholomorphism around the regular part S' of S and, if needed, S' sits into M in a particular nice way, then it is possible to define a 1-dimensional holomorphic (possibly singular) foliation on S' and partial holomorphic connections on certain holomorphic vector bundles on S'. As a consequence, we are able to localize suitable characteristic classes and thus to get index theorems.

-Anna Miriam Benini (University of Roma “Tor Vergata”)
Title: A landing theorem for hairs and dreadlocks of entire functions with bounded post-singular sets.
Abstract. The  Douady-Hubbard landing theorem for periodic external rays is one of the cornerstones of the successful study of polynomial dynamics. It states that, for a complex polynomial f with bounded postcritical set, every  periodic external ray lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic   point is the landing point of at least one periodic external ray. We prove an  analogue of the theorem for entire functions with bounded postsingular set.  If such f  additionally has finite order of growth, then our result states precisely that every periodic hair of f lands at a repelling or parabolic point, and again conversely every repelling or parabolic point is the landing point  of at least one periodic hair.  (Here a "periodic hair" is a curve consisting of escaping points of f  that is invariant under an iterate of  f).  For general f with bounded postsingular set, but not necessarily of finite order, the role of hairs is taken by more general connected sets of  escaping points,  which we call "dreadlocks".  This is joint work with Lasse Rempe-Gillen. 

-Fabrizio Bianchi (University "Paul Sabatier", Toulouse & University of Pisa)
Title: Parabolic implosion in two complex variables.
Abstract: The theory of parabolic implosion - i.e., the study of perturbations of parabolic fixed points - has been at the center of the research in (one-dimensional) holomorphic dynamics in the last couple of decades. Recently, Bedford-Smillie-Ueda and Dujardin-Lyubich started an analogous study in the 2-dimensional semi-parabolic setting, when one direction at the fixed point is attracting and the other one is parabolic. In this talk I will present a generalization of their work to the completely parabolic setting, when both directions at the fixed point are parabolic. As an application, we give estimates for the discontinuity of dynamically defined sets at the parabolic parameter.

-Cinzia Bisi (University of Ferrara)
Title: Localized intersection of currents and the Lefschetz coincidence point theorem.
Abstract: We introduce the notion of a Thom class of a current and define the localized intersection of currents. In particular, we consider the situation where we have a smooth map of manifolds and study localized intersections of the source manifold and currents on the target manifold. We then obtain a residue theorem on the source manifold and give explicit formulas for the residues in some cases. These are applied to the problem of coincidence points of two maps. We define the global and local coincidence homology classes and indices. A representation of the Thom class of the graph as a Čech–de Rham cocycle immediately gives us an explicit expression of the index at an isolated coincidence point, which in turn gives explicit coincidence classes in some non-isolated components. Combining these, we have a general coincidence point theorem including the one by S. Lefschetz.

-Ko Fujisawa (Hokkaido University)
Title: Equivariant Cech-de Rham theory and equivariant Thom form.
Abstract: As well known,  Atiyah-Bott-Berline-Vergne’s localization formula is a very useful tool in computations  of integrals of closed equivariant differential forms. This formula is proved by using the universal equivariant Thom form in the theory of Mathai-Quillen (the fermionic integral and supersymmetry arguments).  In this talk, we introduce the equivariant Cech-de Rham theory and define the universal equivariant Thom form in our framework; in particular, the Thom form is obtainedby localizing a certain equivariant Chern form. An emphasis is that our expression is more elementary and simpler than that of Mathai-Quillen method.

-Takeshi Izawa (Hokkaido University of Science)
Title: Sato hyperfunctions via relative Dolbeault cohomolgy.
Abstract: The Cech-Dolbeault cohomology is the hypercohomology of the Dolbeault complex. We define the relative Dolbeault cohomology by the subgroup of Cech-Dolbeault cocycles (hyperforms, for short) vanishing outside the exceptional set. Through the localization method,  the relativeversion of the Kodaira-Serre duality is discussed (it is like the Alexander duality in the Chech-de Rham theory). As an application, we revisit the theory of Sato hyperfunctions,  an analytic counterpart of Schwartz's distribution theory; in fact, a hyper function on a real analytic manifold is interpreted as a localised Dolbeaelt-hyperform and some properties of hyper functions can be treated more concisely in our framework. This is a joint work with N. Honda (HU) and T. Suwa (HU).

-Toru Ohmoto (Hokkaido University)
Title: Degeneracy loci and Schubert using Chech-de Rham theory. 
Abstract: For a complex vector bundle morphism f from E to F on a complex manifold X, the k-th degeneracy locus is defined as the locus of points x in X at which the rank of f_x is less than or equal to k.  The Giambelli-Thom-Porteous formula expresses the locus as a certain Schur function of Chern classes.  Here we present its localized form at the level of Cech-de Rham (or Cech-Dolbeault) cocyle. Also equivariant theory provides a perspective on degeneracy formulae and Schubert calculus. A joint work of T. Suwa and K. Fujisawa.

-Tatsuo Suwa (Hokkaido University)
Title: Relative Bott-Chern cohomology.
Abstract: The Bott-Chern cohomology of a complex manifold refines both the de Rham and Dolbeault cohomologies. In this talk, I present a theory of relative Bott-Chern cohomology and discuss possible applications in the refined residue theory. A joint work with M. Correa. 


Monday, September 12:
9:30-10:30 Fujisawa
10:40-11:40 Ohmoto
14:30-15:30 Benini
15:40-16:40 Bisi

Tuesday, September 13:
9:30-10:30 Bianchi
10:40-11:40 Arcangeli
14:30-15:30 Izawa
15:40-16:40 Suwa

List of participants:

Marco Abate (University of Pisa)
Paolo Arcangeli (University of Roma "Sapienza")
Anna Miriam Benini (University of Roma "Tor Vergata")
Fabrizio Bianchi (University "Paul Sabatier", Toulouse & University of Pisa)
Cinzia Bisi (University of Ferrara)
Filippo Bracci (University of Roma "Tor Vergata")
Ko Fujisawa (University of Hokkaido)
Takeshi Izawa (University of Hokkaido)
Samuele Mongodi (University of Pisa)
Toru Ohmoto (University of Hokkaido)
Jasmin Raissy (University "Paul Sabatier", Toulouse)
Tatsuo Suwa (University of Hokkaido)
Francesca Tovena (University of Roma "Tor Vergata")

-Scientific Committee: Marco Abate (University of Pisa), Filippo Bracci (University of Roma "Tor Vergata"), Toru Ohmoto (University of Hokkaido), Jasmin Raissy (University "Paul Sabatier", Toulouse). 
-Organizing Committee: Marco Abate (University of Pisa), Simona Settepanella (University of Hokkaido).
-Contact: Marco Abate (marco.abate [at] unipi.it).

This program is supported by: Centro de Giorgi, FIRB2012 grant  "Differential geometry and geometric function theory",RBFR12W1AQ 002.