Hungarian Academy of Sciences Mathematical Institute:
Financial Support; In-kind Support; Facilities; Collaborative Research; Personnel Exchanges
This grant was a joint NSF-OTKA grant. Its purpose was to stimulate
research between Hungarian and American mathematicians. Our project
was a joint research proposal, with the American PI to travel to
Hungary to work with the Hungarian participant. The American PI
travelled to Hungary twice a year. On one occasion, an undergraduate
researcher also travelled to Hungary to take part in the research.
This is the PI's home institution, where most of the research between
the undergraduate student and the professor took place. The
university provided faculty/student research funding, and supplemented
the funding from this grant, in addition to providing resources
necessary for completing the papers written as part of the grant.
Alice Fialowski and the PI are collaborating with Marilyn Daily, who
was a graduate student, and now is a recent Phd on a paper. The PI
gave a talk at the University of Debrecen, in Debrecen, Hungary, at
the invitation of Victor Bodi, who also came and visited the
collaborators in Budapest, in order to discuss our work. Victor Bodi
is considering applying for a grant to visit the PI at his
institution. Alice Fialowski and the PI met with James Stasheff in
Vienna to discuss our work with him. The PI has met other potential
collaborators as a result of some talks given at AMS meetings at will
be meeting with Lucian Ionescu to discuss possible collaboration.
Activities and findings:
Research and Education Activities:
With 3 undergraduate students, have investigated 1 and 2 dimensional
A-infinity algebras. With Alice Fialowski, investigated L-infinity
algebras of dimensions 0|4, 0|3, 1|2 and 2|1, with an undergraduate
student participating in the research of the 2|1 dimensional case.
Gave several talks at conferences demonstrating how to construct
miniversal deformations and extend codifferentials to more complex
infinity algebra structures. Applied the methods to some problems
related to the homology of graph complexes.
Talks given by the PI at various conferences and universities are
August 2004-Conference on Differential Geometry and its Applications,
Prague, Czech Republic
'The Orbifold Structure of the Moduli Space of Four Dimensional Lie
April 2004-American Mathematical Society Sectional Meeting,
'Deformations of Infinity Algebras'
November 2003-American Mathematical Society Sectional Meeting,
'Extensions and Deformations of Infinity Algebras'
January 2003-Eotvos Lorand University
'Classification of Low dimensional Infinity Algebras'
January 2003-University of Debrecen,
'Classification of Extensions of Infinity Algebras'
January 2003-Alfred Reyni Institute of Mathematics,
'Deformations of Infinity Algebras'
Conference on Categorical Algebra, Deformation Theory and Field Theory
II, Kyoto, Japan
'Examples of Infinity Algebras and Their Deformations'
Dr. Fialowski gave a talk on the joint results at the Max-Planck
Institute for Mathematics Bonn, Germany in 2003 and at the
International Algebra Conference in Lisboa Portugal in July 2003. In
August 2003 both Dr. Penkava and Dr. Fialowski were invited to the
Schrodinger Institut in Vienna wheer they consulted their recent
results with Prof. James Stasheff, a leading expert in the area.
Our work has recently attracted the attention of other researchers. In
particular, Dr. Penkava and Dr. Fialowski are collaborating with
Marilyn Daily, a PH.D. student of Prof. Thomas Lada, who just
completed successfully her Ph.D. thesis at North Carolina State
University. Marilyn Daily has done some work on Z-graded L-infinity
algebras of small dimension, and the three researchers are writing a
paper on the classification and deformation theory of some low
dimensional examples of Z-graded L-infinity algebras, a natural
addition of the work of Dr. Penkava and Dr. Fialowski on classifying
Z_2-graded L-infinity algebras. Dr. Penkava met with Marilyn Daily in
North Carolina in November 2003, and Marilyn travelled to Budapest in
March 2004 where she worked with Dr. Fialwoski on this project. We
plan to complete the work by the end of this year.
The research between Dr. Fialowski And Dr. Penkava has involved
undergraduate students in a significant manner. Derek Bodin took part
in the research, contributing both to the computational aspect, by
writing programs to compute cohomology using the computer algebra
system Maple, and to the theoretical aspect by helping to classify the
degree two L-infinity structures on a 2|1-dimensional space applying
methods that were developed by Derek Bodin and Dr. Penkava in the
course of a year long effort at the University of Wisconsin-Eau
Claire. Carolyn Otto, a sophomore mathematics major, is just beginning
her research with Dr. Penkava, and has indicated a strong interest in
continuing this work. While it is impossible to involve her in all
aspects of the work, there is every reason to expect that her work
could be of high quality, so her contributions to the research could
We gave a complete classification of all L-infinity algebras of
dimension 0|3 and 1|2, as well as their versal deformations. We
classified L-infinity algebras with nonzero degree 1 or degree 2 term
for 2|1 case. We analyzed the moduli space of Lie algebras of
dimension 4, from the point of view of infinity algebras. We developed
methods of construction for versal deformations by recursive formulas,
and extensions by using cohomological methods. We classified all 1
dimensional A-infinity algebras, and studied moduli space of 0|2 and
1|1 dimensional associative algebras.
So far, we have discovered some very complex patterns in our analysis
of extensions of three dimensional infinity algebras. We have proved
several general results about extensions, and have been led to a
general conjecture which has been verified in several instances that
would aid in the classification of extensions. Our study of the versal
deformations of three dimensional infinity algebras led us to discover
a recursive method for calculating versal deformation directly,
instead of buildingup to the versal deformation by considering
infinitesimal deformations, then second order deformations and so on.
We have developed a systematic approach for presenting an L-infinity
algebra in terms of cochains which makes the verification that a
proposed structure satifies the axioms of an infinity algera easy,
especially with the use of the computer algera tools that were
designed in conjunction with Derek Bodin.
The research of PI and Alice Fialowski has resulted in 2 published
joint papers, 3 accepted joint papers, 2 additional submitted papers,
as well as several papers in preparation. Our collaboration originated
with a grant from the National Research Council, which enabled the PI
to travel to Hungary to map out a plan of research in 1997. Later,
through grants from the University of Wisconsin, the Eotvos Lorand
University, and the Alfred Renyi Institute of Mathematics in Budapest,
we continued our work until we obtained the current travel grant from
the NSF, which provided us with the opportunity to meet for longer
periods of time and more frequently, as well as to bring an
undergraduate into our collaboration.
The original problem we studied was how to construct miniversal
deformations of infinity algebras. This problem had been studied by
Dmitry Fuchs and Alice Fialowski for Lie algebras. The PI and Dr.
Fialowski extended the construction to the case of infinity algebras.
Because the cohomology of an infinity algebra is in general, infinite
dimensional, an appropriate notion of finiteness needed to be
developed in order to extend the construction technique to infinity
Later, James Stasheff asked us if we knew any finite dimensional
examples of infinity algebras, which led us to begin studying low
dimensional examples of infinity algebras. Our success in the
application of the constructive method of determining versal
deformations to 1 and 2 dimensional infinity algebras led us to be
interested in studying more examples, in order to understand what
general methods could be derived.
The work for this grant was mostly with three dimensional L-infinity
algebras, although we also studied four dimensional Lie algebras. In
addition, the PI worked with undergraduate students on one and two
dimensional A-infinity algebras. Each of the different types of three
dimensional vector spaces reuqired different techniques of analysis.
Later, when Marilyn Daily wrote about three dimensional Z-graded
infinity algebras, it was easy to see how her examples fit into the
cases we had studied, which has led to a collaboration with
M. Daily on the classification of these algebras and the relation to
Z_2-graded infinity algebras.
Training and Development:
Pi and coresearcher have developed considerable understanding of
versal deformations and extensions of infinity algebras. Undergraduate
students have gone on to successful careers as graduate students, and
teachers. By working closely with a professor, students developed an
understanding of what research in mathematics consists of and were
able to contribute significantly to the development of computer
programs and to help construct deformations and extensions.
Now the PI and Alice Fialowski are expanding their activities to
applications and are bringing their results to the attention of the
research community through talks and papers.
One of the purposes of the joint Hungarian-US program was to encourage
interaction between researchers from different cultures. Through the
PI's interactions with the public in Budapest, the importance of
mathematical research was showcased. Reports of this research were
covered in the local press in Wisconsin, giving the public a sense of
the value of mathematical research, and how the research between
students and faculty enriches their education.
Other Specific Products:
Maple software for computation of brackets of coderivations for both
the tensor and symmetric coalgebras. This software also will compute
brackets of deformed products.
This software will be made available to other researchers and can be
used on the Maple computer algebra system. Most likely, these
programs will be posted on the PI's web page eventually.
Contributions within Discipline:
The techniques we developed for computation of deformations and
extensions of infinity algebras can be used in some applied problems.
For example, the PI and an undergraduate student computed all
A-infinity algebras of dimension 1. Using this classification, one
can give a canonical orientation to the highest dimensional cells in
the complex of oriented Ribbon graphs.