Project Reporting FINAL REPORT FOR AWARD # 0120676

Michael R Penkava ; U of Wisconsin Eau Claire
U.S.-Hungary Mathematics Research on Cohomology and Deformations of Infinity and Lie Algebras

Participant Individuals:
Senior personnel(s) : Alice Fialowski
Undergraduate student(s) : Derek Bodin

Partner Organizations:
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.

University of Wisconsin-Eau Claire: Financial Support; In-kind Support; Facilities; Collaborative Research; Personnel Exchanges
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.

Other collaborators:

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 listed below. August 2004-Conference on Differential Geometry and its Applications, Prague, Czech Republic 'The Orbifold Structure of the Moduli Space of Four Dimensional Lie Algebras' April 2004-American Mathematical Society Sectional Meeting, Lawrenceville NJ 'Deformations of Infinity Algebras' November 2003-American Mathematical Society Sectional Meeting, Raleigh, NC 'Extensions and Deformations of Infinity Algebras' January 2003-Eotvos Lorand University Budapest, Hungary 'Classification of Low dimensional Infinity Algebras' January 2003-University of Debrecen, Debrecen, Hungary 'Classification of Extensions of Infinity Algebras' January 2003-Alfred Reyni Institute of Mathematics, Budapest, Hungary 'Deformations of Infinity Algebras' December 2002- 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 be substantial.

Findings:
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 algebras. 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.

Outreach Activities:
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.

Journal Publications:
A. Fialowski and M. Penkava, "Versal Deformations of Three Dimensional Lie algebras as L-infinity Algebras", Communications in Contemporary Mathematics, vol. , (), p. . Accepted
A. Fialowski and M. Penkava, "Extensions of L-infinity algebras on a 2|1 dimensional space", International Journal of Mathematics, vol. , (), p. . Submitted
A. Fialowski and M. Penkava, "Strongly Homotopy Lie Algebras of One Even and Two Odd Dimensions", Journal of Algebra, vol. , (), p. . Accepted
A. Fialowski and M. Penkava, "Examples of Miniversal Deformations of Infinity Algebras", Reine und Angewandte Mathematik, vol. , (), p. . Submitted
D. Bodin, A. Fialowski and M. Penkava, "Classification and versal deformations of L-infinity algebras on a 2|1 dimensional space", Homology, Homotopy and its Applications, vol. , (), p. . Accepted
A. Fialowski and M. Penkava, "Examples of Infinity and Lie Algebras and their Versal Deformations", Banach Center Publications, vol. 55, (2002), p. 27. Published

Book(s) of other one-time publications(s):

Other Specific Products:

Software (or netware)
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:

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.


Categories for which nothing is reported:
Products: Book or other one-time publication
Products: Internet Dissemination
Contributions to Other Disciplines
Contributions to Education and Human Resources
Contributions to Resources for Science and Technology
Contributions Beyond Science and Engineering


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