Katrina D. Barron  Office: ATTENTION!!! My office has changed to 276C Hurley!!!!!! Phone: 574-631-3981 E-mail: kbarron@nd.edu Mailing Address: 255 Hurley Hall Department of Mathematics University of Notre Dame Notre Dame, IN 46556

i miei bambini

Current Courses:

Spring 2008 Semester:
    Math 10560 Section 03 Calculus II. Click here for the course webpage.
    Math 80220 Section 01 Topics in Algebra II: Vertex Operator Algebras and Conformal Field Theory. Click here for the course webpage.

Research: My research focuses on vertex operator superalgebras and the algebraic and geometric foundations of superconformal field theory.

Conformal field theory (CFT), or more specifically, string theory, and related superconformal field theories (SCFTs) are the most promising attempts at developing a physical theory that combines all fundamental interactions of particles, including gravity.  The ``super" refers to an assumed symmetry between bosons (integral spin particles with symmetric wave functions) and fermions (half integral spin particles with anti-symmetric wave functions).

The geometry of CFT and SCFT extends the use of Feynman diagrams, describing the interactions of point particles whose
propagation in time sweeps out a line in space-time, to one-dimensional strings or superstrings whose propagation in time
sweeps out a two-dimensional surface or supersurface called a ``worldsheet".

Much of my research involves the study of relationships between the worldsheet geometry of CFT and SCFT and properties of the algebras of correlation functions of the particle interactions.  For genus-zero holomorphic CFT and SCFT these algebras are called vertex operator superalgebras.

Selected Publications:

K. Barron, On axiomatic aspects of N=2 vertex superalgebras with odd formal variables, and deformations of N=1 vertex superalgebras, submitted.

K. Barron, The moduli space of N=2 super-Riemann spheres with tubes, Commun. in Contemp. Math., vol. 9 (2007), 857-940.

K. Barron, Y.-Z. Huang, J. Lepowsky, An equivalence of two constructions of permutation-twisted modules for lattice vertex operator algebras, J. Pure and Appl. Algebra, vol. 210 (2007), 797-826.

K. Barron, Superconformal change of variables for N=1 Neveu-Schwarz vertex operator superalgebras, J. of Algebra, vol. 277 (2004), 717-764.

K. Barron, The notion of N=1 supergeometric vertex operator superalgebra and the isomorphism theorem, Commun. in Contemp. Math., vol. 5 (2003), 481-567.

K. Barron, The moduli space of N=1 superspheres with tubes and the sewing operation, Memoirs of the AMS, vol. 162, no. 772, (2003).

K. Barron, C. Dong, G. Mason, Twisted sectors for tensor product vertex operator algebras associated to permutation groups, Commun. in Math. Phys., vol. 227 (2002), 349-384.

K. Barron, Y.-Z. Huang, J. Lepowsky, Factorization of formal exponentials and uniformization, J. of Algebra, vol. 228 (2000), 551-579.

K. Barron, N=1 Neveu-Schwarz vertex operator superalgebras over Grassmann algebras and with odd formal variables, in ``Representations and Quantizations: Proceedings of the International  Conference on Representation Theory, 1998", ed. by J. Wang and Z. Lin, China Higher Education Press & Springer-Verlag, Beijing, 2000, 9-36.

Graduate Ph.D. and Undergraduate Honors Students:

Keith Hubbard, Ph.D. Student.
   -Ph.D. Spring 2005.
   -Dissertation: The notion of vertex operator coalgebra: A construction and geometric interpretation.
   -Papers: The notion of vertex operator coalgebra and a geometric construction, Commun. in Alg., vol. 34, (2006), no. 5, 1541-1589.
            Constructions of vertex operator coalgebras via vertex operator algebras, J. of Algebra, vol. 294, (2005), 278-293.

Andrew Lynch, Senior Honors Student.
    -Honors B.A. Spring 2003.
    -Senior Honors Thesis: Aspects of the classification of finite simple groups.
 

Conferences:

AMS 2005 Fall Eastern Sectional Meeting #1009, October 8-9, Annandale-on-Hudson, NY, Special Session on Theory of Infinite-Dimensional Lie Algebras, Vertex Operator Algebras, and Related Topics

Groups and Algebras in M-Theory, Rutgers University, NJ, 31 May - 4 June, 2005.

Lie Algebras, Vertex Operator Algebras, and Applications: A Conference in Honor of Robert L.Wilson and James Lepowsky on Their 60th Birthdays. NC State University,  Raleigh, NC, 17-21 May 2005.

AMS 2003 Spring Central Sectional Meeting #985, April 4-6, Bloomington, IN, Special Session on Representations of Infinite Dimensional Lie Algebras and Mathematical Physics

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