Katrina D. Barron Associate Professor Office: 276C Hurley. Phone: 574-631-3981 E-mail: kbarron@nd.edu Mailing Address: 255 Hurley Hall |
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i miei bambini |

Lie Algebras, Vertex Operator Algebras and Related Topics, in honor of James Lepowsky and Robert Wilson. University of Notre Dame, August 14–18, 2015.

Special Session on Representation Theory, Vertex Operator Algebras, and Related Topics , AMS Special Session, November 14–15, 2015, Rutgers University.

Vertex Algebras and Quantum Groups, February 7–12, 2016, Banff International Research Station.

Special Session on Vertex Algebra and Related Algebraic and Geometric Structures , AMS Special Session, March 19–20, 2016, SUNY-Stonybrook.

Special Session on Representation Theory and Algebraic Mathematical Physics, AMS Special Session, March 11–12, 2017, Charleston, SC.

Representation Theory XV, Inter University Center, Dubrovnik, Croatia, 18–25 June 2017.

Affine, Vertex and W-algebras, INdAM Roma, Italy, December 2017.

Vertex operator algebras, number theory, and related topics, A conference in honor of Geoffrey Mason, Sacramento, California, 11-13 June 2018.

Geometric and Categorial Aspects of CFTS, Casa Matematica, Oaxaca, Mexico, 23-28 September 2018.

AMS Fall Eastern Sectional Meeting, Special Session on Representations of Infinite Dimensional Lie Algebras and Applications, University of Delaware, 28-30 September 2018.

Beyond Rationality 2: Indecomposability and Post-Rational Conformal Field Theory, Woudschoten Zeist, Netherlands, 16-17 May 2019.

The Mathematical Foundations of Conformal Field Theory and Related Topics, International Conference in Honor of Yi-Zhi Huang, (co-organizer) Chern Institute, Nankai, China, 10-14 June 2019.

Representation Theory XVI, Dubrovnik, Croatia, 23-29 June 2019.

AMS Fall Central Sectional Meeting, Special Session on Supergeometry, Poisson Brackets, and Homotopy Structures, Madison, WI, 14-15 September 2019.

AMS Fall Eastern Sectional Meeting, Special Session on Representations of Lie Algebra, Vertex Operators, and Related Topics, Binghamton, NY, 12-13 October 2019.

QuaSy-Con: Workshop on Quantum Symmetries in the Midwest, UIUC, 9-10 November 2019.

AMS Joint Meetings, Special Session on Mathematical Aspects of Conformal Field Theory, Denver, 15-18 January 2020.

Rio, Brazil, June 2020. (Cancelled -- rescheduled for March 2022)

Canberra, Australia, July 2020. (Cancelled)

MSRI SWiM, Berkeley, July 2020. (Cancelled -- postponed to Summer 2021 virtual.)

WoMaP2020, Banff, Canada, 20-25 September 2020. Scaled back to virtual two day workshop, 21-22 September 2020.

Virtual Workshop on Vertex Operator Algebras and Related Topics, 9-10 April 2021.

**Upcoming Conferences: **

Subfactors, Vertex Operator Algebras, and Tensor Categories, BIRS-Hangzhou China and Virtual, 19-24 September 2021.

Workshop on Vertex Algebras, Instituto de Matematica Pura e Aplicada, Rio de Janeiro, Brazil, 21-25 March 2022.

Fields Insitute Workshop: The pursuit of symmetry: A conference in honour of the 80th birthday of Robert V. Moody, 25-29 April 2022.

**Math 20750 ** - Ordinary Differential Equations

**Spring 2022 Courses: **

**Math 40480 ** - Complex Analysis

**Math 70220 ** - Graduate level Lie Groups and Lie Algebras

**Research: **My research focuses on vertex operator
algebras and the algebraic and geometric foundations of
conformal field theory.

Conformal field theory (CFT), or more specifically, string theory, and related superconformal field theories (SCFTs) are an attempt at developing a physical theory that combines all fundamental interactions of particles, including gravity. The ``super" refers to certain assumed symmetries 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 the algebras governed by the worldsheet geometry of CFT, their representation theory, and category and number theoretic aspects. For genus-zero holomorphic CFT these algebras are called vertex operator algebras.

In addition to being a physical model for particle interactions, vertex operator algebras have important and deep links to the theory of finite simple groups, number theory, topology, etc. My research often also touches upon or has applications to such other branches of mathematics.

**Selected Publications (last updated 2014):**

K. Barron and N. Vander Werf, Permutation-twisted modules for even order cycles acting on tensor product vertex operator superalgebras, Internat. Jour. of Math, 25 (2014), no. 2, 1450018 (35 pages).

K. Barron, On the correspondence between mirror-twisted sectors for N=2 supersymmetric vertex operator superalgebras of the form (V tensor V) and N=1 Ramond sectors of V , in: ``Proceedings of the Xth International Workshop on Lie Theory and Its Applications in Physics", June 2013, Varna, Bulgaria. ed. V. Dobrev. Springer Proceedings in Mathematics & Statistics, Vol. 111, 2014.

K. Barron, Twisted modules for tensor product vertex operator superalgebras and permutation automorphisms of odd order, in: "Proceedings of the Southeastern Lie Theory Workshop Series 2012-2014", ed. by K. Misra, D. Nakano and B. Parshall, Conf. Proc. Series of the AMS, 2015.

K. Barron and N. Vander Werf, On permutation-twisted free fermions and two conjectures, in: ``Proceedings of the XXIst International Conference on Integrable Systems and Quantum Symmetries", June 2013, Prague, Czech Republic; ed. C. Burdik, O. Navratil and S. Posta; Jour. of Physics: Conference Series, vol. 474 (2013), 012009; 35 pages.

K. Barron, On twisted modules for N=2 supersymmetric vertex operator superalgebras, in: ``Proceedings of the IXth International Workshop on Lie Theory and Its Applications in Physics", June 2011, Varna, Bulgaria; ed. V. Dobrev, Springer 2013, 411--420. Longer preprint: Twisted modules for N=2 supersymmetric vertex operator superalgebras arising from finite automorphisms of the N=2 Neveu-Schwarz algebra.

K. Barron, On uniformization of N=2 superconformal and N=1 superanalytic DeWitt super-Riemann surfaces, submitted.

K. Barron, Automorphism groups of N=2 superconformal super-Riemann spheres, J. Pure Appl. Algebra, vol. 214 (2010), 1973-1987.

K. Barron, Axiomatic aspects of N=2 vertex superalgebras with odd formal variables, Commun. in Alg., vol. 38 (2010), 1199-1268.

K. Barron, Alternate notions of N=1 superconformality and deformations of N=1 vertex superalgebras, in ``Vertex Operator Algebras and Related Areas", Commun. in Math., Amer. Math. Soc., Vol. 497, (2009), 33-51.

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.

K. Barron, A supergeometric interpretation of vertex operator superalgebras, Internat. Math. Res. Notices 1996, no. 9, 409--430.

Last updated August 2021.