Katrina D. Barron  Associate Professor Office: 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

Women in Mathematical Physics at AWM at JMM, 2026.

Women in Mathematical Physics III, tentatively scheduled for 2025.

Vertex Algebra, Tensor Category and Related Topics, Shanghai Jiao Tung University, Shanghai, China, 21-25 June, 2024.

International Conference on Lie Algebra and Number Theory, National Institute of Technology, Calicut, India, 10-14 June 2024.

QuaSy-Con II (Conference on Quantum Symmetries in the Midwest), University of Notre Dame, 24-26 May 2024.

Conferences in the last 10 years:

Women in Mathematical Physics II, Banff, Canada, 18-23 August 2023.

Women in Algebra and Combinatorics: Northeast Conference Celebrating the Association for Women in Mathematics: 50 Years and Counting, University of Albany, New York, 28-30 April, 2023.

Representation Theory XVII, Dubrovnik, Croatia, 2-9 October 2022.

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

Virtual Workshop on Vertex Operator Algebras and Related Topics, hosted by CUNY, New York, 9-10 April 2021.

Women in Mathematical Physics, Banff, Canada, 20-25 September 2020. Scaled back to virtual two day workshop, 21-22 September 2020.

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

Canberra, Australia, July 2020. (Cancelled)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 field theories are an attempt at developing a physical theory that combines all fundamental interactions of particles, including gravity. 

The geometry of CFT 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, quantum groups, etc. My research includes many of these interactions with other branches of mathematics.

Selected Publications and Preprints:

D. Addabbo and K. Barron, On generators and relations for higher level Zhu algebras and applications, J. Alg., 623 (2023), 496-540.

D. Addabbo and K. Barron, The level two Zhu algebra for the Heisenberg vertex operator algebra, Commun. in Alg., (2023), 1-60.

K. Barron, K. Batistelli, F. Orosz Hunziker, V. Pedic Tomic, and G. Yamskulna, On rationality of C-graded vertex algebras and applications to Weyl vertex algebras under conformal flow, J. Math. Phys., 63 (2022), Paper No. 091706, 25 pp.

K. Barron, N. Vander Werf, and J. Yang, The level one Zhu algebra for the Virasoro vertex operator algebra, in: Vertex operator algebras, number theory and related topics, 17–43, Contemp. Math., 753, Amer. Math. Soc., (2020).

K. Barron, N. Vander Werf, Classification of screening systems for lattice vertex operator algebras, Lett. Math. Phys. 109 (2019), no. 7, 1573–1610.

K. Barron, N. Vander Werf, and J. Yang, The level one Zhu algebra for the Heisenberg vertex operator algebra, in: Affine, Vertex and W-algebras, Springer INdAM Series 37 (2019), 37-64.

K. Barron, N. Vander Werf, Nathan, and J. Yang, Higher level Zhu algebras and modules for vertex operator algebras, J. Pure Appl. Algebra 223 (2019), no. 8, 3295–3317.

K. Barron, E. Jurisich, A. Milas, and K. Misra, Lie Algebra, Vertex Operator Algebras, and Related Topics , Proceedings of the Conference Aug. 14-18, 2015, University of Notre Dame, Contemp. Math, Amer. Math. Soc. 695 (2017), 274 pp.

K. Barron, Twisted modules for tensor product vertex operator superalgebras and permutation automorphisms of odd order, in: ``Lie algebras, Lie superalgebras, vertex algebras and related topics," 45–79, Proc. Sympos. Pure Math., 92, Amer. Math. Soc., Providence, RI, 2016.

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 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.