Lomelí, Luis

Langlands Program and L-functions

Langlands functoriality conjectures predict generalized reciprocity laws between two areas of current research, namely, Number Theory and Representation Theory. Classically, Artin reciprocity and Tate’s thesis are covered by the abelian case of Class Field Theory. The Langlands Program addresses, among other aspects, the open problem posed by the non-abelian case. Artin L-functions on the Galois side and automorphic L-functions on the representation theoretic side carry important arithmetic information that is to be preserved by Langlands correspondences. In particular, we study automorphic L-functions via the Langlands-Shahidi method over function fields and have obtained new cases of Langlands functoriality and important applications.

Programa de Langlands y funciones L

Las conjeturas de Langlands consisten en leyes de reciprocidad generalizadas entre dos áreas de investigación actual, la teoría de números moderna y la teoría de representaciones. En la teoría de cuerpos de clases se estudia el caso abeliano, tesis de Tate y reciprocidad de Artin. El programa de Langlands aborda, entre otros aspectos, el problema abierto que nos plantea el caso no abeliano. Las funciones L asociadas a representaciones de Galois, por parte de la teoría de números, y las funciones L automorfas, por parte de la teoría de representaciones, contienen información aritmética importante la cual debe ser preservada por las conjeturas de Langlands. En particular, hacemos un estudio de las funciones L automorfas por medio del método de Langlands-Shahidi sobre un cuerpo de funciones y obtenemos casos y aplicaciones importantes en la funtorialidad de Langlands.

CV

Work experience:

October 2020 – Present: Associate Professor, Instituto de Matemáticas PUCV.

March 2016 – September 2020: Assistant Professor, Instituto de Matemáticas PUCV.

Prof. Lomelí has held visiting and postdoctoral positions at the following institutions:

IHÉS, MPIM, MSRI, University of Oklahoma, Purdue University and the University of Iowa.

Ph.D.:

2007 Purdue University.

Coordinator:

Ph.D. Research Line in Number Theory, Instituto de Matemáticas PUCV.

Ph.D. Program:

Director of the Ph.D. Program in Mathematics, Instituto de Matemáticas PUCV.

Master’s Program:

Director of the Master’s Program in Mathematics, Instituto de Matemáticas PUCV.

Grants:

2021 — 2024: “Number Theory and Representation Theory: interconnections stemming from the Langlands Program”. FONDECYT Standard Research Grant 1212013, Chile. Principal Investigator.

2020 — 2021: “Number Theory: Interconnections with Algebra, Combinatorics and Representation Theory”, MATH-AmSud 20-MATH-06. CNRS, CONICYT, COLCIENCIAS. One of four international coordinators (Argentina, Chile, Colombia, France).

2018: “Local Langlands correspondence for GL(n) and the Langlands-Shahidi method”. CONICYT MEC Grant No. 80170039, Chile. Sponsoring Scientist.

2017 — 2020: “Langlands Program and L-functions”. FONDECYT Standard Grant No. 1171583, Chile. Principal Investigator.

Academic year 2016: Project VRIEA/PUCV 039.367, Chile. Principal Investigator.

Publications

L. Gutiérrez, L. Lomelí and J. Pantoja, SL* over local and adèle rings: *-Euclideanity and Bruhat generators, Journal of Algebra 589 (2022), 1-28.

L. Lomelí, “Langlands Program and Ramanujan Conjecture: a survey”, Contemporary Mathematics A.M.S. 775 (2021), 161-203.

G. Henniart and L. Lomelí, “Asai cube L-functions and the local Langlands conjecture”, Journal of Number Theory 221 (2021), 247-269.

L. Lomelí, “Rationality and holomorphy of Langlands-Shahidi L-functions over function fields”, Mathematische Zeitschrift 291 (2019), 711-739.

W. T. Gan and L. Lomelí, Globalization of supercuspidal representations over function fields and applications, Journal of the European Mathematical Society, 20 (2018), 2813-2858.

Lomelí, L. A., On automorphic L-functions in positive characteristic, Annales de l’Institut Fourier 66 (2016), 1733-1771.

R. Ganapathy and L. Lomelí, On twisted exterior and symmetric square gamma factors, Annales de l’Institut Fourier 65 (2015), 1105-1132.

L. Lomelí, The LS method for the classical groups in positive characteristic and the Riemann Hypothesis, American Journal of Mathematics 137 (2015), 473-496.

G. Henniart and L. Lomelí, Uniqueness of Rankin-Selberg factors, Journal of Number Theory 133 (2013), 4024-4035.

G. Henniart and L. Lomelí, Characterization of gamma factors: the Asai case, International Mathematics Research Notices Vol. 2013, No. 17, pp. 4085-4099.

G. Henniart and L. Lomelí, Local-to-global extensions for GL(n) in non-zero characteristic: a characterization of symmetric and exterior square gamma factors, American Journal of Mathematics, 133, (2011), pp. 187-196.

L. Lomelí, Functoriality for the classical groups over function fields, International Mathematics Research Notices, Vol. 2009, No. 22, pp. 4271-4335.

Courses

Current courses for 2023:

Number Theory, First Semester 2023.

Courses taught at IMA PUCV

Measure Theory, Second Semester 2022.

Algebraic Number Theory, First Semester 2022.

Abstract Algebra, First Semester 2022.

Class Field Theory, Second Semester 2021.

Number Theory, Second Semester 2021.

Abstract Algebra, First Semester 2021.

Algebra II, Second Semester 2020.

Algebraic Number Theory, First Semester 2020.

Algebra II, Second Semester 2019.

Topics in Number Theory, First Semester 2019.

Number Theory, Second Semester 2018.

Trees and Buildings, First Semester 2018.

Introduction to Linear Algebra and Differencial Equations, First Semester 2017.

Analytic Number Theory, Second Semester 2016.

Measure Theory, First Semester 2016.

Students

Ph.D. Students:

Javier Navarro.

Emilio Améstica (2023).

Héctor del Castillo (2021).

Master’s Students:

Cristóbal Torres (2020).

Cristian Pérez (2019).

Nicolás Álamos (2018).

Conferences / Workshops

Banff CMO Workshop:

November 27 – December 2, 2022: “Langlands Program: Number Theory and Representation Theory”. Banff International Research Station, Casa México Oaxaca.

Online Conference:

Dec 15 – 18, 2020: “Number Theory and Representations in Valparaíso”. Instituto de Matemáticas PUCV.

Number Theory and Representation Theory Day:

Jan 15, 2020: “Jornada de Teoría de Números y Representaciones”, Universidad de Santiago de Chile.

Study Group / Mini Courses

Study Group (En Español)

Representations and Number Theory, First Semester 2023. Hybrid mode (PUCV-USACH).

Galois Cohomology, Second Semester 2020. Online.

Modular Forms, First Semester 2020. Online.

Mini Course (In English):

Jan 13 and 17, 2020: Anne-Marie Aubert (CNRS, Sorbonne Université), “Local Langlands correspondence for depth zero representations of the classical groups”, Instituto de Matemáticas PUCV, Valparaíso.

Study Group

Chevalley Groups, Second Semester 2019.

p-adic Fourier Analysis and Representation Theory of p-adic Groups, First Semester 2019.

Representation Theory, Second Semester 2018.

Mini Course:

July 3, 5, 10 and 12, 2018: Guy Henniart (Université d’Orsay), “Carayol representations and ramification”, Instituto de Matemáticas PUCV, Valparaíso.

Study Group

The Local Langlands Conjecture for GL(2), First Semester 2018.

The Local Langlands Conjecture for GL(2), Second Semester 2017.

Pi and Tau

“Día de Pi 2023”, Instituto de Matemáticas PUCV, Valparaíso.

“Día de Pi 2022”, Instituto de Matemáticas PUCV, Valparaíso.

“Día de Tau 2021”, Instituto de Matemáticas PUCV, Valparaíso.

Divulgación:

DIMAT