2019, Mar 6 Slides (Parts 1 & 2)
Video 1
Video 2
Simons Center for Geometry and Physics
Two-parts overview lecture in workshop: Automorphic Structures in String Theory
2018, Dec 18 StanfordAlgebraic Geometry and Number Theory Seminar
2018, Dec 7 RutgersLie Groups/Quantum Mathematics Seminar
2018, Nov 28 StanfordQuantum Groups Learning Seminar
2017, Dec 20 ChalmersAlgebraic Geometry and Number Theory Seminar
2017, Jul 18 KIASFour lectures during the program: Arithmetic Geometry and Quantum Field Theory

Automorphic String Amplitudes

Automorphic forms arise naturally when studying scattering amplitudes in toroidal compactifications of string theory. In this talk, I will summarize the conditions on four-graviton amplitudes from the literature required by U-duality, supersymmetry and string perturbation theory, which are satisfied by certain Eisenstein series on exceptional Lie groups. Physical information, such as instanton effects, are encoded in their Fourier coefficients on parabolic subgroups, which are, in general, difficult to compute. I will demonstrate a method for evaluating certain Fourier coefficients of interest in string theory. Based on arXiv:1511.04265, arXiv:1412.5625 and work in progress.

2017, Jan 30 Slides Oxford UniversityString theory seminar
2016, Dec 6 Slides Albert Einstein Institute, PotsdamSeminar
2016, Nov 15 Slides ULB, BrusselsHEP seminar
2016, Nov 3 Slides DAMTP, CambridgeString Theory Seminar
2016, Oct 11 Slides Paris (IPhT, Saclay)Séminaire de matrices, cordes et géométries aléatoires
2016, Oct 5 Slides Stanford (SITP)SITP seminar

Eisenstein series attached to small automorphic representations

In this talk we study certain Fourier coefficients of Eisenstein series attached to small automorphic representations motivated by open problems in string theory. Whittaker coefficients on the Borel subgroup can be computed using the Casselman-Shalika formula and Langlands' constant term formula, but no similar tools are available for Fourier coefficients on other parabolic subgroups. We therefore demonstrate a method to compute parabolic Fourier coefficients of Eisenstein series in terms of known Whittaker coefficients on the Borel and show how they simplify for small automorphic representations. This is joint work with Dmitry Gourevitch, Axel Kleinschmidt, Daniel Persson and Siddhartha Sahi.

2016, Oct 4 Slides Stanford (math department)Representation theory seminar
2016, Sep 30 Slides Rutgers Lie Group/Quantum Mathematics Seminar
2016, Sep 14 Slides Video Simons Center, Stony Brook
Program: Automorphic forms, mock modular forms and string theory

Small automorphic representations

In this talk we study maximal parabolic Fourier coefficients of automorphic forms in small automorphic representations motivated by string theory. We start with an introduction to how automorphic forms enter in string theory scattering amplitudes, discussing the moduli space of string theory compactified on tori, U-duality, and the differential equations for the automorphic forms that arise from supersymmetry. We see that the scattering amplitudes may be described using certain Eisenstein series in the minimal and next-to-minimal representations, and that physical information may be extracted from them by Fourier expansions. For higher rank groups we show how to compute maximal parabolic Fourier coefficients in terms of Whittaker vectors for the minimal and ntm representations for SL(3) and SL(4) using methods from Ginzburg and Miller-Sahi. We test similar relations for $E_6$, $E_7$ and $E_8$ by comparing the factorization of global (adelic) Whittaker vectors with known local spherical vectors. Based on arXiv:1412.5625 [math.NT] and arXiv:1511.04265 [math.NT]

2016, Jan 26 Slides Rutgers Number Theory Seminar

Instantons and Automorphic Representations

The low energy expansion of four-graviton scattering amplitudes in string theory on toroidal compactifications is given by automorphic forms on the moduli space attached to certain automorphic representations. These automorphic forms contain all perturbative corrections in the string coupling constant as well as non-perturbative, instanton corrections. Physical information, such as the instanton measure, can be extracted by studying their Fourier coefficients on certain parabolic subgroups of the underlying group G corresponding to taking different limits on the moduli space. Building on recent work by Miller-Sahi and Ginzburg, we present a method for calculating such Fourier coefficients by associating them with nilpotent orbits which are deeply connected to automorphic representations. We provide evidence that, for the minimal representation related to 1/2 BPS corrections, the above Fourier coefficients are determined by single, G-translated, maximally degenerate Whittaker vectors and propose a generalization for 1/4 BPS. Based on arXiv:1412.5625 [math.NT]

2015, June 22 Poster session Strings 2015, Bangalore
2015, June 11 Gong-show Advanced Strings School 2015, Bangalore
2015, Mar 17 Albert Einstein Institute, Potsdam Seminar
2015, Mar 10 Chalmers, Gothenburg Fundamental physics group seminar