## Xinchen Miao : Local Integrability of Bessel functions on GL(n)

- Uploaded by schrett ( 1 Views )The study of Bessel functions plays an important role in number theory, automorphic forms and Langlands program. In my talk, we will focus on the Bessel functions over non-archimedean local fields. I will report on my recent work which proves that the Bessel function is locally integrable on GLn(Qp) for all n>=2, where Qp is a non-archimedean local field. The proof involves various tools in number theory and representation theory.

## Gene Kopp : The Shintani-Faddeev modular cocycle

- Uploaded by schrett ( 12 Views )We ask the question, "how does the infinite q-Pochhammer symbol transform under modular transformations?" and connect the answer to that question to the Stark conjectures. The infinite q-Pochhammer symbol transforms by a generalized factor of automorphy, or modular 1-cocycle, that is analytic on a cut complex plane. This "Shintani-Faddeev modular cocycle" is an SL_2(Z)-parametrized family of functions generalizing Shintani's double sine function and Faddeev's noncompact quantum dilogarithm. We relate real multiplication values of the Shintani-Faddeev modular cocycle to exponentials of certain derivative L-values, conjectured by Stark to be algebraic units generating abelian extensions of real quadratic fields.

## Stephan Huckemann : Statistical challenges in shape prediction of biomolecules

- Uploaded by schrett ( 17 Views )The three-dimensional higher-order structure of biomolecules
determines their functionality. While assessing primary structure is
fairly easily accessible, reconstruction of higher order structure is
costly. It often requires elaborate correction of atomic clashes,
frequently not fully successful. Using RNA data, we describe a purely
statistical method, learning error correction, drawing power from a
two-scale approach. Our microscopic scale describes single suites by
dihedral angles of individual atom bonds; here, addressing the
challenge of torus principal component analysis (PCA) leads to a
fundamentally new approach to PCA building on principal nested spheres
by Jung et al. (2012). Based on an observed relationship with a
mesoscopic scale, landmarks describing several suites, we use Fréchet
means for angular shape and size-and-shape, correcting
within-suite-backbone-to-backbone clashes. We validate this method by
comparison to reconstructions obtained from simulations approximating
biophysical chemistry and illustrate its power by the RNA example of
SARS-CoV-2.

This is joint work with Benjamin Eltzner, Kanti V. Mardia and Henrik
Wiechers.

Literature:

Eltzner, B., Huckemann, S. F., Mardia, K. V. (2018):
Torus principal component analysis with applications to RNA
structure. Ann. Appl. Statist. 12(2), 1332?1359.

Jung, S., Dryden, I. L., Marron, J. S. (2012):
Analysis of principal nested spheres. Biometrika, 99 (3), 551-568

Mardia, K. V., Wiechers, H., Eltzner, B., Huckemann, S. F. (2022).
Principal component analysis and clustering on manifolds. Journal of
Multivariate Analysis, 188, 104862,
https://www.sciencedirect.com/science/article/pii/S0047259X21001408

Wiechers, H., Eltzner, B., Mardia, K. V., Huckemann, S. F. (2021).
Learning torus PCA based classification for multiscale RNA backbone
structure correction with application to SARS-CoV-2. To appear in the
Journal of the Royal Statistical Society, Series C,
bioRxiv https://doi.org/10.1101/2021.08.06.455406

## Kim Klinger-Logan : A shifted convolution problem arising from physics

- Uploaded by schrett ( 14 Views )Physicists Green, Russo, and Vanhove have discovered solution to differential equations involving automorphic forms appear at the coefficients to the 4-graviton scattering amplitude in type IIB string theory. Specifically, for \Delta the Laplace-Beltrami operator and E_s(g) a Langlands Eisenstein series, solutions f(g) of (\Delta-\lambda) f(g) = E_a(g) E_b(g) for a and b half-integers on certain moduli spaces G(Z)\G(R)/K(R) of real Lie groups appear as coefficients to the analytic expansion of the scattering amplitude. We will briefly discuss different approaches to finding solutions to such equations and focus on a shifted convolution sum of divisor functions which appears as the Fourier modes associated to the homogeneous part of the solution. Initially, it was thought that, when summing over all Fourier modes, the homogeneous solution would vanish but recently we have found an exciting error term. This is joint work with Stephen D. Miller, Danylo Radchenko and Ksenia Fedosova.