## Curtis Porter : CRash CouRse in CR Geometry

- Uploaded by ydai39 ( 1693 Views )CR geometry studies real hypersurfaces in complex vector spaces and their generalizations, CR manifolds. In many cases of interest to complex analysis and PDE, CR manifolds can be considered ``curved versions" of homogeneous spaces according to Elie Cartan’s generalization of Klein’s Erlangen program. Which homogeneous space is the ``flat model" of a CR manifold depends on the Levi form, a tensor named after a mathematician who used it to characterize boundaries of pseudoconvex domains. As in the analytic setting, the Levi form plays a central role in the geometry of CR manifolds, which we explore in relation to their homogeneous models.

## Igor Zelenko (Texas A&M U) : Gromovs h-principle for corank two distribution of odd rank with maximal first Kronecker index

- Uploaded by schrett ( 4 Views )While establishing various versions of the h-principle for contact distributions (Eliashberg (1989) in dimension 3, Borman-Eliashberg-Murphy (2015) in arbitrary dimension, and even-contact contact (D. McDuff, 1987) distributions are among the most remarkable advances in differential topology in the last four decades, very little is known about analogous results for other classes of distributions, e.g. generic distributions of corank 2 or higher. The smallest dimensional nontrivial case of corank 2 distributions is Engel distributions, i.e. the maximally nonholonomic rank 2 distributions on $4$-manifolds. This case is highly nontrivial and was treated recently by Casals-Pérez-del Pino-Presas (2017) and Casals-Pérez-Presas (2017). In my talk, I will show how to use the method of convex integration in order to establish all versions of the h-principle for corank 2 distributions of arbitrary odd rank satisfying a natural generic assumption on the associated pencil of skew-symmetric forms. During the talk, I will try to give all the necessary background related to the method of convex integration in principle. This is the joint work with Milan Jovanovic, Javier Martinez-Aguinaga, and Alvaro del Pino.

## A. Raghuram : Special values of automorphic L-functions

- Uploaded by schrett ( 23 Views )In the first part of the talk I will describe a general context which, in some specific situations, permits us to give a cohomological interpretation to the Langlands-Shahidi theory of L-functions. In the second part of the talk, I will specialize to the context of the general linear group over a totally imaginary base field F, and discuss some recent results of mine on the special values of Rankin-Selberg L-functions for GL(n) x GL(m) over such an F. The talk is based on my preprint: https://arxiv.org/abs/2207.03393

## Thomas Hameister : The Hitchin Fibration for Quasisplit Symmetric Spaces

- Uploaded by schrett ( 26 Views )We will give an explicit construction of the regular quotient of Morrissey-Ngô in the case of a symmetric pair. In the case of a quasisplit form (i.e. the regular centralizer group scheme is abelian), we will give a Galois description of the regular centralizer group scheme using parabolic covers. We will then describe how the nonseparated structure of the regular quotient recovers the spectral description of Hitchin fibers given by Schapostnik for U(n,n) Higgs bundles. This work is joint with B. Morrissey.

## Gene Kopp : The Shintani-Faddeev modular cocycle

- Uploaded by schrett ( 26 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 ( 48 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