The PDE seminars for the Autumn of 2021 will be held every Tuesday from 10:15–12:00

In a famous paper, Geir Ellingsrud and Stein Arild Strømme use the Atiyah-Bott localization theorem in equivariant cohomology to compute the number of complex twisted cubics on a complete intersection. Motivated by results from A1-homotopy theory there is a new way of doing such enumerative counts which works over an arbitrary base field, not only the complex numbers. Recently, Marc Levine proved a version of Atiyah-Bott localization for this new way of counting.

In the talk I will recall the classical Atiyah-Bott localization theorem and explain how one can use it in enumerative geometry. Furthermore, I will explain how this new way of counting works and present some results about twisted cubics on complete intersections counted this way. This is based on joint work with Marc Levine.

Marginal maximum likelihood estimation of longitudinal latent variable models for ordinal observed variables is challenging due to the high latent dimensionality required to accurately model residual dependencies for repeated measurements. We use second-order Laplace approximations to the high-dimensional integrals in the marginal likelihood function for longitudinal item response theory models and implement an efficient estimation method based on the approximations. The method is illustrated with items from the Montreal Cognitive Assessment, administered at four time points in a Hong Kong study of aging and well-being. We discuss the limitations of the proposed estimation method and outline a potential extension to the approach that uses a dimension-reduction technique.

Stable polynomials are a multivariate generalization of real-rooted univariate polynomials. This notion of stability for hypersurfaces can be extended to lower-dimensional varieties, giving rise to positively hyperbolic varieties. I will present results showing that tropicalizations of positively hyperbolic varieties are very special polyhedral complexes with a rich combinatorial structure. This, in particular, generalizes a result of P. Brändén showing that the support of a stable polynomial must be an M-convex set.

In this talk I will discuss the variational form of Bayes theorem by Zellner (1988). This result is the rationale behind the variational (approximate) inference scheme, although it is not always that clear in modern presentations. I will discuss two applications of this results. First, I will show how to do a low-rank mean correction within the INLA framework (with amazing results), which is essential for the next generation of the R-INLA software currently in development. In the second one, I will introduce the
*Bayesian learning rule*, which unify many machine-learning algorithms from fields such as optimization, deep learning, and graphical models. This includes classical algorithms such as ridge regression, Newton's method, and Kalman filter, as well as modern deep-learning algorithms such as stochastic-gradient descent, RMSprop, and Dropout.

The first part of the talk is based on our recent research at KAUST, while the second part is based upon \texttt{arxiv.org/abs/2107.04562} with Dr. Mohammad Emtiyaz Khan, RIKEN Center for AI Project, Tokyo.