Analysis and PDE seminar: Spring 2025

Date and time: Thursday, February 6, at 12.15

Place: Hjørnet

Speaker: René Langøen, Phd. student @ Department of Mathematics, UiB

Title: A Fourier Basis for the Klein Bottle, with Examples

Abstract: TBA

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Date and time: Thursday, February 13, at 12.15

Place: Sigma

Speaker: Erlend Grong, Associate Professor @ Department of Mathematics, UiB

Title: Canonical sub-Riemannian connections 

Abstract: 

The topic is sub-Riemannian manifold; manifolds where there we only have inner products defined in some of the directions. 

The objective is to be able to determine if two such manifolds are the same, i.e. does or does there not exist an isometry between two different sub-Riemannian manifolds

Similar to the Levi-Civita connection, we discuss a canonical way of defining a Cartan connection on sub-Riemannian manifolds, which gives us a canonical choice of complement and affine connections. We present a new construction, which, unlike the earlier normalization condition of Morimoto, these new connections do not generate extra holonomy when considering general loops compared to horizontal loops. 

These results are part of a preliminary work with Jan Slovak. Results are based on a previous work on chain complexes with Francesca Tripaldi.

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Date and time: Thursday, February 27, at 12.15

Place: Sigma

Speaker: Irina Markina, Professor @ Department of Mathematics, UiB

Title: Automorphisms of H-type Lie algebras and how they can be useful

Abstract: H-type Lie algebras is a family of two-step (pseudo) metric Lie algebras with additional symmetries inherited from the Clifford algebras. We will describe the structure of the group of automorphisms of two-step Lie algebras and determine it precisely for H-type Lie algebras. Then, we outline two ongoing projects where the group of automorphisms and their subgroups of isometric automorphisms will be useful. 

This is a joint project with K.Furutani, Yu.Nikonorov, and I.Kath

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Date and time: Thursday, March 13, at 12.15

Place: Sigma

Speaker: Francesco Ballerin, PhD. student @ Department of Mathematics, UiB

Title: Geometric Deep Learning and Building SO(3)-Equivariant Neural Networks for Learning Vector Fields on Spheres

Abstract: 

Geometric deep learning is the field of deep learning which studies how to encode symmetries in a neural network in order to guarantee that certain type of transformations (symmetries) do not impact the result of a neural network. In particular we are interested in vector fields on the sphere.

Analyzing vector fields on the sphere, such as wind speed and direction on Earth, is a difficult task. Models should respect both the rotational symmetries of the sphere and the inherent symmetries of the vector fields. In this work, we introduce a deep learning architecture that respects both symmetry types using novel techniques based on group convolutions in the 3-dimensional rotation group. This architecture is suitable for scalar and vector fields on the sphere as they can be described as equivariant signals on the 3-dimensional rotation group. Experiments show that our architecture achieves lower prediction and reconstruction error when tested on rotated data compared to both standard CNNs and spherical CNNs.

In this talk we introduce the field of geometric deep learning, with applications to sets, graphs, images, and manifolds, and then focus on the specific problem of how to treat vector fields on spheres.

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Date and time: Thursday, April 3., at 12.15

Place: Aud. 4, RFB

Speaker: Anton Klimovsky, Senior Lecturer @ Würzburg University, Germany

Title: Mean-Field Spin Glasses: From Parisi PDE to Machine Learning Landscapes

Abstract: 

Mean-field spin glasses, often epitomized by the Sherrington-Kirkpatrick(SK) model, are paradigmatic disordered systems characterized by complexenergy landscapes and non-trivial phase transitions. This presentationelucidates the theoretical framework pioneered by Giorgio Parisi --recipient of the 2021 Nobel Prize in Physics for his revolutionaryreplica symmetry breaking ansatz. We examine the Parisi variationalformula for the limiting free energy and its profound analyticalimplications, focusing on the Hamilton-Jacobi-Bellman equation (the"Parisi PDE") and its stochastic control interpretations. Themathematical rigour underlying these developments is furthered by MichelTalagrand’s seminal contributions to probability theory and functionalanalysis, recognized by the 2024 Abel Prize.Furthermore, we explore the remarkable connections between spin glasstheory and contemporary machine learning. Drawing inspiration from JohnHopfield’s influential neural network models and the foundational workof Geoffrey Hinton on Boltzmann machines -- both of whom were honouredwith the 2024 Nobel Prize in Physics -- we discuss how spin glassconcepts inform energy-based models and the challenges of optimizingnon-convex, high-dimensional landscapes. This presentation underscoresthe interdisciplinary nature of mean-field spin glass theory and itsenduring relevance to rigorous mathematical analysis of complex systemsacross diverse scientific domains.

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