We discuss on fractional weighted Sobolev spaces with degenerate weights and related weighted nonlocal integrodifferential equations. We provide embeddings and Poincare inequalities for these spaces and show robust convergence when the parameter of fractional differentiability goes to $1$. Moreover, we prove local H\"older continuity and Harnack inequalities for solutions to the corresponding nonlocal equations. The regularity results naturally extend those for degenerate linear elliptic equations presented in [Comm. Partial Differential Equations 7 (1982); no. 1; 77?116] by Fabes, Kenig, and Serapioni to the nonlocal setting. This is a joint work with Linus Behn, Lars Diening and Julian Rolfes from Bielefeld.

Ist lecture: Understanding material microstructure Abstract Under temperature changes or loading, alloys can form beautiful patterns of microstructure that largely determine their macroscopic behaviour. These patterns result from phase transformations involving a change of shape of the underlying crystal lattice, together with the requirement that such changes in different parts of the crystal fit together geometrically. Similar considerations apply to plastic slip. The lecture will explain both successes in explaining such microstructure mathematically, and how resolving deep open questions of the calculus of variations could lead to a better understanding. 2nd lecture: Monodromy and nondegeneracy conditions in viscoelasticity Abstract For certain models of one-dimensional viscoelasticity, there are infinitely many equilibria representing phase mixtures. In order to prove convergence as time tends to infinity of solutions to a single equilibrium, it is necessary to impose a nondegeneracy condition on the constitutive equation for the stress, which has been shown in interesting recent work of Park and Pego to be necessary. The talk will explain this, and show how in some cases the nondegeneracy condition can be proved using the monodromy group of a holomorphic function. This is joint work with Inna Capdeboscq and Yasemin Şengül.

Semi-supervised domain adaptation (SSDA) is a statistical learning problem that involves learning from a small portion of labeled target data and a large portion of unlabeled target data, together with many labeled source data, to achieve strong predictive performance on the target domain. Since the source and target domains exhibit distribution shifts, the effectiveness of SSDA methods relies on assumptions that relate the source and target distributions. In this talk, we develop a theoretical framework based on structural causal models to analyze and compare the performance of SSDA methods. We introduce fine-tuning algorithms under various assumptions about the relationship between source and target distributions and show how these algorithms enable models trained on source and unlabeled target data to perform well on the target domain with low target sample complexity. When such relationships are unknown, as is often the case in practice, we propose the Multi-Start Fine-Tuning (MSFT) algorithm, which selects the best-performing model from fine-tuning with multiple initializations. Our analysis shows that MSFT achieves optimal target prediction performance with significantly fewer labeled target samples compared to target-only approaches, demonstrating its effectiveness in scenarios with limited target labels.

This presentation will consider the Sobolev regularity for solutions to space-time non-local equations. Spatial non-local operators in this presentation are the infinitesimal generator of Levy processes. The relation between the fundamental solution and the transition density of corresponding processes which generates the operator allows one to obtain the estimation of solutions in Lp-spaces. Several results will be introduced with assumptions. The main ingredients are a representation of solutions, heat kernel estimations, and some properties of (singular) integral operators.

This lecture explores the mathematical foundations underlying neural network approximation, focusing on the development of rigorous theories that explain how and why neural networks approximate functions effectively. We talk about key topics such as error estimation, convergence analysis, and the role of activation functions in enhancing network performance. Additionally, the lecture will demonstrate convergence analysis in the context of scientific machine learning, further bridging the gap between empirical success and theoretical understanding. Our goal is to provide deeper insights into the mechanisms driving neural network efficiency, reliability, and their applications in scientific computing.

The objective of the tutorial Computer-Assisted Proofs in Nonlinear Analysis is to introduce participants to fundamental concepts of a posteriori validation techniques. This mini-course will cover topics ranging from finite-dimensional problems, such as finding periodic orbits of maps, to infinite-dimensional problems, including solving the Cauchy problem, proving the existence of periodic orbits, and computing invariant manifolds of equilibria. Each session will last 2 hours: 1 hour of theory followed by 1 hour of hands-on practical applications. The practical exercises will focus on implementing on implementing computer-assisted proofs using the Julia programming language.

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Please bring your laptop to the tutorial and find the attached syllabus for more information. (to download the syllabus) https://saarc.kaist.ac.kr/boards/view/seminars/32