For certain PDEs, it is not always possible to find smooth solutions; hence traditionally, we tend to relax the requirement of smoothness to finding weak (distributional) solutions to those PDEs, using the theory of Sobolev spaces. In dealing with Sobolev spaces, especially for unbounded regions, we encounter functions that are locally integrable but are not globally integrable, but have globally finite Sobolev energy. Such functions are called Dirichlet-Sobolev functions. Constant functions are certainly of this kind, with zero energy. It is natural to ask whether such functions are always, after subtracting a suitable constant that may depend on the function, globally integrable. The focus of this talk is to present results related to this question in the context of complete metric measure spaces (including Riemannian manifolds and Carnot-Caratheodory spaces) equipped with a locally doubling measure supporting a local Poincare type inequality.

With the advent of Large Language Models (LLMs), AI has been marketed as an all-in-one assistant for productivity. This works well for many small-scale use cases, but for domains like science and mathematics, the complexity skyrockets. How far away are we from an assistant that can help find and summarize arXiv preprints? We’ll take a look at ways of reading the minds of LLMs and understanding how they “think” about highly knowledge-based data like math papers. How do they understand the structure? the contents? What are the limitations and what can we do the improve their understanding?

A direct analogue of Velu’s formulas is suggested for a genus two curve. Based on the structure of the differential field of Kleinian ℘-functions associated with the curve, formulas for cyclic n-isogenies are derived. Parameters of the image curve are computed explicitly.