## Saman Habibi Esfahani : Non-linear Dirac operators and multi-valued harmonic forms

- Geometry and Topology ( 0 Views )This talk is based on joint work with Yang Li. I will discuss non-linear Dirac operators and related regularity questions, which arise in various problems in gauge theory, Floer theory, DT theory, and minimal submanifolds. These operators are used to define generalized Seiberg-Witten equations on 3- and 4-manifolds. Taubes proposed that counting harmonic spinors with respect to these operators on 3-manifolds could lead to new 3-manifold invariants, while Donaldson and Segal suggested counting spinors over special Lagrangians to define Calabi-Yau invariants. Similar counts appear in holomorphic Floer theory, where Doan and Rezchikov outlined a Fukaya 2-category for hyperkÃ¤hler manifolds based on such counts. The central question in all of these proposals is whether the space of such harmonic spinors is compact. We address this question in certain cases, proving and disproving several conjectures in the field and, in particular, answering a question raised by Taubes in 1999. The key observation is that multivalued harmonic forms, in the sense of Almgren and De Lellis-Spadaro's Q-valued functions, play a crucial role in the problem.

## Murilo Corato Zanarella : First explicit reciprocity law for unitary Friedbergâ??Jacquet periods

- Number Theory ( 0 Views )In the early 2000's, Bertolini and Darmon introduced a new technique to bound Selmer groups of elliptic curves via level raising congruences. This was the first example of what is now termed a "bipartite Euler system", and over the last decade we have seen many breakthroughs on constructing such systems for other Galois representations, including settings such as twisted and cubic triple product, symmetric cube, and Rankinâ??Selberg, with applications to the Blochâ??Kato conjecture and to Iwasawa theory. For this talk, I'll consider Galois representations attached to automorphic forms on a totally definite unitary group U(2r) over a CM field which are distinguished by the subgroup U(r) x U(r). I'll discuss a new "first explicit reciprocity law" in this setting and its application to the corresponding Blochâ??Kato conjecture, focusing on new obstacles which arise from the lack of local multiplicity one.

## Galen Reeves : Non-asymptotic bounds for approximate message passing via Gaussian coupling

- Probability ( 0 Views )Approximate message passing (AMP) has emerged as a powerful framework for the design and analysis of iterative algorithms for high dimensional inference problems involving regression and low-rank matrix factorization. The basic form of an AMP algorithm consists of a recursion defined on a random matrix. Under suitable conditions, the distribution of this recursion can be well approximated by a Gaussian process whose mean and covariance are defined via a recursive process called state evolution. This talk will briefly summarize some of the key ideas in AMP (no background is assumed). I will then describe a new approach for analyzing these algorithms that constructs an explicit coupling between the AMP iterates and a Gaussian process, Under mild regularity conditions, this coupling argument provides simple and interpretable guarantees on the non-asymptotic behavior of AMP algorithms. Related work can be found in the arXiv papers: https://arxiv.org/abs/2405.08225 and https://arxiv.org/abs/2306.15580