## 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 hyperkähler 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.

## Mohammed Abouzaid : Bordism of derived orbifolds

- Geometry and Topology ( 0 Views )Among the first significant results of algebraic topology is the computation, by Thom, Milnor, Novikov, and Wall among others, of the bordism groups of stably complex and oriented manifolds. After reviewing these results, I will discuss the notion of derived orbifolds, and briefly indicate how the bordism groups of these objects appear as universal recipients of invariants arising in Gromov-Witten theory and symplectic topology. Finally, I will state what is known about them, as well as some conjectures about the structure of these groups.

## Luya Wang : Deformation inequivalent symplectic structures and Donaldsons four-six question

- Geometry and Topology ( 0 Views )Studying symplectic structures up to deformation equivalences is a fundamental question in symplectic geometry. Donaldson asked: given two homeomorphic closed symplectic four-manifolds, are they diffeomorphic if and only if their stabilized symplectic six-manifolds, obtained by taking products with CP^1 with the standard symplectic form, are deformation equivalent? I will discuss joint work with Amanda Hirschi on showing how deformation inequivalent symplectic forms remain deformation inequivalent when stabilized, under certain algebraic conditions. This gives the first counterexamples to one direction of Donaldson??s ??four-six? question and the related Stabilizing Conjecture by Ruan. In the other direction, I will also discuss more supporting evidence via Gromov-Witten invariants.