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1. MONOPOLES AND LANDAU-GINZBURG MODELS II: FLOER HOMOLOGY

   Wang, Donghao (May 2020, ArXivorg)

   This is the second paper of this series. We define the monopole Floer homol- ogy for 3-manifolds with torus boundary, extending the work of Kronheimer-Mrowka for closed 3-manifolds. The Euler characteristic of this Floer homology recovers the Milnor torsion invariant of the 3-manifold by a theorem of Meng-Taubes. 

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2. Cohomological Invariants in Positive Characteristic

   https://doi.org/10.1093/imrn/rnaa321  

   Totaro, Burt (January 2021, International Mathematics Research Notices)

   Abstract We determine the mod $$p$$ cohomological invariants for several affine group schemes $$G$$ in characteristic $$p$$. These are invariants of $$G$$-torsors with values in étale motivic cohomology, or equivalently in Kato’s version of Galois cohomology based on differential forms. In particular, we find the mod 2 cohomological invariants for the symmetric groups and the orthogonal groups in characteristic 2, which Serre computed in characteristic not 2. We also determine all operations on the mod $$p$$ étale motivic cohomology of fields, extending Vial’s computation of the operations on the mod $$p$$ Milnor K-theory of fields. 

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3. Holomorphic Floer theory and Donaldson-Thomas invariants

   Bousseau, Pierrick (April 2024, Proceedings of symposia in pure mathematics)

   We present several expected properties of the holomorphic Floer theory of a holomorphic symplectic manifold. In particular, we propose a conjecture relating holomorphic Floer theory of Hitchin integrable systems and Donaldson-Thomas invariants of non-compact Calabi-Yau 3-folds. More generally, we conjecture that the BPS spectrum of a 4-dimensional N = 2 quantum field theory can be recovered from the holomorphic Floer theory of the corresponding Seiberg-Witten integrable system. 

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4. Title: String-Math 2022 Article: Holomorphic Floer theory and Donaldson-Thomas invariants

   Bousseau, Pierrick (April 2024, AMS)

   Donag, Ron; Langer, Adrian; Sułkowski, Piotr; Wendland, Katrin (Ed.)

   We present several expected properties of the holomorphic Floer theory of a holomorphic symplectic manifold. In particular, we propose a conjecture relating holomorphic Floer theory of Hitchin integrable systems and Donaldson-Thomas invariants of non-compact Calabi-Yau 3-folds. More generally, we conjecture that the BPS spectrum of a 4-dimensional N = 2 quantum field theory can be recovered from the holomorphic Floer theory of the corresponding Seiberg-Witten integrable system. 

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5. Equivariant knots and knot Floer homology

   https://doi.org/10.1112/topo.12312  

   Dai, Irving; Mallick, Abhishek; Stoffregen, Matthew (September 2023, Journal of Topology)

   Abstract We define several equivariant concordance invariants using knot Floer homology. We show that our invariants provide a lower bound for the equivariant slice genus and use this to give a family of strongly invertible slice knots whose equivariant slice genus grows arbitrarily large, answering a question of Boyle and Issa. We also apply our formalism to several seemingly nonequivariant questions. In particular, we show that knot Floer homology can be used to detect exotic pairs of slice disks, recovering an example due to Hayden, and extend a result due to Miller and Powell regarding stabilization distance. Our formalism suggests a possible route toward establishing the noncommutativity of the equivariant concordance group. 

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