Knots, Sheaves, Lagrangians

The broad of this study group is to understand Webster knot homology theories from three perspectives: diagrammatic, algebraic geometry, symplectic geometry. We will start with the diagrammatic perspective, understanding the construction of Khovanov homolog, Webster's discovery that it can be recovered from bimodules over KLRW algebras, and the following generalization to all Dynkin types. We will continue with the algebraic geometry perspective, seeing how KLRW algebras have Morita-equivalent models as coordinate rings of additive Coulomb branches X, with bimodules realized by correspondences between such branches. In the final symplectic geometry perspective, will will discuss Aganagic-Gaiotto-Witten's proposed LG model (Y,W) for X, where Y is the multiplicative partner of X, and review the evidence that homological mirror symmetry DCoh(X)=DFuk(Y,W) holds.

Time: Fridays 10:45-12:15
Location: Columbia University, Mathematics Hall, Room TBD

Date Speaker Title Reference
Feb 13 Ross Akhmechet Khovanov homology and Bar-Natan refinement [8,3]
Feb 20 Fan Zhou Webster homology [10]
Feb 27 Felix Roz Khovanov homology is a particular Webster homology [9]
Mar 06 Tommaso Botta Additive/multiplicative Coulomb branches X/Y [4]
Mar 13 Peter Moody Coulomb branches of quiver type [5]
Mar 27 Ivan Danilenko How to recover Webster homology from DCoh(X) [11]
Apr 03 Elise LePage Aganagic-Gaiotto-Witten LG potential W:Y->C [1, 6]
Apr 10 Johan Asplund Fukaya category Fuk(Y,W) [7]
Apr 17 Filip Zivanovic DCoh(X) embeds into DFuk(Y,W) [2]

References

[1] (2022) Aganagic, Homological knot invariants from mirror symmetry. arXiv:2207.14104
[2] (2024) Aganagic, Danilenko, Li, Shende, Zhou, Quiver Hecke algebras from Floer homology in Coulomb branches. arXiv:2406.04258
[3] (2004) Bar-Natan, Khovanov's homology for tangles and cobordisms. arXiv:0410495
[4] (2016) Braveman, Finkelberg, Nakajima, Towards a mathematical definition of Coulomb branches of gauge theories. arXiv:1601.03586
[5] (2016) Braveman, Finkelberg, Nakajima, Coulomb branches of quiver gauge theories and slices in the affine Grassmannian. arXiv:1604.03625
[6] (2011) Gaiotto, Witten, Knot invariants from four-dimensional gauge theory. arXiv:1106.4789
[7] (2017) Ganatra, Pardon, Shende, Covariantly functorial wrapped Floer theory on Liouville sectors. arXiv:1706.03152
[8] (1999) Khovanov, A categorification of the Jones polynomial. arXiv:9908171
[9] (2013) Webster, Tensor product algebras, Grassmannians and Khovanov homology. arXiv:1312.7357
[10] (2013) Webster, Knot invariants and higher representation theory. arXiv:1309.3796
[11] (2019–2022) Webster, Coherent sheaves and quantum Coulomb branches. arXiv:1905.04623, arXiv:2211.02099