Organized by Mina Aganagic, Sujay Nair, and Peng Zhou

Weekly on **Mondays 2:10 PM** (Pacific Time)

Meetings are in-person, at **402** Physics South

We have a lunch on the 4th floor of Physics South before the seminar.

If you want to be added to the seminar **mailing list**, use this link https://groups.google.com/g/berkeley-string-math

For those joining us remotely, we have a Zoom link: https://berkeley.zoom.us/j/99373923587?pwd=SUhXamdtbHhJMUJERlJ4NVJHL1Jtdz09

Jan 29 | Spencer Tamagni | Quasimaps with monopoles and nonabelian shift operators | |

Feb 5 | Yegor Zenkevich | Quantum toroidal algebras and spiralling branes | |

Feb 12 | Andrey Smirnov | Frobenius structures for quantum differential equations and mirror symmetry. | |

Feb 19 | No Seminar |
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Feb 26 | Roger Casals | Spectral Networks and Morse flow trees | |

Mar 4 | Luca Iliesiu | TBA | |

Mar 13* |
Andrei Okounkov | TBA | |

Mar 18 | Dylan Butson | TBA | |

Mar 25 | No Seminar |
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Apr 1 | Ahsan Khan | TBA | |

Apr 8 | Vasily Krylov | TBA | |

Apr 15 | David Rose | TBA | |

Apr 22 | Andy Neitzke | TBA | |

Apr 29 | |||

May 6 | Yehao Zhou | TBA |

*Special Day/Time/Location.

This is a research seminar, intended for mathematicians and physicists. For the speaker to successfully reach the audience in both fields, it is important to explain, as clearly as possible: the motivations for the work, questions addressed, key ideas. The audience may fail to appreciate the glory of the result, otherwise.

Fall 2023 Spring 2023 Fall 2022 Spring 2022 Fall 2021, Spring 2021, Fall 2020, Summer 2020, Spring 2020, Fall 2019, Spring 2019, Fall 2018, Fall 2017, Spring 2017, Fall 2016

We revisit supersymmetric localization with monopole operators in 3D N=4 gauge theories, arriving at a new point of view on their quantized Coulomb branch algebras. Our construction agrees with and recovers the existing mathematical definition. We apply the machinery to define a nonabelian analog of shift operators in enumerative theory of quasimaps, and illustrate the construction in detail for curve counts in GL(n)/B.

The talk will consist of two parts. In the first, longer, part I will review the basics of quantum toroidal algebras and their representation theory focusing on the simplest example of type gl(1). I will also discuss the identification between the representations of the algebra and branes of Type IIB string theory, how brane interactions manifest themselves in this setup and lead to integrability.

In the second part, if time permits, I will introduce new brane configurations inspired by algebraic considerations which I call spiralling branes. I will compute the corresponding partition functions and show that they coincide with the K-theoretic vertex function of Nekrasov and Okounkov and the magnificent four partition function recently introduced by Nekrasov, elucidating the algebraic meaning of both of them.

There exists a well-known similarity between the Kloosterman sum in number theory and the Bessel differential equation. This connection was explained by B. Dwork in 70s by discovering the Frobenius structures in the p-adic theory of the Bessel differential equation. In my talk I will speculate that this connection extends to the equivariant quantum differential equations for a wide class of varieties, which includes the Nakajima quiver varieties. As the main result, I will give an explicit conjectural description of the Frobenius structure for the equivariant quantum connection. The traces of the Frobenius structures are natural finite field analogs of the mirror integral representations of the J-function in quantum cohomology. As an example, I will discuss how, in the case of Grassmannians Gr(k,n), these considerations bring us to the B-model side of Gr(k,n) discovered earlier by Marsh and Rietsch.

I will explain how to associate a spectral network to a Demazure weave via Floer theory. Specifically, the talk will present how spectral networks, as introduced by Gaiotto-Moore-Neitzke, arise when computing J-holomorphic strips associated to augmentations of Lagrangian fillings of Legendrian links. In particular, this provides a Floer-theoretical description of Stokes lines in higher rank and with arbitrary irregular Stokes data, not necessarily associated to an algebraic braid. The talk will contain motivation and examples, including the Berk-Nevins-Roberts collision lines in higher rank WKB problems from the symplectic viewpoint as well as connections to cluster algebras and wall-crossing.