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# Brauer Lecture: Dr. Mina Aganagic

## April 11, 2018 @ 3:30 pm - 4:30 pm

# The Alfred Brauer Lectures

## April 10 – 12, 2018

## Dr. Mina Aganagic

### Professor of Mathematics and Physics,

### University of California, Berkeley

Dr. Mina Aganagic is Professor of Mathematics and Physics at the University of California, Berkeley. She has a bachelor’s degree and a doctorate from the California Institute of Technology, in 1995 and 1999 respectively. After a postdoctoral position at the Harvard University Physics Department and a faculty position at the University of Washington, she moved to UC Berkeley in 2004. Early in her career, she was named a Sloan Fellow and a DOE Outstanding Junior Investigator. In 2016 she received a prestigious Simons Foundation Investigator award.

**LECTURE 1: “Lesson on Integrability”**

### Tuesday, April 10, 2018 from 3:30 – 4:30 pm* in Phillips Hall, Room 215*

**LECTURE 2: “Lesson on Knot Categorification”**

### Wednesday, April 11, 2018 from 3:30 – 4:30 pm in Phillips Hall, Room 215**

**LECTURE 3: “Lesson on Geometric Langlands”**

### Thursday, April 12, 2018 from 3:00 – 4:00 pm in Phillips Hall, Room 215**

*There will be a reception in the Mathematics Faculty/Student Lounge on the third floor of Phillips Hall, Room 330, 4:45 – 6:00 pm, on Tuesday, April 10^{th}, 2018.

**Refreshments will be available in Phillips Hall, Room 330 from 3:00 – 3:30 pm on Wednesday, April 11^{th}, and 2:30 – 3:00 pm on Thursday, April 12^{th}.

**“Three Math Lessons From a Six Dimensional String Theory”**

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**Lecture 1: ***“Lesson on Integrability”‘*

Quantum Knizhnik-Zamolodchikov Equation is a difference generalization of the famous Knizhnik-Zamolodcikov equation. The equation itself, its solutions, and its monodromies all turn out to emerge from geometry of a certain class of holomorphic symplectic varieties. This connection between geometry and representation theory can be understood starting from a certain string theory in six dimensions. As an application, from the six dimensional perspective we will discover relations between several different approaches to integrable lattice models, including the one of Nekrasov and Shatshvili, and that of Costello with Witten and Yamazaki. (The lecture is based on joint works with Andrei Okounkov, and work to appear with Nikita Nekrasov and Samson Shatashvili).

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**Lecture 2: ***“Lesson on Knot Categorification”*

I will describe three paths to categorification of Reshetikhin-Turaev-Witten invariants of knots, and the relations between them. The first approach is based on derived categories of coherent sheaves on intersections of slices in affine Grassmannians. The second is based on Fukaya-Seidel categories of their mirrors. The third approach is based on counting solutions to five dimensional equations with gauge theory origin, due to Witten. All three approaches have a common starting point in the six dimensional string theory. (The lecture is based on joint works to appear, with Andrei Okounkov and with Dimitrii Galakhov.)

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**Lecture 3: ***“Lesson on Geometric Langlands”*

One manifestation of Langlands duality turns out to be an identification between the q-conformal blocks of the quantum affine algebra and the deformed W-algebra associated to two Langlands dual Lie algebras. The proof of the correspondence (in the simply laced cases) relies on recent results in quantum K-theory of the Nakajima quiver varieties. The “quantum q-Langlands correspondence” is best understood starting with string theory in six dimensions. The six dimensional perspective leads to many extensions of the correspondence, and also explains the relation to the work of Kapustin and Witten. (The lecture is based on joint work with Edward Frenkel and Andrei Okounkov.)

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