## Online: A geometric description of Reidemeister-Turaev Torsion

Zoom meeting ID: 919-9946-8404

Password: see email announcement or contact the seminar organisers:

Tobias Barthel (barthel.tobi[at]gmail.com)

David Gay (dgay[at]uga.edu)

Arunima Ray (aruray[at]mpim-bonn.mpg.de)

Reidemeister-Turaev torsion is an invariant of a 3-manifold with an Euler structure.

An Euler structure is a homotopy class of non-vanishing vector fields on the manifold.

In this talk we give a geometric description of Reidemeister-Turaev Torsion by using

the two point configuration space of the 3-manifold. In this description, an Euler structure

is understood as a framing of the diagonal of the configuration space. This description is deeply related to the Chern-Simons perturbation theory.

## Sites and sheaves

I will define sites, i.e. categories with a Grothendieck topology on them. I will give several examples of sites of topological spaces and of schemes. Sites are the right categorical context for sheaf theory, and I will explain how. Finally, I will sketch a proof of Grothendieck's result that representable functors are sheaves in the fpqc topology - and hence also in the fppf and étale topology. This is mostly based on Vistoli's notes, section 2.3.

https://bbb.mpim-bonn.mpg.de/b/rei-xh2-kg6

Organiser: Reinier Kramer

Please send an email to rkramer@mpim... for the password for the sessions.

## ONLINE: Zagier's polylogarithm conjecture and an explicit 4-ratio

Zoom Online Meeting ID: 919 6497 4060

For password see the email or contact Pieter Moree (moree@mpim...).

In his celebrated proof of Zagier's polylogarithm conjecture

for weight 3 Goncharov introduced a "triple ratio", a projective

invariant akin to the classical cross-ratio. He has also conjectured the

existence of "higher ratios" that should play an important role for

Zagier's conjecture in higher weights. Recently, Goncharov and Rudenko

proved the weight 4 case of Zagier's conjecture with a somewhat indirect

method where they avoided the need to define a corresponding "quadruple

ratio". We propose an explicit candidate for such a "quadruple ratio"

and as a by-product we get an explicit formula for the Borel regulator

of K7 in terms of the tetralogarithm function (joint work with H. Gangl

and D. Radchenko).

## ONLINE: When multiplicativity meets automaticity...

Automatic sequences - that is, sequences computable by finite automata,provide a basic model for computation. The main objective of the present talk is to show how a blend of ideas from number theory, combinatorics and ergodic theory can be used to characterize automatic sets with some

multiplicative properties.

Zoom Online Meeting ID: 919 6497 4060

For password see the email or contact Peter Moree (moree@mpim...).

## ONLINE: A glimpse on 3d modularity

https://bbb.mpim-bonn.mpg.de/b/gae-a7y-hhd

Since the 1980s, the study of invariants of 3-dimensional manifolds has benefited from the connections between topology, physics and number theory. Recently, a new 3-manifold invariant has been discovered: this is known as the homological block. In this talk, I will describe the appearance of quantum modular forms, false theta functions and I will explain how to predict certain topological data of 3-manifolds via the quantum modular properties of the associated

homological blocks. The talk will be based on the article 1809.10148 and work in progress with Cheng, Chun, Feigin, Gukov, and Harrison.

## Online: Recap of scheme theory

I will recall the definition of sheaves and schemes and many of their properties, such as e.g. properness, smoothness, &c. This is all material from Hartshorne, parts II and III, with less of a focus on sheaf cohomology (already treated in the previous reading group, on DT invariants), and making use of the category theory background.

https://bbb.mpim-bonn.mpg.de/b/rei-xh2-kg6

Organiser: Reinier Kramer

Please send an email to rkramer@mpim... for the password for the sessions.

## Online: Category background for stacks

I will recall often-used categorical constructions, such as the Yoneda lemma, categorical limits, and adjunctions. Most examples will be algebraic or topological in nature, with more geometric examples coming in the next session. I will also introduce group objects and discrete group objects in a category.

https://bbb.mpim-bonn.mpg.de/b/rei-xh2-kg6

Organiser: Reinier Kramer

Please send an email to rkramer@mpim... for the password for the sessions.

## Computability of the Minimal Genus on Second Homology

Surface representatives of second homology classes can be used to give geometric invariants for second homology classes, the most prominent examples are the genus and the Euler characteristic. In this talk I will introduce the minimal genus problem, explain why determining the minimal genus of a given homology class is in general undecidable, and how to compute it for a large class of "negatively-curved" spaces including 2-dimensional CAT(-1)-complexes. This will need a normal form result proven by me and Mark Pedron, that extends a theorem by Edmonds on maps between surfaces.

Zoom meeting ID: 919-9946-8404

Password: see email announcement or contact the seminar organisers:

Tobias Barthel (barthel.tobi[at]gmail.com)

David Gay (dgay[at]uga.edu)

Arunima Ray (aruray[at]mpim-bonn.mpg.de)

## ONLINE: A short journey through indefinite theta series

Online link will be send in e-mail announcement.

To start the summer season we give an informal introduction to the theory of indefinite theta series and their role in arithmetic and geometry. In particular, the talk will be colloquium style directed at the entire mathematical community of the MPI.

## ONLINE: Perfect points on abelian varieties

Let k be a field which is finitely generated over the algebraic closure of F_p, L be its perfection and let A be a k-abelian variety. The main goal of this talk is to provide some new result on the structure of the torsion free part of A(L). These results are motivated by their application to the "full" Mordell-Lang conjecture.The main tool is the study of various p-adic incarnation of certain 1-motives attached to L-rational points of A.

https://bbb.mpim-bonn.mpg.de/b/gae-a7y-hhd

## ONLINE: Special cycles on orthogonal Shimura varieties

Extending on the work of Kudla-Millson and Yuan-Zhang-Zhang, together with Yott we are constructing special cycles for a specific orthogonal Shimura variety. We further construct a generating series that has as coefficients the cohomology classes corresponding to the special cycle classes on the orthogonal Shimura variety and show the modularity of the generating series in the cohomology group over the complex numbers.

## Virtual: Floer and Khovanov homologies of band sums

Given a nontrivial band sum of two knots, we may add full twists to the band to obtain a family of knots indexed by the integers. In this talk, I'll show that the knots in this family have the same Heegaard and instanton knot Floer homology but distinct Khovanov homology, generalizing a result of M. Hedden and L. Watson. A key component of the argument is a proof that each of the three knot homologies detects the trivial band. The main application is a verification of the generalized cosmetic crossing conjecture for split links.

Zoom meeting ID: 919-9946-8404

Password: see email announcement or contact the seminar organisers:

Tobias Barthel (barthel.tobi[at]gmail.com)

David Gay (dgay[at]uga.edu)

Arunima Ray (aruray[at]mpim-bonn.mpg.de)

## WKB expansions, resurgence and BPS structures (after Iwaki-Nakanishi)

Link: https://bbb.mpim-bonn.mpg.de/b/gae-nhq-dzk

We introduce the WKB method, which was originally initiated as the method for obtaining approximate solutions of the Schrödinger equation in the semiclassical limit in quantum mechanics, and later evolved via the Borel resummation method into the so-called exact WKB analysis. We describe how the Voros symbols encode the monodromy data and how that evolves along mutations of the Stoke graphs. On the other hand, the moduli spaces of meromorphic quadratic differentials on Riemann surfaces gives the stability spaces on a triangulated category of quivers with potential. The associated BPS structures count finite-length trajectories. Our main goal is to show that the corresponding Riemann—Hilbert problem can be partially solved via exact WKB analysis. If time permits, we will briefly describe how the method of topological recursion reconstructs WKB solutions.

## ONLINE: Generalized blowup equations

Blowup equations as the functional equations for the Nekrasov partition function of 4-dimensional gauge theory was proposed by Nakajima-Yoshioka in 2003. Certain K-theoretic versions were soon later found by Göttsche, Nakajima and Yoshioka. I will talk about two further generalizations found in recent three years, an elliptic version from the viewpoint of gauge theory and a most general version from the viewpoint of topological string theory on local Calabi-Yau threefolds. These equations can be used to solve the elliptic genera of certain 6d gauge theories and the refined BPS invariants of local Calabi-Yau.

https://bbb.mpim-bonn.mpg.de/b/gae-a7y-hhd

## Perverse sheaves and the cohomology of regular Hessenberg varieties

Hessenberg varieties are a distinguished family of projective varieties associated to a semisimple complex algebraic group. We use the formalism of perverse sheaves to study their cohomology rings. We give a partial characterization, in terms of the Springer correspondence, of the irreducible representations which appear in the action of the Weyl group on the cohomology ring of a regular semisimple Hessenberg variety. We also prove a support theorem for the universal family of regular Hessenberg varieties, and we deduce that its fibers, though not necessarily smooth, always have the "Kahler package". This is joint work with Peter Crooks.

## Categorification of the Hecke algebra at roots of unity

Categorical representation theory is filled with graded additive categories (defined by generators and relations) whose Grothendieck groups are algebras over $\mathbb{Z}[q,q^{-1}]$. For example, Khovanov-Lauda-Rouquier (KLR) algebras categorify the quantum group, and the diagrammatic Hecke categories categorify Hecke algebras. Khovanov introduced Hopfological algebra in 2006 as a method to potentially categorify the specialization of these $\mathbb{Z}[q,q^{-1}]-algebras$ at $q = \zeta_n$ a root of unity. The schtick is this: one equips the category (e.g. the KLR algebra) with a derivation d of degree 2, which satisfies d^p = 0 after specialization to characteristic p, making this specialization into a p-dg algebra. The p-dg Grothendieck group of a p-dg algebra is automatically a module over $\mathbb{Z}[\zeta_{2p}]$... but it is NOT automatically the specialization of the ordinary Grothendieck group at a root of unity!

Upgrading the categorification to a p-dg algebra was done for quantum groups by Qi-Khovanov and Qi-Elias. Recently, Qi-Elias accomplished the task for the diagrammatic Hecke algebra in type A, and ruled out the possibility for most other types. Now the question is: what IS the p-dg Grothendieck group? Do you get the quantum group/hecke algebra at a root of unity, or not?

This is a really hard question, and currently the only techniques for establishing such a result involve explicit knowledge of all the important idempotents in the category. These techniques sufficed for quantum sl_n with $n \le 3$, but new techniques are required to make further progress.

After reviewing the theory of p-dg algebras and their Grothendieck groups, we will present some new techniques and conjectures, which we hope will blow your mind.

Everything is joint with You Qi.

## Conjectures on p-cells, tilting modules, and nilpotent orbits

For quantum groups at a root of unity, there is a web of theorems (due to Bezrukavnikov and Ostrik, and relying on work of Lusztig) connecting the following topics: (i) tilting modules; (ii) vector bundles on nilpotent orbits; and (iii) Kazhdan–Lusztig cells in the affine Weyl group. In this talk, I will review these results, and I will explain a (partly conjectural) analogous picture for reductive algebraic groups over fields of positive characteristic, inspired by a conjecture of Humphreys. This is joint work with W. Hardesty and S. Riche.

## K-Motives and Koszul Duality

Koszul duality, as conceived by Beilinson-Ginzburg-Soergel, describes a remarkable symmetry in the representation theory of Langlands dual reductive groups. Geometrically, Koszul duality can be stated as an equivalence of categories of mixed (motivic) sheaves on flag varieties. In this talk, I will argue that there should be an an 'ungraded' version of Koszul duality between monodromic constructible sheaves and equivariant K-motives on flag varieties. For this, I will explain what K-motives are and present preliminary results.

## A categorification of the Lusztig-Vogan module

Admissible representations of real reductive Lie groups are a key player in the world of unitary representation theory. The characters of irreducible admissible representations were described by Lusztig-Vogan in the 80’s in terms of a geometrically-defined module over the associated Hecke algebra. In this talk, I’ll describe a categorification of this module using Soergel bimodules, with a focus on examples. This is work in progress.

## An extension of Suzuki's functor to the critical level

Suzuki's functor relates the representation theory of the affine Lie algebra to the representation theory of the rational Cherednik algebra in type A. In this talk, we discuss an extension of this functor to the critical level, t=0 case. This case is special because the respective categories of representations have large centres. Our main result describes the relationship between these centres, and provides a partial geometric interpretation in terms of Calogero-Moser spaces and opers.

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