## The Hilbert scheme of points of affine 3-space

We describe the scheme structure of the Hilbert scheme of points of affine 3-space,

in terms of representations of the Jacobi algebra of a quiver with potential. This exhibits the Hilbert scheme of points as the critical locus of a regular function on a smooth variety.

We discuss the torus action on the Hilbert scheme and its Euler characteristic.

## Representation theory learning seminar

## Quadratic Relations in Feynman Categories

I will start with a brief introduction to Feynman categories, and then explore the consequences of

quadratic relations for the morphisms in Feynman categories.

In the first part of the talk, I show how this leads to cubical complexes and prove that in this

way both the complex for moduli spaces of curves and the cubical complexes for Cutkosky rules

and calculations for Outer space arise naturally from push-forwards.

In the second part, I study generalizations of the quadradtic relations, which naturally lead

## New guests at the MPIM

## Doubling Properties of Solutions to Elliptic PDE

## What is the category of representations of classical Lie algebras of "infinite rank"?

The classical Lie algebras sl, so, sp "of infinite rank" in the naive sense have uncountably many nonconjugate Borel subalgebras. Therefore the definition of categories of their representations involves choices. A natural choice is a Dynkin Borel subalgebra. In this talk, I will advocate for another choice, namely that of a perfect Borel subalgebra. In addition, I will impose the condition of large local annihilator on the objects of the category. The resulting category is a nice highest-weight category with standard objects, but without costandard objects.

## Rational homology cobordisms of plumbed manifolds and arborescent link concordance

We investigate rational homology cobordisms of 3-manifolds with non-zero first Betti number.

This is motivated by the natural generalization of the slice-ribbon conjecture to multicomponent links.

We introduce a systematic way of constructing rational homology cobordisms between plumbed 3-manifolds and concordances between arborescent links.

We then describe a sliceness obstruction based on Donaldson's diagonalization theorem that leads to a proof of the slice-ribbon conjecture for 2-component Montesinos' links up to mutation.

## On the periodicity of geodesic continued fractions

In this talk, we present some generalizations of Lagrange's periodicity theorem in the classical theory of continued fractions. The main idea is to use a geometric interpretation of the classical theory in terms of closed geodesics on the modular curve. As a result, for an extension F/F' of number fields with rank one relative unit group, we construct a geodesic multi-dimensional continued fraction algorithm to``expand'' a basis of F over the rationals, and prove its periodicity. Furthermore, we show that the periods describe the relative unit group.

## Hecke's integral formula and Kronecker's limit formula for an arbitrary extension of number fields

The classical Hecke's integral formula expresses the partial zeta function of real quadratic fields as an integral of the real analytic Eisenstein series along a certain closed geodesic on the modular curve. In this talk, we present a generalization of this formula in the case of an arbitrary extension E/F of number fields. As an application, we present the residue formula and Kronecker's limit formula for an extension E/F of number fields, which gives an integral expression of the residue and the constant term at s=1 of the``relative'' partial zeta function associated to E/F.

## On Sequences of Integers of Quadratic Fields and Relations with Artin’s Primitive Root Conjecture

I will consider the integers $\alpha$ of the quadratic field $ \mathbb{Q} (\sqrt[]{d})$ with $d$ is a square-free integer. Using the embedding into $ \text{GL}(2,\mathbb{R})$ we obtain bounds for the smallest positive integer $\nu$ such that $\alpha^\nu\equiv 1\bmod p.$ More generally, if $\mathcal{O}_{f}$ is a number ring of conductor $f$, we study the first integer $n=n(f)$ such that $\alpha^n\in\mathcal{O}_{f}$. We obtain bounds for $n(f)$ and for $n(fp^{k})$.

## tba

## A ribbon obstruction and derivatives of knots

We define an obstruction for a knot to be Z[Z]-homology ribbon, and use this to provide restrictions on the integers that can occur as the triple linking numbers of derivative links of knots that are either homotopy ribbon or doubly slice. Our main application finds new non-doubly slice knots.

## Anticyclotomic p-adic spinor L-functions for PGSp(4)

The Boecherer conjecture is a generalization of the Waldspurger formula and relates squares of Bessel periods

of genus two Siegel cusp forms to the central L-values.

This conjecture was currently proved by Furusawa and Morimoto for the special Bessel period, and the general

case is a work-in-progress. In this talk I will construct a square root of an anticyclotomic p-adic L-function with

explicit interpolation formulas for Siegel cusp forms of genus 2 and scalar weight greater than 1 with respect to

## Theta integrals and generalized error functions

Recently Alexandrov, Banerjee, Manschot and Pioline constructed generalizations of Zwegers theta functions for lattices of signature $(n-2,2)$. They also suggested a generalization to the case of arbitrary signature $(n-q,q)$, and this case was subsequently proved by Nazaroglu. Their functions, which depend on certain collections $\mathcal{C}$ of negative vectors, are obtained by 'completing' a non-modular holomorphic generating series

by means of a non-holomorphic theta type series involving generalized error functions.

## Tamagawa numbers in the function field case

The Tamagawa number of a semi-simple simply connected group over a function field is equal to 1, and the space of connections on a principal bundle is contractible. In the 1980s, Professor Harder pointed out to me the intriguing but mysterious relationship between these two statements. A recent proof of Weil's Tamagawa number conjecture in the function field case by Gaitsgory and Lurie using derived geometry sheds new light on this old conundrum. I will report on this and related work.

## Algebraic groups with good reduction

Let $G$ be an absolutely almost simple algebraic group over a field $K$. Assume that $K$ is equipped with a "natural" set $V$ of discrete valuations. We are interested in the $K$-forms of $G$ that have good reduction at all $v$ in $V$. In the case $K$ is the fraction field of a Dedekind domain, a similar question was considered by G.~Harder; the case where $K = \mathbb{Q}$ and $V$ is the set of all $p$-adic places was analyzed in detail by B.H.~Gross and B.~Conrad.

## On the automorphic spectrum of non-quasi-split groups

Langlands' conjectures provide a description of the discrete automorphic representations of connected reductive groups defined over global fields, as well as of the irreducible admissible representations of such groups defined over local fields. When the group in question is quasi-split, a precise form of these conjectures has been known for a long time and important special cases have recently been proved.

## Some conjectures on Weil cohomology theories over $\overline{F}_p$

tba.

## Towards Harder-Narasimhan filtrations for Fukaya-Seidel type categories with coefficients

We report on progress in our joint work with F. Haiden, L. Katzarkov, and P. Pandit on a program of extending the Bridgeland-Smith construction of stability conditions to the case of $SL(3)$ spectral curves. We consider Fukaya-Seidel categories of graph Lagrangians with coefficients in a constant category, in our case of type $A2-CY2$, on a contractible flat Riemann surface. As in the recent theory of "perverse schobers", objects involve putting triangles at the threefold vertices of the underlying graph.

## Eisenstein cohomology and automorphic L-functions

Günter Harder has pioneered the theory of Eisenstein cohomology over the last few decades. This involves my own work with Harder on rank one Eisenstein cohomology for $GL(N)$ over a totally real field and the arithmetic of Rankin-Selberg $L$-functions for $GL(n) \times GL(m)$. Since then I have been involved in several projects which have the common theme of Eisenstein cohomology of some ambient reductive group and the special values of certain automorphic $L$-functions.

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