## Introduction to Homotopy Algebras

## IMPRS seminar on various topics: Homotopy algebras

## New guests at the MPIM

## An overview of the barycenter method in nonpositively curved manifolds II

## Podium discussion on topology

## Talk coaching for Postdocs

## Fulton MacPherson compactifications

## K-theory as a tool for classification of crystalline

Topological K-theory provides a great tool to study topological phases in systems of free fermions. We start by reviewing an argument by Freed & Moore on the appearance of the K-theory of tori in the context of quantum mechanics. After the mathematical essence has been extracted, the focus will be on computational methods using the Atiyah-Hirzebruch spectral sequences. We will focus on the example of a 2-dimensional system with a rotational symmetry of order two.

## Cohomology of arithmetic groups and Eisenstein series (I) - some geometric aspects at infinity

## A new approach to gaps between zeta zeros

In this talk we look at the value-distribution of Dirichlet polynomials

hypothesis on the distribution of zeroes of the Riemann zeta function.

## Derived Categories and the Genus of Curves

A 19th century problem in algebraic geometry is to understand the relation between the genus and the degree of a curve in complex projective space. This is easy in the case of the projective plane, but becomes quite involved already in the case of three dimensional projective space. I will talk about generalizing classical results on this problem by Gruson and Peskine to other threefolds. This includes principally polarized abelian threefolds of Picard rank one and some Fano threefolds.

## Representations up to homotopy and their cohomology

Representations up to homotopy of Lie groupoids were introduced by Arias Abad and Crainic offering a conceptually clear setting to define the adjoint representation of a Lie groupoid. A representation up to homotopy (ruth) is defined as a certain differential graded module, so one can talk about the cohomology of a ruth. A result of Gracia Saz and Mehta says that any ruth on a 2-term graded vector bundle corresponds a geometric object called a VB-groupoid.

## Geometric approach to quantum theory and inclusive scattering matrix

I give a formulation of quantum theory where the starting point is a convex set of states. I show that

the formulas for probabilities can be derived from first principles. One can define particles as

elementary excitations of ground state and quasiparticles as elementary excitations of translation invariant stationary state. The collisions of (quasi)particles in the geometric approach are described by inclusive scattering matrix. (Inclusive cross-sections can be expressed in terms of inclusive scattering matrix. Only

## Vertex Algebras and Factorization Algebras

There are several mathematical approaches to chiral two-dimensional conformal field theory. One of them is the notion of a factorization algebra on a Riemann surface, introduced by Costello and Gwilliam and inspired by Beilinson and Drinfeld's algebro-geometric notion of factorization algebra. Costello and Gwilliam construct vertex algebras from holomorphic factorization algebras on C. We construct holomorphic factorizations algebras from vertex algebras using geometric vertex algebras as an intermediate notion.

## Counting lattice points via Hirzebruch–Riemann–Roch

Given a lattice polytope P, one can construct a toric variety X, together with an ample line bundle L on X. It turns out that its Euler characteristic is equal to the number of lattice points contained in P. Moreover, the Hirzebruch–Riemann–Roch theorem tells us how to calculate this Euler characteristic in terms of the Todd class of the toric variety X. This yields an efficient method for counting the lattice points in P, because there is a polynomial time algorithm that computes the Todd class of X given the polytope P.

## Stable equivariant homotopy theory

## Denominators of Eisenstein cohomology classes - continued -

## New guests at the MPIM

## An overview of the barycenter method in nonpositively curved manifolds

## The popularity of values of Euler’s function

For each positive integer $m$, let $N(m)$ denote the number of $\varphi$-preimages of $m$, where $\varphi$ is Euler’s totient function. For example, $N(12) = 6$, corresponding to the six preimages 13, 21, 26, 28, 36, and 42. We discuss several statistical questions concerning $N(m)$ — for instance, its average size, its maximal order, and the typical size of

$N(\varphi(k))$ as $k$ varies.

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