## Homotopy versus Isotopy: Spheres with Duals in 4-Manifolds, II

This will be an experimental “Jazz session” where both of us will be explaining our recent results together. We’ll also be encouraging audience participation and are aiming not only at 4-manifolds experts.

## Homotopy versus Isotopy: Spheres with Duals in 4-Manifolds, I

This will be an experimental “Jazz session” where both of us will be explaining our recent results together. We’ll also be encouraging audience participation and are aiming not only at 4-manifolds experts.

## "Pentagramma Mirificum". Hirzebruch lecture by Sergey Fomin on Friday, November 8, University Club Bonn

**Pentagramma Mirificum** (the miraculous pentagram) is a beautiful geometric construction studied by Napier and Gauss. Its algebraic description yields the simplest instance of cluster transformations, a remarkable family of recurrences which arise in diverse mathematical contexts, from representation theory and enumerative combinatorics to theoretical physics and classical geometry (Euclidean, spherical, or hyperbolic). The lecture will explore some of the most basic and concrete examples of cluster transformations, and briefly discuss their properties such as periodicity, integrability, Laurentness, and positivity.

**Sergey Fomin** studied at St. Petersburg State University where he received his doctorate in 1982 with Leonid Ossipov and Anatoli Verschik. He was a lecturer at St. Petersburg Electrotechincal University (LETI) from 1982 to 1991. From 1992 to 2000 he was on the faculty at the Massachusetts Institute of Technology (Assistant Professor 1993, Associate Professor 1996) before moving to a position at the University of Michigan in 1999, where he has been Robert M. Thrall Collegiate Professor since 2007. He was also a scientist at the Institute of Computer Science and Automation of the Russian Academy of Sciences in St. Petersburg from 1991 to 2005. He was a visiting scientist at the Mittag-Leffler-Institut (1992, 2005), MSRI, the Isaac Newton Institute in Cambridge, the University of Strasbourg (IRMA), the Hausdorff Institute in Bonn, and the Erwin Schrödinger Institute for Mathematical Physics in Vienna. In 2012 he became a fellow of the American Mathematical Society. He was invited speaker at the International Congress of Mathematicians 2010 in Hyderabad. In 2018 he received the Leroy P. Steele Prize of the American Mathematical Society.

His research is on combinatorics with applications to geometry, algebra, and representation theory, where he introduced cluster algebras with Andrei Zelevinsky. He has also worked in ennumerative geometry (Schubert calculus) and mathematical physics (e.g. Yang-Baxter equation, Bethe ansatz).

## Spaces, Categories, and the Cobar Construction

In topology, there are two ways to think about fiber bundles on connected spaces. Generally speaking, parallel transport along loops starting and ending at a basepoint gives rise to a map of groups from the loop space (i.e., the space of loops) to the group of automorphisms of the fiber. This map determines, and is determined by, the bundle. A bundle also determines, and is determined by, a map of spaces from its base to the classifying space of the automorphisms of its fibre. Thinking of fiber bundles in two ways establishes a duality between connected spaces and groups. Every space has a loop space, which is a group, and every group has a classifying space, which is a space. Moreover, the classifying space of the loop space of a space is the space itself, and the loop space of the classifying space of the group is the group itself. In algebra, one half of this duality assigns to a coalgebra (an avatar of a space) an algebra (an avatar of a group). This assignment, known as the cobar construction, was introduced in a 1956 paper of Frank Adams under the assumption of simply-connectedness. A deeper look at the categorical meaning of the construction reveals that the simply-connectedness assumption can be removed. This is joint work with Manuel Rivera which started when he visited the Max-Planck-Institute for a week in 2016. Some newer results are with Manuel Rivera and Felix Wierstra whom I met at Max-Planck last year.

## The Hodge and de Rham Chern characters of holomorphic connections, II

Joint with Cheyne Glass, Micah Miller, and Thomas Tradler.

In the first part of the talk, I will describe the formulae for the Hodge and de Rham characteristic classes of holomorphic bundles solely in terms of their clutching functions. To do so, I define a map of simplicial presheaves, the Chern character, that assigns to every sequence of composable isomorphisms of vector bundles with holomorphic connections that do not necessarily preserve the connections, an appropriate sequence of holomorphic forms. We then apply this map of simplicial presheaves to the Cech nerve of a cover of a complex manifold and assemble the data by passing to the totalization. In this way, we obtain a map of simplicial sets that on the vertices produces an explicit formula for the Hodge Chern character of a bundle in terms of its clutching functions. On the edges, we obtain similar invariants associated to isomorphisms between bundles. Similarly, we obtain suitable Hodge and de Rham Chern characters in the equivariant setting.

## The Hodge and de Rham Chern characters of holomorphic connections, I

Joint with Cheyne Glass, Micah Miller, and Thomas Tradler.

In the first part of the talk, I will describe the formulae for the Hodge and de Rham characteristic classes of holomorphic bundles solely in terms of their clutching functions. To do so, I define a map of simplicial presheaves, the Chern character, that assigns to every sequence of composable isomorphisms of vector bundles with holomorphic connections that do not necessarily preserve the connections, an appropriate sequence of holomorphic forms. We then apply this map of simplicial presheaves to the Cech nerve of a cover of a complex manifold and assemble the data by passing to the totalization. In this way, we obtain a map of simplicial sets that on the vertices produces an explicit formula for the Hodge Chern character of a bundle in terms of its clutching functions. On the edges, we obtain similar invariants associated to isomorphisms between bundles. Similarly, we obtain suitable Hodge and de Rham Chern characters in the equivariant setting.

## tba

## Persistent homology (from a mathematical point of view)

The goal of the talk is to present the basics of persistent homology which is one of the primary tools in applied algebraic topology and topological data analysis.

We hope to explain the mathematical ideas behind it, some of its current challenges and relations to sheaf theory.

## Fibre surfaces, knots, and elastic strings

## tba

## On a mean value result for a product of L-functions

The asymptotic behaviour of moments of L-functions is of special interest to number theorists and there

are conjectures that predict the shape of the moments for families of L-functions of a given symmetry type.

However, only some results for the first few moments are known. In this talk we will consider the asymptotic

behaviour of the first moment of the product of a Hecke L-function and a symmetric square L-function

in the weight aspect. This is joint work with O. Balkanova, G. Bhowmik, D. Frolenkov.

## Higher structures in rational homotopy theory, II

## Higher structures in rational homotopy theory, I

## From almost purity to functional analysis

Faltings' almost purity theorem asserts that, in p-adic geometry, things become simpler when passing to certain highly ramified, largely non-geometric, coverings. The coverings in question are somewhat akin to covering an interval by a Cantor set. I will show how embracing this idea has led us (with Dustin Clausen) to recast the basic notions of topology, leading to a better setup in which to do algebra with topological groups.

## Perturbing an isoradial triangulation

The theory of random Delaunay triangulations of the plane has been proposed by

David-Eynard and others as a discrete model for 2-dimensional quantum gravity: In this model

the role of a continuous metric is played by a Delaunay triangulation while flat metrics

correspond to isoradial triangulations (on which one can define a theory of discrete analyticity).

Like the continuous case, the partition function for this discrete theory is given by a suitably

normalized determinant of a Beltrami-Laplace operator which varies with the choice of

triangulation. An elegant formula of Richard Kenyon expresses this determinant as a finite sum

of local contributions when the triangulation is both isoradial and periodic. In joint work with

François David we consider smooth perturbations of a periodic isoradial triangulation and obtain

an asymptotic expansion for the second variation of the log-determinant of the discrete

Beltrami-Laplace operator. This result can be interpreted as a discretization of the formula for

the second variation (of the logarithm) of the continuous partition function know from conformal

field theory; using this interpretation we can identify the central charge in this discrete setting.

## "surprise"

## Algebraic geometry of the Lagrangian correspondence of Gaiotto

## A simplicial and categorical background for Hopf algebras in number theory and physics

We will discuss how simplicial and categorical structures lead to Hopf algebras.

his type of Hopf algebracomes from a categorical setup, where the product is a monoidal product and the co-product is

dual to the partial productthat is given by composition of morphisms, providing a simple "raison d'être" for these

"prima vista" complicated co-products. In this setting, we see the analogous structure of the Hopf algebra of Goncharov

and those of Connes and Kreimer.

## On Hecke eigenvalues of Maass cusp forms

For a Hecke-Maass newform F of level N and with pth Hecke eigenvalue a(p) where p is a prime, the generalized Ramanujan conjecture

asserts that a(p) is bounded by 2 in the absolute value for ( p, N) = 1. In this talk we will survey the recent results and developments

centered on this intriguing problem.

## Rationality of Fano 3-folds over nonclosed fields

The rationality problem for smooth Fano threefolds over algebraically closed fields is basically solved.

In this talk I will discuss rationality of forms of these Fanos over nonclosed fields of characteristic 0.

I will concentrate on the case where the Picard number equals 1.

The talk is based on joint work with Alexander Kuznetsov (in progress).

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