Developments in Modern Mathematics:
a WiMGo conference

Plenary Talk:

Sylvie Paycha (University of Potsdam)
Title: Mathematical reflections on locality
Abstract: Starting from the principle of locality in quantum field theory, which
states that an object is influenced directly only by its immediate surroundings, I shall first briefly review some features of the notion of locality arising in physics and mathematics. These are then encoded in locality relations, given by symmetric binary relations whose graph consists of pairs of "mutually independent elements". Locality morphisms, namely maps that factorise on products of such pairs of elements, play a key role in the context of renormalisation in multiple variables. They include "locality evaluators", which are used to consistently evaluate meromorphic germs in several variables at their poles. If time allows, I will report on recent joint work with Li Guo and Bin Zhang which gives a classification of locality evaluators on certain classes of algebras of meromorphic germs. It uses previous joint work with these two authors and Pierre Clavier.

Thematic Talks:

Sylvie Paycha (University of Potsdam)


Gandalf Lechner (University of Erlangen–Nuremberg)
Title: Twisted Araki-Woods Factors in QFT: Modular Data and Inclusions
Abstract: Twisted Araki-Woods Algebras form a large family of von Neumann algebras that include second quantization factors (Bosonic/Fermionic), von Neumann algebras arising from Free Probability, q-deformed factors, and many more. In general, such a von Neumann algebra L(T,H) is defined in terms of two objects - a standard subspace (that is, a specific real linear subspace H of a complex Hilbert space) and a twist (that is, a specific selfadjoint operator T on the tensor square of the Hilbert space), and is represented on a kind of Fock space with a vacuum vector. From the point of view of applications in quantum field theory, H encodes a localization region in some spacetime and T encodes a two-particle interaction. After explaining the general construction, this talk will focus on two main questions: 1) under which conditions is the vacuum vector cyclic and separating for L(T,H)? This turns out to be closely related to concepts from physics, namely the Yang-Baxter (braid) equation and an abstract version of "crossing symmetry" (a concept from scattering of elementary particles). 2) under which conditions is an inclusion of two twisted Araki-Woods factors non-singular (non-trivial relative commutant)? This question is connected to constructive QFT and currently under control in some situations.
This is joint work with Ricardo Correa da Silva.

Alessandra Frabetti (University of Lyon)
Title: Non-associative renormalization group
Abstract: Renormalization Hopf algebras represent the renormalization group of (perturbative) quantum field theory as a functor. In practice, this is a group of formal diffeomorphisms in the powers of the coupling constant with coefficients given by the couterterms of divergent graphs, and the "functorial representation" makes sense only for commutative coefficients and commutative Hopf algebras. However, in quantum electrodynamics and in quantum chromodynamics the coefficients of the series and the natural renormalization Hopf algebras are not commutative. In this talk I explain how we can still use them to represent a renormalization action, if we renounce to the associativity of the "group" and see it as a "loop" (a non-associative group). You can download the Slides here

John Barrett (University of Nottingham)
Title: Quantum gravity from integration over Dirac ensembles
Abstract: The talk will outline a programme to define quantum gravity using integration over a space of Dirac operators and the corresponding fermionic variables in the framework of non-commutative finite spectral triples. I will summarise briefly some progress we have made in understanding some simple cases of random Dirac operators. You can download the Slides here

Talks contributed by participants:

Samuel Hannah (University of Cardiff)
Title: Classifying Frobenius Algebras in Dijkraaf-Witten Categories
Abstract: Algebra objects are interesting objects in category theory, with particular uses in classifying modules over tensor categories. Detecting and classifying algebra objects provides an important goal in understanding the representation theory of such categories. Dijkraaf-Witten categories are an example of tensor categories associated to a finite group G and a 3-cocycle on G. During this talk, I will present a classification of Frobenius algebras in these categories, generalising existing results by Davydov-Simmons to a field of arbitrary characteristic. This is done by means of a Frobenius Monoidal Functor. I will present as an example the case when G is a dihedral group of odd degree. Joint work with Ana Ros Camacho (Cardiff University) and Robert Laugwitz (University of Nottingham).

Tim Henke (University of Southern Denmark)
Title: Dynamic CS/WZNW duality: the missing case
Abstract: In physics, CS/WZNW duality is an equivalence between Chern–Simons gauge theory and WZNW conformal field theory first proposed by Witten in 1989. This statement was formalised by Beauville, Laszlo, and Pauly over the next decade by proving the vector space isomorphism between the geometric quantisation of the moduli space of flat connections, representing the Hilbert space of gauge fields between charges on a surface, with the Tsuchiya–Ueno–Yamada (TUY) space of conformal blocks, representing the conformal vacua between operator insertions corresponding to the charges. In quantum Chern–Simons theory, changes to the complex structure change the state via the (projectively flat) Hitchin connection, while the conformal blocks are governed by the TUY connection. In positive genus, these have been proven to be equivalent, but the genus 0 case was not covered. In this talk, I will present joint work with J. E. Andersen for projective equivalence of the two connections for the missing case of genus 0, where the TUY connection corresponds to the Knizhnik–Zamolodchikov connection. After a short introduction to the topic, I will discuss our result, some of the ideas and obstacles, and conclude with some applications and future perspectives. You can download the Slides here

Plenary Talk:

Nadia Larsen (University of Oslo)
Title: Equilibrium states for C*-algebras of right LCM monoids
Abstract: C*-algebras associated to monoids are constructions where categorical features of the monoid reflect properties of their isometric representations and provide an important example class in the theory. This interplay is prominent in the study of equilibrium, or KMS states. For a C*-algebra and a one-parameter automorphism group, a KMS state at a real parameter, known as an inverse temperature, is a positive linear functional on the C*-algebra that obeys a trace-like condition involving a twisting of elements by the one-parameter group. Whether KMS states exist at some inverse temperature is pending on internal structure of the C*-algebra and points back to the monoid. I will survey work on these KMS states, with emphasis on cancellative monoids that admit right least common multiples. One tractable class includes Noetherian monoids such as right-angled and finite-type Artin monoids, and I will end with some recent results obtained jointly with L.E. Gazdag and M. Laca. You can download the Slides here

Thematic Talks:

Nadia Larsen (University of Oslo)
Title: Spectral triples for noncommutative solenoids
Abstract: Noncommutative solenoids are C*-algebras introduced in work of Latrémolière and Packer. They admit both a realisation as inductive limits of rotation algebras, and as twisted group C*-algebras. We shall present a construction of odd finitely summable spectral triples for noncommutative solenoids based on length functions of bounded doubling. We further show that there is an alternative realisation as inductive limits of spectral triples on rotation algebras. This is joint work with Carla Farsi, Therese Basa Landry and Judith Packer. You can download the Slides here

Kristin Courtney (University of Southern Denmark)
Title: A C*-structure on images of completely positive order zero maps
Abstract: In the context of C*-algebras, *-homomorphisms are the most natural maps to work with. Unfortunately they are not always as prolific as we would like. In lieu of *-homomorphisms, we often turn to linear maps which preserve positivity and orthogonality: completely positive order zero maps. These enjoy a rich structure, which has made them extremely important in the study of nuclear C*-algebras. In this talk, we consider the structure of the image of a completely positive order zero map. When the domain is separable and unital, we can define multiplication and a corresponding pre-C*-norm on the image. Generalizing this construction, we are able to characterize when a self-adjoint linear subspace of a C*-algebra is the image of a unital C*-algebra under a completely positive order zero map. In both directions, the argument is constructive. This is joint work with Wilhelm Winter.

Bram Mesland (University of Leiden)
Title: Groupoid C*-algebras and solid state physics
Abstract: In this talk I will give an overview of some recent uses of groupoid C*-algebras in solid state physics. A solid material can be modelled by an abstract discrete point set embedded in a topological group. The ambient group encodes the physical symmetries of the system under study. Because of the absence of any internal structure in the discrete point set, the mathematical notion of a groupoid naturally presents itself. The associated C*-algebra encodes the physical observables and its K-theory classifies observable physical quantities. These techniques give insight into certain classes of architected materials as well interacting particle systems. Based on joint worked with E. Prodan. You can download the Slides here

Talks contributed by participants:

Natalia Maślany (Jagiellonian University)
Title: Differential embeddings into algebras of topological stable rank 1
Abstract: We identify a class of smooth Banach *-algebras that are differential subalgebras of commutative C*-algebras whose openness of multiplication is completely determined by the topological stable rank of the target C*-algebra. Finally, we completely characterize in the complex case (uniform) openness of multiplication in algebras of continuous functions in terms of the covering dimension. The talk will be based on recent joint work with Tomasz Kania. You can download the Slides here

Masayoshi Kaneda
Title: Open projections and Murray-von Neumann projections
Abstract: We characterize the C*-algebras for which openness of projections in their second duals is preserved under Murray–von Neumann equivalence. They are precisely the extensions of commutative C*-algebras by annihilator C*-algebras. We also show that the annihilator C*-algebras are precisely the C*-algebras for which all projections in their second duals are open.

Plenary Talk:

Cecília Salgado (University of Groningen)
Title: Arithmetic and geometry of algebraic surfaces: the place of elliptic fibrations
Abstract: In this talk, I will go over some of the recent progress on the arithmetic of algebraic surfaces. In particular, I will discuss the role of elliptic fibrations in the study of rational points on surfaces, and present open problems summarising the state of the art. You can download the Slides here

Thematic Talks:

Cecília Salgado (University of Groningen)
Title: Mordell-Weil rank jumps on families of elliptic curves
Abstract: We will discuss recent advances around the variation of the Mordell-Weil rank in families of elliptic curves. The first part of the talk will be dedicated to introducing the theme, motivating, presenting the state of the art, and a brief view of the different techniques used to deal with the problem. The second part will cover recent results on rank jumps on rational and K3 elliptic surfaces. You can download the Slides here

Min Lee (University of Bristol)
Title: Murmurations of holomorphic modular forms in the weight aspect
Abstract: In April 2022, He, Lee, Oliver, and Pozdnyakov made an interesting discovery using machine learning – a surprising correlation between the root numbers of elliptic curves and the coefficients of their L-functions. They coined this correlation 'murmurations of elliptic curves.' Naturally, one might wonder whether we can identify a common thread of 'murmurations' in other families of L-functions. In this talk, I will introduce a joint work with Andrew R. Booker and Jonathan Bober, demonstrating murmurations in holomorphic modular forms.

Talks contributed by participants:

William Bernardoni (Case Western Reserve University)
Title: Applications of Generalized Universal Valuations
Abstract: In their paper on a scheme theoretic version of tropicalization, Jeffrey and Noah Giansiracusa used an extended notion of a valuation into non-totally ordered but commutative idempotent semirings to generalize the idea of tropicalization. They introduced the idea of a universal valuation semiring which could classify these generalized valuations. In this talk we will show that this construction can be extended further to capture features of non-commutative rings. We will give an explicit characterization of this generalized universal valuation semiring, as well as show how this characterization implies the Non-Archimedean case of Ostrowski's theorem on the characterization of absolute values of rational numbers. We will also show how a further generalization of this construction gives a method to characterize the existence of representations of a ring into ultrametric spaces. This construction gives a non-commutative method of tropicalization that we hope may be further used to help capture unique aspects of non-commutative geometry. You can download the Slides here

Konrad Zou (University of Bonn)
Title: Categorical local Langlands for GL_n: the irreducible case with integral coefficients
Abstract: Fargues and Scholze conjecture a Hecke-equivariant equivalence of categories between certain coherent sheaves on the stack of Langlands parameters and compact objects in the category of lisse-etale sheaves on Bun_G. We will discuss how to prove this conjecture for irreducible parameters for GL_n, even with integral coefficients. It turns out that this needs surprisingly little knowledge about the spaces involved, the non-formal input is the cardinality of the Fargues-Scholze L-packets and genericity of their members. The formal input is about localizations of categories over schemes, which we will discuss.

Leo Schäfer (University of Göttingen)
Title: A Weyl-Type Inequality for Certain Sums
Abstract: An application of the circle method requires bounds on exponential sums. We take a look at exponential sums of a certain form to which the classical inequality generalizes nicely.

Plenary Talk:

Elise Goujard (University of Bordeaux)
Title: Geometry of large genus flat surfaces.
Abstract: Gluing the opposite sides of a square give a flat torus: a torus endowed with a flat metric induced by the Euclidean metric on the square. Similarly, one can produce higher genus surfaces by gluing parallel sides of several squares. These "square-tiled surfaces" inherit from the squares a flat metric with conical singularities. In this talk we will present several recent results and conjectures on the large genus asymptotics of these surfaces, and more generally of some families of flat surfaces (joint work with V. Delecroix, P.Zograf and A. Zorich).
We will also see how these results can be interpreted in the language of closed curves on surfaces and meanders. You can download the Slides here

Thematic Talks:

Elise Goujard (University of Bordeaux)
Title: Random square-tiled surfaces of large genus and random multicurves on surfaces of large genus
Abstract: In this talk, I will present in more details the ingredients of the results about large genus square-tiled surfaces presented in the plenary talk. I will introduce background results on the count of simple closed geodesic multicurves on hyperbolic surfaces, by Mirzakhani, and on the count of metric ribbon graphs, by Kontsevich and Norbury, as well as the uniform large genus asymptotics of intersection numbers of Witten-Kontsevich correlators proven by Aggarwal. All these resultats allow us to describe the structure of a random multi-geodesic on a hyperbolic surface of large genus and of a random square-tiled surface of large genus. For instance we show that the number of components of a randon multi-geodesic asymptotically behaves as the number of cycles of a random permutation of 3g-3 elements taken with respect to a very explicit probability distribution (joint work with V. Delecroix, P.Zograf and A. Zorich). You can download the Slides here

Nguyen-Thi Dang (Université Paris-Saclay)
Title: Equidistribution of periodic tori
Abstract: Bowen and Margulis independently proved in the 70s that closed geodesics on compact hyperbolic surfaces equidistribute towards the Liouville measure.
From a homogeneous dynamics point of view, this measure is the quotient of the Haar measure on PSL(2,R) modulo some discrete cocompact sugroup.
In a joint work with Jialun Li, we investigate the higher rank setting of this problem by taking a higher rank Lie group (like SL(d, R) for d ≥ 3) and by studying the dynamical properties of ''geodesic flows in higher rank'': the so-called Weyl chamber flows and their induced diagonal action. We obtain an equidistribution formula of periodic tori (instead of closed orbits of the geodesic flow).

Talks contributed by participants:

Thorsten Hertl (University of Freiburg)
Title: Moduli Spaces of Positive Curvature Metrics
Abstract: In the last decade the observer moduli space of Riemannian metrics with positive curvature conditions have become more and more popular. So far, results in this direction only work if the dimension of the underlying manifold is bigger than 5 or if the manifold is spin. I will present a different construction that works in dimension 4 and does not rely on spin geometry. You can download the Slides here

Plenary Talk:

Cornelia Drutu (University of Oxdorf)
Title: Understanding infinite groups via their actions on Banach spaces
Abstract: One way of studying infinite groups is by analysing their actions on classes of interesting spaces. This is the case for Kazhdan's property (T) and for Haagerup's property (also called a-T-menability), formulated in terms of actions on Hilbert spaces and relevant in many areas (e.g. for the Baum-Connes conjectures, in combinatorics, for the study of expander graphs, in ergodic theory, etc.) Recently, these properties have been reformulated for actions on Banach spaces, with very interesting results. This talk will overview some of these reformulations and their applications. Part of the talk is on joint work with Ashot Minasyan and Mikael de la Salle, and with John Mackay.

Thematic Talks:

Cornelia Drutu (University of Oxdorf)
Title: Connections between hyperbolic geometry and median geometry
Abstract: In this talk, I shall explain how groups endowed with various forms of hyperbolic geometry, from lattices in rank one simple groups to acylindrically hyperbolic groups, present various degrees of compatibility with the median geometry. The latter geometry can be seen as a non-discrete version of the geometry of CAT(0) cube complexes. This is on joint work with Indira Chatterji, and with John Mackay.

Irene Pasquinelli (University of Bristol)
Title: Complex hyperbolic lattices
Abstract:Complex hyperbolic space is the still mysterious sibling of real hyperbolic space. In particular, it is the only symmetric space of non-compact type where the relation between arithmeticity and lattices is not completely understood. In my talk, I will introduce complex hyperbolic space and I will talk about lattices in its group of holomorphic isometries, the group PU(n,1), with a particular focus on the non-arithmetic case I will give an overview of some of the known constructions and I will talk about some future directions of research, which we hope will lead to new constructions of non-arithmetic lattices.

Vera Tonić (University of Rijeka)
Title: Asymptotic dimension and some applications to geometric (approximate) group theory
Abstract: The notion of asymptotic dimension (asdim) was introduced by M. Gromov in the early 1990's as a tool to study finitely generated groups through their representations as geometric objects. In this talk we will cover the definition and basic properties of asdim, as well as some applications of asdim in coarse geometry of groups and approximate groups. You can download the Slides here

Talks contributed by participants:

Cesar Hilario (Heinrich-Heine Universität Düsseldorf)
Title: Regular but non-smooth curves of genus 3
Abstract: Regular but non-smooth curves represent a unique feature of geometry in positive characteristic, that results from the fact that over an imperfect field the notion of regularity is weaker than the notion of smoothness. Such curves can easily occur as the generic fibres of fibrations by singular curves, as was the case in the program by Bombieri and Mumford to extend the classification of surfaces to arbitrary characteristic, where they encountered quasielliptic fibrations. In this talk I will present a precise description of regular but non-smooth rational curves of genus 3, which admit a non-smooth point that is a canonical divisor. From a fibration point of view these correspond to fibrations by singular quartics with a canonical moving singularity. I will show that in certain cases such a fibration can be seen as a surface fibered over the projective line, i.e., a pencil, which at the same time is a purely inseparable degree 2 cover of a quasielliptic pencil.

Plenary Talk:

Birgit Richter (University of Hamburg)
Title: Gluing algebras to points
Abstract: Hochschild homology of an associative algebra A can be thought of the result of gluing A to the points of (a simplicial model of) the circle. Here, we have a cyclic ordering of the points in that model which allows us to work with associative algebras. If we go to higher dimensions, levels of commutativity might be needed because points may merge together from different directions. This is the basic idea behind higher order Hochschild homology or, more generally, general Loday constructions and factorization homology. You can download the Slides here

Thematic Talks:

Birgit Richter (University of Hamburg)
Title: Loday constructions for Tambara functors
Abstract: If we want to glue algebras to spaces with an action of a finite group G, then points are not the smallest entities any more, but we need to glue algebras to G-orbits in a meaningful way. In joint work with Ayelet Lindenstrauss and Foling Zou we investigate how that might work for G-Tambara functors; these are analogues of commutative algebras in the G-equivariant setting. I'll give the definition, and then present examples. You can download the Slides here

Charlotte Kirchhoff-Lukat (Massachusetts Institute of Technology)
Title: The Fukaya category of a log symplectic surface
Abstract: Floer theory and Fukaya categories constitute powerful invariants of symplectic manifolds. As a first step in the effort to extend these techniques to Poisson structures with degeneracies, I will present the construction of the Fukaya category for log symplectic Poisson structures on oriented surfaces, with a focus on the additional features of the theory arising from the degeneracy locus.

Leonid Ryvkin (University of Lyon)
Title: Differentiation of simplicial manifolds
Abstract: Kan simplicial manifolds provide with a very explicit model for $L_\infty$-groupoids. Pavol Severa proposed a procedure of differentiation for these objects, yielding an $L_\infty$-algebroid, (an NQ-manifold). In the talk I will report on joint work with Du Li, Arne Wessel and Chenchang Zhu, where we have proven that this procedure always works.

Talks contributed by participants:

Francesca Leonardi (University of Leiden)
Title: Logarithmic Hochschild (co)homology through derived intersection theory
Abstract: Hochschild (co)homology is a broadly investigated tool in algebraic geometry, for its relation with Hodge-de-Rham theory and its non-commutative interest. It has a natural description in the language of derived self intersections. In 2012, Arinkin and Căldăraru proved a formality result for smooth schemes, providing a nice decomposition of it and generalising the celebrated HKR isomorphism in derived-categorical terms. Following Olsson's definition of logarithmic Hochschild (co)homology, we prove a similar statement for log smooth schemes, extending the result for instance to some not-too-badly-behaving singularities. Joint work (in progress) with Márton Hablicsek and Leo Herr.

Stefano Ronchi (University of Würzburg / University of Göttingen)
Title: Higher Contangent Groupoids
Abstract: We define a dualization construction for VB 2-groupoids that, when applied to the tangent 2-groupoid of a Lie 2-groupoid, produces a cotangent 2-groupoid that is 2-shifted symplectic with respect to a canonical form. This is analogous to the cotangent groupoid of a Lie groupoid being a symplectic groupoid with respect to the canonical symplectic form on the cotangent of the space of arrows. Our construction is inspired by the simplicial mapping space construction for simplicial vector spaces. This allows certain properties commonly associated with dual spaces to be satisfied naturally. Applications of this construction include integration of a class of Courant algebroids and coadjoint representations up to homotopy for 2-groupoids. This project is joint work in progress with Miquel Cueca and Chenchang Zhu.

Closest airports: Frankfurt Main International, Hannover

Göttingen is well connected via fast trains to most major cities in Germany. The travel time from Frankfurt is about 2h and from Hannover less than 1h.

Walking from the train station to the Mathematical Institut takes no more than 20 minutes passing through the town centre. Most hotels are in walking distance from the main building, however there is also a chance of using the buses.

Here is a list of recommended Hotels in Göttingen:

Leine Hotel
Eden Hotel
Hotel Central
G Hotel
Novostar
Hotel Stadt Hannover

Here is a map where you can find bus stops, sightseeing points and other information about Göttingen. The link is set to the Autumn School room, you can move freely within the map.