# Developments in Modern Mathematics:

the 2nd WiMGo conference

## September 2nd to 5th, 2024

**Registration is closed!**

Our conference ”Developments in Modern Mathematics” is now in its second edition. Once again, we will bring together leading researchers active in the various fields represented at the Mathematical Institute of Göttingen University: Mathematical Physics, Operator Theory, Higher Structures, Geometry, Topology, and Number Theory. This unique event aims to offer an exploration of the latest advancements across these diverse areas of mathematics. With its synergistic approach, this event will provide a broad and rich perspective on recent developments in modern mathematics.

## The Aim

This conference actively pursues two major goals: first, by gathering experts from different yet related research fields, we aim to create a stimulating and vibrant atmosphere that fosters interdisciplinary collaborations and interactions. Second, offering a stage to the most recent developments in modern mathematics means also to play an active role in helping raising awareness of the gender dynamics existing in academia. To this end, we are incorporating lectures on gender equality alongside our mathematical presentations. All our invited speakers have been selected based on excellence, with all plenary speakers being leading female experts. This provides them with well-deserved visibility and demonstrates that mathematics is not solely a male domain; successful female researchers should be recognized as outstanding mathematicians first and foremost.

## The Poster

You can download the pdf-version here

Please help us advertising our conference by hanging this poster at your university!

## The Structure

This conference aims to offer an overview of significant recent advancements obtained in various fields of mathematics. By gathering leading experts from the diverse research areas represented at the Department of Mathematics at the University of Göttingen, we aim to foster dynamic scientific interaction and collaboration among speakers and participants, bridging disciplines that are traditionally viewed as distinct. To pursue this goal, we will offer, in particular to the young participants, the opportunity to present their results in front of a very broad and expert audience. In details, the conference will consist of:

*6 Plenary talks:*these talks provide a broad picture of the field of expertise of the speaker, presenting the main achievements and open questions still present in their research area. These talks should be accessible to the entire audience and hence could be seen as colloquium talks.

Duration: 1 hour

*17 Thematic talks:*as they will take place in parallel sections, these talks will be research level talks, at the level of a

__seminar talk__, given by senior experts in the field.

Duration: 45 minutes

*Participants’ talks:*as also these contributed talks will take place in parallel sections, the participants will be given the possibility of presenting their results to a more specialized audience. We expect that most of the contributed talks will be given by junior participants.

Duration: 20 minutes + 5 minutes for questions

*Poster Session:*the participants will be offered the possibility of presenting their research on a poster. The posters will be visible during all the coffee breaks, for the entire duration of the conference.

## Research Areas and Confirmed Speakers

Each area will have one plenary speaker and at least two additional invited experts speaking during the thematic session in the afternoon.**Mathematical Physics**

The concept of quantisation, from noncommutative field theories to topological recursion. Some geometrical and analytical approaches.

*Plenary Speaker:*

**Patrizia Vitale**(University of Neaples)

*Thematic Speakers:*

**Giovanni Landi**(University of Trieste)**Raimar Wulkenhaar**(University of Münster)

*Distinguished Contributed Talk by:*

**Pierre Bieliavsky**(Université Catholique de Louvain)**Thomas Krajewski**(CPT, Aix-Marseille University)

*Scientific Advisor:*

- Dorothea Bahns

**Operator Theory**

*Plenary Speaker:*

**Magdalena Musat**(University of Copenhagen)

*Thematic Speakers:*

**Sara Azzali**(University of Bari)**Runlian Xia**(University of Glasgow)

*Scientific Advisor:*

- Ralf Meyer

**Higher Structures**

Higher structures in differential geometry. Interactions between higher Lie theory, Symplectic and Poisson geometry, and quantization.

*Plenary Speaker:*

**Eva Miranda**(Technical University of Catalonia and University of Cologne)

*Thematic Speakers:*

**Stefan Waldmann**(University of Würzburg)**Marco Zambon**(University of Leuven)

*Scientific Advisor:*

- Chenchang Zhu

**Geometry**

Geometry of groups and dynamics.

*Plenary Speaker:*

**Indira Chatterji**(Côte d’Azur University)

*Thematic Speakers:*

**Claire Burrin**(University of Zürich)**Valentina Disarlo**(University of Heidelberg)

*Scientific Advisor:*

- Federico Vigolo

**Topology**

Stable homotopy theory and index theory.

*Plenary Speaker:*

**Kate Ponto**(University of Kentucky)

*Thematic Speakers:*

**Nora Doll**(University of Halle Wittenberg)

*Scientific Advisor:*

- Thomas Schick

**Number Theory**

*Plenary Speaker:*

**Vivian Kuperberg**(ETH Zürich)

*Thematic Speakers:*

**Christian Bernert**(Leibniz University Hannover)**Simon Myerson**(University of Warwick)

*Scientific Advisor:*

- Jörg Brüdern

## Initiatives towards Equality in Mathematics

In addition to its mathematical content, the conference will actively pursue the goal of exposing a broad audience of active researchers in mathematics to the problem of gender dynamics in academia. To reach this goal, a series of activities will approach the problem from different perspectives.

__Monday, 2nd Sep: Plenary talk on gender bias__

The objectivity of mathematics makes it difficult to convince mathematicians of the subjectivity of their judgments. This is why we strongly believe that the first topic to be addressed in the direction of gender awareness in the context of mathematics at academic level is the problem of gender bias. This is the topic which will be treat by Lara Gildehaus in her talk.

*Plenary Speaker:*

**Lara Gildehaus**(University of Klagenfurt)

__Tuesday, 3rd Sep: Panel discussion together with our plenary speakers__

Young female researchers tend to think that the obstacles they experience in their academic path should be solely traced back to their lack of competence. This is why during this panel discussion our plenary speakers, all leading researchers in their fields, will be encouraged to report on the difficulties they also had to face, including the problem of finding a balance between work and private life. Our panelists will also be asked for ideas and various possible solutions that can be implemented. A long coffee break is planned afterwards, so that graduate students can get to know and interact with the panelists.

*Moderator:*

**Katrin Wodzicki**(University of Göttingen)

__Wednesday, 4th Sep: Interactive plenary talk__

Over the last years several efforts have been made to promote diversity. The next step to take now is to learn how to work together effectively.

*Plenary Speaker:*

**Andrea Blunck**(University of Hamburg)

__Day 4: Research talks in gender studies in mathematics: towards possible solutions__

Some experts have conducted research to investigate the existence of a stereotypical image of a mathematician in clusters of excellence, and its effects. They will present their results, interpretations and possible solutions.

*Plenary Speaker:*

**Anna Ransiek**(Free University of Berlin)

## The Exhibition

During the conference, the Mathematical Institute will host the exhibition ”Women in Mathematics from around the world. A galley of portraits”. Next to the permanent collection of portraits and historical images remembering some of the most outstanding mathematicians associated to the University of Göttingen, this exhibition will give the stage also to female mathematicians of the present: 33 posters with pictures and interviews of women in mathematics will be placed around the department during the event.

In the words of Sylvie Paycha, one of the editors and first initiator of this project:

”Entering the field of mathematics can be tough, and women often encounter specific obstacles. The exhibition offers a glimpse into the world of mathematics through photographs and excerpts of interviews of thirty-three women mathematicians from around the world. The women mathematicians portrayed here share with us their experience, thus serving as role models to stimulate young women scientists to trust their own strength.”

You can download the poster here

## Programme

## Titles and Abstracts

Here you will find all titles and abstracts of plenary, thematic and contributed talks, divided by sessions.

Mathematical Physics

__Plenary Talk:__

**Patrizia Vitale**(University of Neaples)

*Title:*Noncommutative field and gauge theory

*Abstract:*In this introductory lecture I will start by giving some motivations for considering space-time noncommutativity. Then, I will shortly review quantum mechanics as the prototypical noncommuta- tive theory, as well as the geometric language of standard gauge theory of fundamental interactions. Hence, I will describe a natural generalization of these structures to the noncommutative setting, rely- ing on the introduction of a derivations-based differential calculus. Two main models will be consid- ered: the φ4 scalar field theory and the U(1) gauge theory of electrodynamics. Both will be discussed within two types of noncommutativity, the Moyal spacetime and the Lie algebra type noncommu- tativity. Finally, I will review some aspects of noncommutative spacetime symmetries and twisted noncommutativity.

__Thematic Talks:__

**Giovanni Landi**(University of Trieste)

*Title:*Hopf algebroids of noncommutative principal bundle and gauge theory

*Abstract:*To an equivariant noncommutative principal bundle one associates an Atiyah sequence of twisted derivations whose splittings give (gauge) connections on the bundle. There is an explicit action of vertical twisted derivations as infinitesimal gauge transformations on connections. From the sequence one derives a Chern—Weil homomorphism and twisted Chern—Simons terms. On the principal bundle of orthonormal frames over the quantum sphere 𝑆2𝜃𝑛, the splitting of the sequence leads to a Levi-Civita connection on the corresponding module of twisted derivations. The connection is torsion free and compatible with the ‘round’ metric. We work out the corresponding Riemannian geometry.

**Patrizia Vitale**(University of Neaples)

*Title:*Electrodynamics as a Poisson gauge theory

*Abstract:*I will discuss U(1) gauge theory on Poisson manifolds as semiclassical limit of fully non- commutative spacetimes. The gauge potentials of Poisson electrodynamics are described as sections of a symplectic realization of the spacetime manifold and infinitesimal gauge transformations as a representation of the associated Lie algebroid acting on the symplectic realization. Finite gauge trans- formations are obtained by integrating the sections of the Lie algebroid to bisections of a symplectic groupoid, which form a one-parameter group of transformations

**Raimar Wulkenhaar**(University of Münster)

*Title:*Noncommutative geometry meets topological recursion

*Abstract:*I will try to outline two problems in noncommutative geometry, both related to combi- natorics and enumerative geometry, where tools from topological recursion permitted recently some significant progress. The first problem concerns higher order freeness in free probability theory. The second problem is the exact solution, order by order in their 1/𝑁-expansion, of correlation function in scalar QFT-models on noncommutative spaces.

__Distinguished Contributed Talks:__

**Pierre Bieliavsky**(Université Catholique de Louvain)

*Title:*New classes of locally compact quantum groups

*Abstract:*Locally compact quantum groups (LCQG) were introduced in the beginning of the century by S. Vaes and J. Kustermans as a von Neumann (i.e. measurable) version of quantum groups in the context of operator algebra. Since then the theory has been widely developed. However, it appears that, besides compact quantum groups, the theory suffers from a lack of new non-compact examples.

In a joint work with V. Gayral, S. Neyshveyev and L. Tuset, we constructed non-formal deformation quantizations of Lie group of Fröbenius type which provide classes of Lie group based examples. The construction is mainly geometrical. In the talk, I will first introduce the notion of locally compact quantum group and then explain the construction. I will also discuss some open questions in that context.

**Thomas Krajewski**(Aix-Marseille University)

*Title:*Finite spectral triples and the tenfold way

*Abstract:*The tenfold way is a an extensions of Dyson's threefold way and appeared in condensed matter physics as a way to classify the behavior of fermionic Hamiltonians under the combined effect of time reversal and charge conjugation. It further classifies other mathematical objects : Morita equivalence classes of Clifford algebras, super division algebras, compact symmetric spaces,... We concisely review this construction and show how this fits naturally with the Dirac operator of finite spectral triples.

__Talks contributed by participants:__

**Irina Bobrova**(Max-Planck-Institute for Mathematics in the Sciences)

*Title:*Towards non-commutative discrete Painlevé systems

*Abstract:*It is well-known that, starting from the Affine Weyl groups (or their extension), one can define a discrete dynamic, by using translation operators (see the paper by M. Noumi and Y. Yamada, CMP, 1998). In fact, a proper extension of a birational representation of the Affine Weyl group acting on a parameter space leads to a discrete system for some dynamical variables. We will introduce a generalization of this construction to a non-abelian case. Moreover, regarding the Painlevé equations, such | birational representations naturally arise from the Bäcklund transformations. Since the latter have non-commutative analogs, we will also discuss an application of the Affine Weyl groups for deriving non-abelian versions of the discrete Painlevé equations with additive dynamics. This talk is based on the paper ”Affine Weyl groups and non-abelian discrete systems: an application to d-Painlevé equations” (arXiv:2403.18463).

**Louis De Man**(Université Catholique de Louvain)

*Title:*Smooth Wave Front Set and Deformation of Distribution Algebras

*Abstract:*The smooth wave front set is a microlocal object introduced by Hörmander in 1989, which simultaneously encodes the singular behaviour at finite points of a tempered distribution and its growth at infinity. In 2019, D. Bahns and R. Schulz observed that the smooth wave front set provides a sufficient condition to define the Weyl star-product of two tempered distributions in a non-formal manner. Using this criterion, they constructed non-formal algebras of tempered distributions under the Weyl star-product. We show that their work relies on the interaction between the smooth wave front set and the twist associated to the Weyl star-product (in the sense of Drinfel’d construction). This approach enables to extend their result to other star-products on 𝑅2𝑛, including the Kohn-Nirenberg star-product. The irreducible unitary representations of the Heisenberg group play an important role in this framework.

**Tim Henke**(University of Southern Denmark)

*Title:*Very stable parabolic Higgs bundles

*Abstract:*Higgs bundles are algebraic principal bundles equipped with a Lie algebra-valued one-form known as the Higgs field. Parabolic Higgs bundles are in addition equipped with parabolic structures that are preserved by the residue of the Higgs field. The moduli space of semi-stable Higgs bundles has a rich geometry with connections to mirror symmetry and the very stable locus provides insight into these phenomena from mirror symmetry. I will present some results about the very stable locus in the parabolic case and discuss what we can learn from it.

**Jean Thibaut**(CPT, Aix-Marseille University)

*Title:*Topological gravity and dynamical dark energy

*Abstract:*We describe a 4d topological gauge theory of gravity by building an action from the Euler and Pontryagin characteristic numbers of a manifold with Cartan geometry. Properties of characteristic classes imply that the action depends on the choice of connection only up to the integral of an exact term. Two Cartan geometries are studied: Lorentzian (including dS and AdS) and conformal. In the case of a Lorentzian geometry we retrieve the Holst action with a bare cosmological constant and with the Nieh-Yan, Pontrjagin and Euler terms with coupling constants only depending on the 3 free parameters corresponding to Newton’s constant G, the B-I parameter γ and the bare cosmological constant. Computing the equations of motion with an additional Dirac matter action leads to an effective cosmological constant Λ corresponding to the sum of the bare cosmological constant and the vacuum contribution to it. In case the vacuum contribution is larger than the currently observed value of Λ, we show that agreement with experiment can be reached via an AdS spacetime. For a Conformal geometry one can recover similar equations of motion with an additional source term for curvature depending on variations of the spin density of matter and torsion. Finally we look at a generalization of the previous work which leads to dynamical Λ (dark energy) and G depending on scalars of the curvature and the matter content of the theory. This talk is based on joint work with S. Lazzarini https://arxiv.org/pdf/2403.05284.pdf and a forthcoming paper (dynamical Λ and G part).

Operator Theory

__Plenary Talk:__

**Magdalena Musat**(University of Copenhagen)

*Title:*The Connes-Kirchberg Problem and Infinite dimensional phenomena in quantum information theory

*Abstract:*I will discuss recent developments in the analysis of quantum correlations and their deep interconnections with the multi-faceted Connes-Kirchberg Problem in operator algebras, leading also to infinite dimensional phenomena in quantum information theory.

__Thematic Talks:__

**Sara Azzali**(University of Bari)

*Title:*Traces, KK-theory, and the Godbillon-Vey invariant

*Abstract:*Traces on C*-algebras play an important role in index theory, particularly in extracting numerical invariants from classes defined in K-theory.

By introducing real coefficients in Kasparov bivariant K-theory (KK-theory), traces can be regarded as classes in $KK_\mathbb{R}$. The process of applying a trace then corresponds to taking the Kasparov product.

In this talk, we shall explain these constructions and their applications. Specifically, we will investigate a natural $KK_\mathbb{R}$-class that represents the Godbillon-Vey invariant of a foliation of codimension one. We shall see how the Godbillon-Vey invariant relates to a (densely defined) infinite trace, and its connection to the index theorem for measured foliations. This work is in collaboration with Paolo Antonini (Università del Salento) and Georges Skandalis (Université Paris Cité).

**Magdalena Musat**(University of Copenhagen)

*Title:*Factorizable maps, traces, and quantum correlations

*Abstract:*Factorizable completely positive maps, introduced by C. Anantharaman-Delaroche within the framework of operator algebras, have proven to have interesting applications in the analysis of quantum information theory, leading also to reformulations of the Connes Embedding Problem. Re- cent work with M. Rordam (as discussed in the plenary talk), shows that (infinite dimensional) von Neumann algebras are, indeed, needed to describe such channels. In this talk, we further establish a new viewpoint on factorizable channels, leading to central questions in C*-algebra theory, including the convex structure of the simplex of traces on free products of matrix algebras

**Runlian Xia**(University of Glasgow)

*Title:*Fourier multipliers coming from group actions on tree-like structures

*Abstract:*In this talk, we introduce a generalised Cotlar identity and a geometric form of some Lp-bounded Fourier multipliers for groups acting on R-trees. This class of groups includes free groups, amalgamated free products, HNN extensions, totally ordered groups and many others. The pioneering work in this direction is due to Mei and Ricard who proved Lp-boundedness of Hilbert transforms on free group von Neumann algebras using a Cotlar identity. This work is followed by the paper of Mei, Ricard and Xu focusing on more general Fourier multipliers.

Joint work with Adri´an Gonz´alez and Javier Parcet.

__Talks contributed by participants:__

**Ioannis Apollon Paraskevas**(National and Kapodistrian University of Athens)

*Title:*Fock covariance for product systems and the Hao-Ng isomorphism problem

*Abstract:*We provide a characterisation of equivariant Fock covariant injective representations for product systems. We square this characterisation with established results concerning Fock covariance, on compactly aligned product systems over right LCM semigroups and on product systems with one-dimensional fibers. Using our characterisation we resolve the reduced Hao-Ng isomorphism problem for generalised gauge actions by discrete groups. This is a joint work with Evgenios Kakariadis.

Higher Structures

__Plenary Talk:__

**Eva Miranda**(Technical University of Catalonia and University of Cologne)

*Title:*From Alan Turing to Fluid Computers via Contact Geometry

*Abstract:*What are the limits of computation? Can physical systems compute?

Cris Moore associated universal Turing machines with transformations of the square Cantor set. Using Poincaré sections, this procedure can be extended to dimension 3. When connected with a contact structure, it provides a compelling answer to the questions above; its associated Reeb vector field can "mirror" a particular solution of the Euler equation, creating a universal Turing machine capable of simulating fluid motions. The undecidability of the halting problem, proved by Alan Turing, entails the existence of undecidable fluid paths, introducing a revolutionary notion of "chaos" in the logical sense ([1] and [2]).

By "contact-ing" 2D Turing complete systems, we have designed an abstract model of a fluid computer, which we call "flubit." In collaboration with Ángel González-Prieto and Daniel Peralta-Salas, we have refined this model to design a hybrid machine where flubits serve as the fundamental units of computation. Inspired by Feynman's visionary ideas, we formalize this assembly process as a Topological Field Theory, which we’ve dubbed TKFT (Topological Kleene Field Theory) in honor of the pioneering logician Stephen Kleene.

We will explore this more sophisticated construction [3] in the parallel session on "Higher Structures" using techniques from TQFT and higher categories.

[1] Robert Cardona, Eva Miranda, Daniel Peralta-Salas, and Fran Presas, Constructing Turing Complete Euler Flows in Dimension 3, Proc. Natl. Acad. Sci. USA 118 (2021), no. 19, Paper No. 2026818118, 9 pp.

[2] Renzo Bruera, Robert Cardona, Eva Miranda, Daniel Peralta-Salas, Topological Entropy of Turing Complete Dynamics, arXiv:2404.07288.

[3] Ángel González-Prieto, Eva Miranda and Daniel Peralta-Salas, Computability and Topological Kleene Field theories, preprint, 2024.

__Thematic Talks:__

**Eva Miranda**(Technical University of Catalonia and University of Cologne)

*Title:*From Alan Turing to Higher categories

*Abstract:*In 1936, Alan Turing proved that the halting problem is undecidable, defining one of the first models of computation. In this talk, we will use the construction of a Turing-complete dynamical system that simulates the movement of a particle in a fluid (described in my plenary talk), as a step towards developing a sophisticated computational design using higher structures.

We employ 2-categories to outline an abstract design for a "hybrid computer". Drawing on ideas from Topological Quantum Field Theory (TQFT), we will investigate how to quantize Turing-complete dynamical systems, including those related to fluid dynamics. Our goal is to design a computational field theory conceived as a Topological Field Theory and named TKFT in honor of Stephen Kleene’s work on partial recursive functions. This hybrid computer aims to compute and "quantize" physical systems. TKFTs will be constructed from basic elements, which we call "flubits," associated with 3D Turing-complete Euler flows. It is also possible to design other non-computational field theories that connect the categories of cobordisms and partial functions.

This talk will provide an overview of these ideas and is based on joint works (some ongoing) with Ángel González-Prieto, Stanislav Krysmskii, and Daniel Peralta-Salas.

**Stefan Waldmann**(University of Würzburg)

*Title:*Constraint algebras and reduction

*Abstract:*In this talk I will explain the notion of constraint algebras with some motivation from phase space reduction. I will discuss the intrinsic reduction procedure for constraint algebras and their modules. Finally, I will indicate the deformation theory of constraint algebras with the corresponding Hochschild cohomologies. The talk is based on joint work with Marvin Dippell and Chiara Esposito.

**Marco Zambon**(University of Leuven)

*Title:*Lie 2-algebras of functions of quasi-Poisson and quasi-symplectic groupoids

*Abstract:*The functions on symplectic and Poisson manifolds, together with the Poisson bracket, form a Lie algebra. In this talk we consider Lie groupoids endowed with weak versions of symplectic and Poisson structures, respectively. On quasi-Poisson groupoids there is a well-behaved Lie 2-algebra of functions. By contrast, on quasi-symplectic groupoids we are only able to either associate 1) a Lie 2-algebra of functions which is not Morita invariant, and 2) a graded Lie algebra structure on the truncated differentiable cohomology, which turns out to arise from the “inverse” quasi-Poisson groupoid structure. This talk is based on work in progress with Cristian Ortiz and Gabriele Sevestre.

__Talks contributed by participants:__

**Myriam Mahaman**(Technische Universität Dresden)

*Title:*Rings of differential operators and Hopf algebroids

*Abstract:*In algebraic geometry, it is known that the ring of differential operators over a smooth algebra A has a canonical Hopf algebroid structure. Using some descent techniques for coalgebras, my coauthor Ulrich Krähmer and I determined a class of algebras which are not smooth yet whose rings of differential operators still carry a canonical Hopf algebroid structure.

Geometry

__Plenary Talk:__

**Indira Chatterji**(Côte d’Azur University)

*Title:*Groups through the lense of median geometry

*Abstract:*Geometric group theory aims at understanding groups through their actions on nice geometric objects. We will focus our attention on median spaces, which are metric spaces generalizing products of trees. We will see examples of such spaces and investigate the information one can extract from a group action on a median space.

__Thematic Talks:__

**Claire Burrin**(University of Zürich)

*Title:*Pairs of saddle connections

*Abstract:*: In this talk, I will discuss ‘optimal dynamics’ for the geodesic flow on flat surfaces, starting from the torus and moving to open questions on completely periodic orbits on Veech surfaces. This is joint work with Samantha Fairchild and Jon Chaika.

**Indira Chatterji**(Côte d’Azur University)

*Title:*Roughly median spaces

*Abstract:*We will study the class of roughly median spaces and groups admitting a geometric action on such a space.

**Valentina Disarlo**(University of Heidelberg)

*Title:*The model theory of the curve graph

*Abstract:*The curve graph of a surface of finite type is a graph that encodes the combinatorics of isotopy classes of simple closed curves. It is a fundamental tool for the study of the geometric group theory of the mapping class group. In 1987 N.K. Ivanov proved that the automorphism group of the curve graph of a finite surface is the extended mapping class groups. In the following decades, many people proved analogue results for many ”similar” graphs, such as the pants graph, the arc graph, etc. In response to the many results, N.V. Ivanov formulated a metaconjecture. which asserts that any ”natural and sufficiently rich” object associated to a surface has automorphism group isomorphic to the extended mapping class group. In this talk, I will present a joint work with Thomas Koberda (Virginia) and Javier de la Nuez Gonzalez (KIAS) where we provide a model theoretical framework for Ivanov’s metaconjecture and we conduct a thorough study of curve graphs from the model theoretic point of view, with particular emphasis in the problem of interpretability between different ”similar” geometric complexes. In particular, we will prove that the curve graph of a surface of finite type is w-stable. This talk does not assume any prior knowledge in model theory.

__Talks contributed by participants:__

**Oscar Cosserat**(University of Göttingen)

*Title:*Tracking Liouville tori on a symplectic foliation

*Abstract:*I will introduce in this talk the definition of an integrable Hamiltonian system on a symplectic manifold in the sense of Arnol’d-Liouville. I will then explain a bit of perturbative theory of such dynamics and relations with numerical analysis. In my PhD, I constructed some numerical methods for Hamiltonian systems on Poison manifolds, being foliated by symplectic leaves. The end of the presentation will therefore be dedicated to a discussion on perturbation theory of Hamiltonian systems on Poison manifolds.

**Leolin Nkuete**(University of Bonn)

**Daniele Taufer**(KU Leuven)

*Title:*The regularity of Hessian graphs

*Abstract:*Given a projective hypersurface defined by F=0, its Hessian is defined as the zero locus of the determinant of the Hessian matrix of F. When F is a cubic form, its Hessian also defines a cubic in the same projective space, so it is natural to study the dynamical system this transformation determines, which we call the Hessian graph. When restricting to smooth cubics, i.e. elliptic curves, it was known that this transformation corresponds to a rational function on a moduli space, which led to understanding some local properties of the Hessian graph. In this talk, we will see that we can choose a better model to describe such objects, which turns out to be an elliptic curve again. Under this identification, the Hessian transformation can be seen as a degree-3 endomorphism of this new model, which allows us to prove many novel regularities of Hessian graphs, and even to prescribe the global structure over some finite base fields. This is joint work with M. Mula and F. Pintore.

Topology

__Plenary Talk:__

**Kate Ponto**(University of Kentucky)

*Title:*Representations and Euler characteristics

*Abstract:*The experience of having to work really, really hard to prove a theorem that seems like it shouldn’t be that painful is so common. Occasionally we are extremely lucky and have the opposite experience. I’ll talk about one of my favorite results of this type and how it connects the familiar formulas for the characters of restricted and inducted representations to Euler characteristics for fibrations.

__Thematic Talks:__

**Nora Doll**(University of Halle Wittenberg)

*Title:*Skew localizer for real index pairings

*Abstract:*In this talk index pairings of a projection and a unitary where both, the projection and the unitary fulfill real symmetry relations are considered. For a given combination of symmetries the integer-valued index of the pairing vanishes, but there may be a 𝑍2-index given by the dimension of its kernel, modulo 2. The aim is then to construct a finite dimensional real skew-adjoint matrix called the skew localizer for these pairings and to show that the 𝑍2-index can be computed as the sign of the Pfaffian of the skew localizer. The main tool to prove the connection of the 𝑍2-index to the sign of the Pfaffian of the skew localizer is the orientation flow of paths of bounded real skewadjoint Fredholm operators. This orientation flow might be of independent interest and is introduced in the second part of this talk.

**Kate Ponto**(University of Kentucky)

*Title:*Homotopical invariants for lifting and extensions

*Abstract:*: Motivated by question in fixed point theory, Klein and Williams defined a homotopical obstruction to lifts and extensions that is not the familiar cohomological one. While they considered only a topological setting, their approach is remarkably flexible. I’ll describe how to make sense of it using a model category and a Blakers-Massey theorem.

__Talks contributed by participants:__

**Benjamin Bruske**(Technische Universität Hamburg)

*Title:*Approaching Condensed Mathematics through Compact Bounded Structures

*Abstract:*The theory of condensed mathematics, as developed by Peter Scholze and Dustin Clausen, studies algebraic objects equipped with a topological structure by category-theoretic methods using the language of sheaves and topoi.

There is, however, a classical structure equivalent to quasiseperated condensed objects, in the guise of compactological objects, treated by the functional analyst Lucien Waelbroeck in 1971. This structure permits an alternative approach to understanding the objects described by Scholze's and Clausen's theory, in essence as the free completion of compactological objects by missing quotients. This presentation aims to provide a short introduction to the concept of compactological spaces and its intimate categorical relation to condensed mathematics.

Number Theory

__Plenary Talk:__

**Vivian Kuperberg**(ETH Zürich)

*Title:*Approaches to consecutive primes

*Abstract:*About two hundred years ago, Dirichlet proved that there are infinitely many primes ending in each of 1, 3, 7, and 9. But what if we want to understand the last digits of pairs of consecutive primes? As of a couple decades ago, we know that there are infinitely many primes ending in 1 such that the next prime ends in 1. And we know that there are infinitely many primes ending in 1 such that the next prime ends in one of 3, 7, or 9. However, for each of those three cases, we cannot show that it occurs infinitely often!

In this talk, we will discuss this problem and several approaches to understanding this problem and consecutive primes in general. In particular, we’ll discuss how this problem relates to more general approaches to understanding the distribution of primes.

__Thematic Talks:__

**Christian Bernert**(University of hannover)

*Title:*The astronomy of cubic equations -- Do we really need a telescope to observe their solutions?

*Abstract:*Building on work of Davenport--Lewis and Heath-Brown, we discuss the solubility of (not necessarily homogeneous) cubic equations in many variables. Following an initiative of Browning--Dietmann--Elliott, we also investigate upper bounds on their smallest solutions. Our main detection device is the circle method, but the attempt to provide good uniform lower bounds for the local densities poses an interesting challenge in its own right.

**Vivian Kuperberg**(ETH Zürich)

*Title:*Sums of odd-ly many fractions

*Abstract:*In this talk, I will discuss new bounds on constrained sets of fractions. Specifically, I will discuss the answer to the following question, which arises in several areas of number theory: for an integer 𝑘 ≥2, consider the set of 𝑘-tuples of reduced fractions 𝑞𝑎11, . . . , 𝑞𝑎𝑘𝑘 ∈ 𝐼, where 𝐼 is an interval around 0. How many 𝑘-tuples are there with ∑𝑖 𝑞𝑎𝑖𝑖 ∈Z?

When 𝑘 is even, the answer is well-known: the main contribution to the number of solutions comes from “diagonal” terms, where the fractions 𝑞𝑎𝑖𝑖 cancel in pairs. When 𝑘 is odd, the answer is much more mysterious! In work with Bloom, we prove a near-optimal upper bound on this problem when 𝑘 is odd. I will also discuss applications of this problem to estimating moments of the distributions of primes and reduced residues.

**Simon Myerson**(University of Warwick)

*Title:*Forms with real coefficients and differing degrees

*Abstract:*We explore approaches to systems of forms with differing degrees which use the ‘repulsion’ technique. This allows for Diophantine inequalities with real coefficients to be studied in the general style of Browning and Heath-Brown (2017).

Gender Studies

__Plenary talk on gender bias:__

**Lara Gildehaus**(University of Klagenfurt)

*Title:*The invisible boundaries – Understanding and reducing gender biases in mathematics

*Abstract:*Although there are now hardly any formal barriers to accessing mathematics, and theoretically anyone can be enthusiastic about mathematics, current studies, and participation figures repeatedly indicate that actual access to mathematics is not equal. Therefore, in my presentation, I will try to shed light on existing gender biases in mathematics from school to university to the academic field and discuss their effects. Practical implications for more gender-sensitive mathematics teaching and learning will then be addressed

__Interactive plenary talk:__

**Andrea Blunck**(University of Hamburg)

*Title:*Diversity in Mathematics: Who belongs to Mathematics?

*Abstract:*When it comes to the inclusion of women and other marginalised groups in mathematics, the question arises as to the access to mathematics and, moreover, how mathematics should be defined in this context. In the talk we will discuss historical and current aspects of these topics. In addition, best practice examples for inclusion will be presented.

__Research talk:__

**Anna Ransiek**(Free University of Berlin)

*Title:*Gender as a Barrier in Mathematics? Perspectives of Scientists in Leadership Positions on PhD Students and Postdocs and its Influence on Gendered Gatekeeping.

*Abstract:*Up to now, equality between women and men in mathematics in academia in Germany has not been achieved. The proportion of women, although on a parity basis among first-year students for years, decreases steadily with each academic career level. Female mathematicians are still underrepresented in top academic/scientific positions. The image of the leaky pipeline is therefore still valid in this discipline. In our qualitative sociological research project, we investigate possible causes and mechanisms of the reproduction of gender disparities in a mathematical cluster of excellence. The research setting offers the unique opportunity to gain access to an excellent research environment and examine different career stages as well as status transitions within the cluster and in the academic field. The results presented in the talk are based on the analysis of 45 semi-structured interviews with scientists in leadership positions in the cluster. I will highlight ways of thinking and acting on side of these scientists in leadership position that may create (possible) barriers for female mathematicians.

## The Venue

The conference will take place at the Mathematical Institute, Bunsenstraße 3-5, 37073 Göttingen

## Registration

To register at our event, please fill in the following form. The registration will close on

**Sunday, 11th August**. Some limited financial support is available for young participants in the form of accommodation. When registering, all participants are strongly encouraged to submit an abstract for a contributed talk and/or poster. Professional childcare will be available for all speakers and participants at the conference venue. In case you would like to use our childcare service, please complete your registration by

**Sunday, 4th August**.

**Registration is closed!**

## On-site Registration

The Hilbert Raum is the entrance hall (not a room) accessible from the main staircase at Bunsenstraße 3, 37073 Göttingen

## Conference Dinner

All participants to the conference “Developments in Modern Mathematics” are invited to join us to our conference dinner.

**When:**Wednesday 4th September 2024, 19:30

**Where:**Goa India -> Kurze-Geismar-Straße 43, 37073 Göttingen

The restaurant will offer a buffet dinner, including vegetarian and vegan options. Please let us know in case you have any particular allergy: we will immediately inform the restaurant.

REMARK: The dinner costs will be covered by RTG for all the participants to the conference. However, everyone is expected to cover for her/his own drinks.

## Travel and Accommodation

Here you will find information on how to get to and your stay in Göttingen.

Arriving in Göttingen

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.

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.

Hotels

Here is a list of recommended Hotels in Göttingen:

Leine Hotel

Eden Hotel

Hotel Central

G Hotel

Novostar

Hotel Stadt Hannover

Leine Hotel

Eden Hotel

Hotel Central

G Hotel

Novostar

Hotel Stadt Hannover

Map

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.