# Mathematical Concepts in the Sciences and Humanities

## Abstracts for the talks

**Sergio Albeverio ***University of Bonn*

(joint work with * Francesco De Vecchi (University of Bonn)*)

**On relations between functional integrals, quantum fields, singular pde's and stochastic analysis**

We first recall some basic historic relations between the areas of the title as an introduction to new developments in constructing models of relativistic quantum fields via singular stochastic pde's.We then illustrate the methods for the case of models involving exponential interactions (that occurr also in geometric contexts).

**David Chavalarias ***Centre d'Analyses de Mathématiques Sociales (CAMS), and École des Hautes Études en Sciences Sociales (EHESS)***How social networks modeling can help us understand the inevitable democratic disruption under the impact of Big Tech**

Digital spaces are increasingly present and influential in contemporary societies, to the point that what happens online can no longer be ignored when it comes to understanding their evolution. From politics to global warming debates, through vaccination debates, what happens online, and in particular on social networks, has more and more influence on what happens offline.

This virtualization of social interactions brings as many new research questions as it does major challenges for our democracies. On the one hand, we have never had so much data on social systems, which should be conducive to a better conceptual understanding of their dynamics. On the other hand, the acceleration of information, the amplification of disinformation and misinformation phenomena, as well as the multiplication of digital interference between competing countries, give us the impression of a disruption and a weakening of the social fabric, victim of fragmentation and polarization.

We will illustrate these phenomena from the observation of six years of political activism in France within the framework of the Politoscope project, a social macroscope using Twitter data. Then we will analyze what, in a systemic way, can lead to these democratic disruptions and, in order to stop it, the necessary evolutions of the organization of our digital spaces.

**Vincenzo De Risi ***CNRS***Leibniz and Newton on a Geometry of Space**

The talk considers the evolution of Leibniz's philosophy of space, and the genesis of his famous relational theory that he opposed Newton in the course of the Leibniz-Clarke Correspondence. A detailed analysis of Leibniz' various definitions of space shows that we need to reconsider in depth the chronology of Leibniz's philosophy of space. Finally, the relationship between Leibniz and Newton on the dispute between absolute and relational space is reconsidered in a new light, an unknown source of Leibniz's conception of space is discussed, and the possibility for a different reading of the Leibniz-Clarke Correspondence is opened up.

**Tobias Elze ***Harvard Medical School***Identifying Disease Subtypes from Complex Medical Data**

Many contemporary fields of medicine produce increasingly high-dimensional data. "Traditional" data analysis methods trained in clinical research education are insufficient to investigate such highly complex data. Therefore, various machine learning techniques are increasingly finding their way into medical data analysis. However, disease diagnosis itself has traditionally been defined on simple rules based on easily observable low-dimensional data. Traditional clinical diagnoses can therefore be inappropriate as ground truths to train machine learning classifiers. Here, using the eye disease of glaucoma as an example, we demonstrate how patterns identified by unsupervised machine learning on complex medical data can help to refine and quantitatively re-define disease diagnosis. Our patterns of vision loss due to glaucoma enable a whole field of quantitative research which was not available by previously existing diagnostic approaches.

**James Evans ***The University of Chicago***The Curvature of Culture and Society: Analyzing meaning and connection through curved embeddings of text and networks**

Text and network embedding models have become a powerful tool for the study of culture and society, respectively, in recent years. Word embeddings may be understood as representing semantic relations between words as relationships between vectors in a high-dimensional semantic space, specifying a relational model of meaning consistent with many contemporary theories of culture. Similarly, network embeddings may be understood as representing social relations between persons or collectives as relationships between vectors in a high-dimensional space. In this paper, we theorize the independent ontology and value of discrete textual expressions and network ties and curved manifolds built from them, where linguistic expression trace communicative action and network topologies trace social interaction, while continuous manifolds capture the space of probable actions and interactions. Then we introduce a unified, geometric characterization of system-level structure in textual and social networks that simplifies description while improving prediction of system-level network processes. By viewing words in an expression and nodes in graphs as embedded in a latent manifold, geometric curvature of the manifold shapes action and interaction patterns. Regions embedded in positive curvature manifest dense thickets of words and ties through which information pools and cycles; regions embedded in negative curvature are characterized by infrequent communication and sparse ties representing ridges over which information spreads and flows. In this way, curvature combines local and global perspectives, providing a continuous characterization of semantic and network structure that eliminates the need for discrete distinctions between roles and communities to enable accurate modeling of system-level processes like semantic and network evolution and information diffusion with increased accuracy. We develop statistical tests for network curvature estimation, show how they link and validate emerging approaches to curvature measurement in discrete mathematics and machine learning, and demonstrate the utility of our approach for characterizing semantic and social networks, predicting network evolution and deepening geometric understanding of semantic and social constraint using simulated and observed networks across domains of multi-ethnic community life, adolescent school interactions, and scientific innovation.

**Shaoming Fei ***Capital Normal University***Geometry in quantum computation and information processing**

We introduce the geometry of quantum computation and some related researches in quantum information processing, including quantum coherence, quantum entanglement, quantum nonlocality, quantum information masking, quantum uncertainty relations, as well as tensor network compressed sensing and machine learning.

**Peter Gärdenfors ***Lund University***Determining cognitive categories by their invariances**

The world as we perceive it is structured into objects, actions and places. In this talk my aim is to explain why these categories are cognitively primary. From an empiricist and evolutionary standpoint, it is argued that the reduction complexity of sensory signals is based on the brain’s capacity to identify various types of invariances that are evolutionarily relevant for the activities of the organism. The aim of the article is to explain why places, object and actions are primary cognitive categories in our constructions of the external world. Following Breidbach and Jost (2006), it is shown that the invariances that determine these categories have their separate characteristics and that they are, by and large, independent of each other. This separation is supported by what is known about the neural mechanisms. The invariances for the category of numbers is also briefly discussed.

**Daniela Horak ***AIG***Stability analysis of intersection graphs and it’s application for hyperparameter tuning in TDA Mapper.**

TDA Mapper (also referred to as topological data visualisation or topological clustering), is a method that yields an approximation of a Reeb graph of a data manifold, and has been used successfully in applications across different fields over several last decades.

In spite of TDA Mapper showing great promise, it's wide spread adoption has been limited. This is primarily due to practioners' ad-hoc approach to parameter tuning.

This is unsurprising, since all unsupervised learning algorithms face the same issue due to absence of ground truth labels, and TDA Mapper even more so, due to existence of hyper parameters along with parameters. Model selection is, therefore, highly non-trivial.

In this talk I will present stability analysis as a systematic tool for hyper-parameter tuning on Mapper Graphs, discuss appropriate distance metrics on these mathematical structures, and conclude with few applications.

**Enno Kessler ***Max Planck Institute for Mathematics***Super J-holomorphic curves**

J-holomorphic curves or pseudoholomorphic curves are maps from Riemann surfaces to symplectic manifolds satisfying the Cauchy-Riemann equations. J-holomorphic curves are of great interest because they allow to construct invariants of symplectic manifolds and those invariants are deeply related to topological superstring theory. A crucial step towards Gromov–Witten invariants is the compactification of the moduli space of J-holomorphic curves via stable maps which was first proposed by Kontsevich and Manin.

In this talk, I want to report on a supergeometric generalization of J-holomorphic curves and stable maps where the domain is a super Riemann surface. Super Riemann surfaces have first appeared as generalizations of Riemann surfaces with anti-commutative variables in superstring theory. Super J-holomorphic curves couple the equations of classical J-holomorphic curves with a Dirac equation for spinors and are critical points of the superconformal action. The compactification of the moduli space of super J-holomorphic curves via super stable maps might, in the future, lead to a supergeometric generalization of Gromov-Witten invariants.

Based on arXiv:2010.15634 [math.DG] and arXiv:1911.05607 [math.DG], joint with Artan Sheshmani and Shing-Tung Yau.

**Thorbjørn Knudsen ***University of Southern Denmark***Robustness and Complexity: A New Perspective for Strategy.**

This paper develops a novel approach to analyze the relation between complexity and robustness in social organizations. To this end, we develop a measure of complexity that can be used to quantify organizational complexity from empirical data. From a theoretical perspective, we outline how our complexity measure can be used to identify gains from improving the balance between an organization's differentiation (division of labor) and integration (of the divided effort). Our complexity measure also serves as a stepping-stone for analyzing robustness in complex organizations. We show that complexity not only represents a liability of change, but also enhances the organization’s ability to withstand turbulence in its value network. Surprisingly, perhaps, we can demonstrate that complexity controls robustness: the higher the complexity of a system, the greater its potential robustness. We argue that this insight reconciles literatures on resilience and complexity and opens a new perspective on the drivers of performance in business ecosystems and other complex organizations.

**Hông Vân Lê ***Czech Academy of Sciences***Categorical language and geometrical methods in Machine Learning**

In Machine Learning we develop mathematical methods for modeling data structures, which express the dependency between observables, and design efficient algorithms for estimation of such dependency. The most advanced part of Machine Learning is statistical learning theory that takes into account our incomplete information of observables, using measure theory and functional analysis. In this way we not only unveil hidden structure of data but also make a prediction for the future. In my lecture I shall consider basic problems in statistical learning theory and demonstrate the efficiency of categorical language, manifested, in particular, in terms of probabilistic morphisms, and the use of geometric constructions, e.g. diffeological Fisher metric, in solving these problems.

**Johannes Lenhard ***TU Kaiserslautern***Created by Prediction. On the history, ontology, and computation of the Lennard-Jones Fluid**

**Volker Lindenstruth ***Frankfurt Institute for Advanced Studies - FIAS***Quarks und Bytes**

At the Large Hadron Collider (LHC) at CERN in Geneva very large scale experiments are conducted. In these experiments each nucleus-nucleus collision can generate thousands of new elementary particles. For the understanding of the dynamics and the underlying physics it is paramount to identify those particles. They are being measured with a variety of detectors in order to identify their various properties. In addition the particles traverse a large magnetic field in order to determine their momentum. Typically those detectors measure space points, which have to be connected to the particles trajectory, which generates a combinatorial background with makes the computing cost prohibitively expensive. Novel algorithms have been developed which are based on cellular automata and Kalman filters in order to minimize the computational overhead. Those algorithms exhibit a linear dependence of the tracking time on the number of space points. Further the Kalman filter was optimized to equally perform very fast. To date those algorithms are the baseline for many new detector developments. The ALICE experiment at the LHC has undergone a significant upgrade and will start again with beam time in a few weeks. The data rates exceed 600 GB/s and all data of the experiment has to be processed on-line. An appropriate compute farm with 16.000 CPU cores and 2000 GPUs has been deployed, which is capable to handle the task. The presentation will outline the requirements and the algorithms used and the particular solution as an example for this research field.

**Shiping Liu ***University of Science and Technology of China***Geometric and spectral theory of signed graphs**

A signed graph is a graph whose edges are labelled by a signature. It serves as a simple model of discrete vector bundle. The fundamental ideas of balance and switching of signed graphs often lead to more systematic understanding of various parts of graph theory. In this talk, I will explain two such cases: a unification of Cheeger inequality and Bauer-Jost dual Cheeger inequality, and a unification of the discrete nodal domain theorem due to Davis, Galdwell, Leydold and Stadler, and Fiedler's approach on eigenvectors of acyclic matrices.

This talk is based on joint works with Fatihcan Atay, Chuanyuan Ge.

**Joachim Lohkamp ***University of Münster***The Geometry of Positive Mass Theorems**

We give an overview over geometric tools and methods we developed to solve problems in higher dimensional scalar curvature geometry and general relativity.

**Wiktor Mlynarski ***Institute of Science and Technology Austria***Adaptation of large-scale neural populations to natural scene statistics**

The idea that sensory neurons exploit the statistical structure of natural stimuli to efficiently transmit information has been a guiding principle in neuroscience for over half a century. This conceptual framework, known as the efficient coding hypothesis, has provided successful theoretical accounts of sensory coding across species and sensory systems with the retina being the paramount example. The spatial organisation of the retina has, however been assumed to be uniform and translation invariant. Here we challenge this view. By combining theory and novel experimental approaches, we demonstrate that large populations of retinal neurons exploit the inhomogeneous structure of natural scenes across the visual field to increase the efficiency of neural coding.

**Michael Röckner ***University of Bielefeld***Equilibria of nonlinear distorted Brownian motions**

Joint work with: Viorel Barbu (Romanian Academy, Iasi)

This talk will review the connection of nonlinear Fokker-Planck-Kolmogorov (FPK) equations and McKean-Vlasov SDEs, with special emphasis on the case where the coefficients depend Nemytskii-type on the time marginal laws. A class of examples are nonlinear distorted Brownian motions. Recent results on their asymptotic behaviour, obtained through their corresponding nonlinear FPK equations, will be presented both in the non-degenerate and degenerate case.

References

Barbu, Viorel; Röckner, Michael Probabilistic representation for solutions to nonlinear Fokker-Planck equations. SIAM J. Math. Anal. 50 (2018), no. 4, 4246–4260.

Barbu, Viorel; Röckner, Michael From nonlinear Fokker-Planck equations to solutions of distribution dependent SDE. Ann. Probab. 48 (2020), no. 4, 1902–1920.

Barbu, Viorel; Röckner, Michael Solutions for nonlinear Fokker-Planck equations with measures as initial data and McKean-Vlasov equations. J. Funct. Anal. 280 (2021), no. 7, 108926, 35 pp.

Barbu, Viorel; Röckner, Michael The invariance principle for nonlinear Fokker-Planck equations. J. Differential Equations 315 (2022), 200–221.

Barbu, Viorel, Röckner, Michael Uniqueness for nonlinear Fokker-Planck equations and for McKean-Vlasov SDEs: The degenerate case arXiv:2203.00122

Barbu, Viorel, Röckner, Michael The evolution to equilibrium of solutions to nonlinear Fokker-Planck equations arXiv:1904.08291 , to appear in Indiana

Univ. Math. J., 36 pp.

**Areejit Samal ***The Institute of Mathematical Sciences***Minimum complexity drives regulatory logic in Boolean models of living systems**

Boolean modeling is an established framework for studying gene regulatory networks. In a Boolean network, different molecular components correspond to the nodes that receive inputs and relay outputs according to node-specific input-output rules, determining how their states change with time. In this talk, I will present our work where we have examined the rules arising in curated Boolean models of diverse regulatory networks. We have studied the complexity of these logic rules using two definitions in computer science: Boolean complexity based on string lengths in formal logic which is yet unexplored in the biological context, and the average sensitivity. We find that an overwhelming majority of the rules in these biological models minimize the two complexities, pointing to complexity as a major force in selecting regulatory logic in living systems. We provide quantitative support for the long-standing hypothesis that logic rules in gene regulatory networks are likely to be ‘simple’, or in other words, possess ‘minimum complexity’. These observations have implications for ongoing efforts to build predictive models of biological systems.

**Klaus Scherrer ***Institute Jacques Monod, CNRS***The Working Genome is RNA - Evolution of a Conceptual Challenge**

**Charlotte Schubert ***Leipzig University***Rethinking Authorship in Classics**

Authorship as a concept remains controversial: death of the author, return of the author, authorial function, collective authorship, are just a few highlights of the discussion, which continues with undiminished acuteness. In the Classics, the question of the author arises in a very extreme form in view of the numerous fragment editions (text corpora of authors whose works are lost or only fragmentarily preserved) and the almost equally numerous attributions of works to "pseudo" authors. Here, computational approaches, operating on a path independent of the classical hermeneutic method, offer the humanities a completely new perspective. Using a striking example (Pseudo-Xenophon's Athenaion Politeia, which is considered the first completely preserved prose writing of antiquity), it will be shown how a new approach can be gained for the controversial topic of authorship.

**Lorenz Schwachhöfer ***TU Dortmund University***Parametrized measure models in classical and quantum information geometry.**

In this talk, we shall give an account of the geometrization of the space of probability measures on a given measure space, and report on recent progress to generalize this to quantum models as well.

**Angela Stevens ***University of Münster***Mathematical models for regeneration in biological model organisms**

Flatworms are among the biological model organisms for regeneration. Severe cutting and grafting experiments still result in the regeneration of a complete and functioning flatworm. So far, most mathematical models could not account for these basic experiments, without adaptations for each setting. A new model is suggested and analyzed, which covers the central regeneration experiments, while not changing the structure of the equations. Dynamic boundary conditions play a crucial role here.

(Joint work with A. Scheel and C. Tenbrock)

**Michael Struwe ***ETH Zurich***Plateau flow**

Using the Millot-Sire interpretation of the half-Laplacian on S^{1} as the Dirichlet-to-Neumann operator for the Laplace equation on the ball B, we devise a classical approach to the heat ﬂow for half-harmonic maps from S^{1} to a closed target manifold N ⊂^{n}, recently studied by Wettstein, and for arbitrary ﬁnite-energy data we obtain a result fully analogous to the classical results for the harmonic map heat ﬂow of surfaces and in similar generality. When N is a smoothly embedded, oriented closed curve Γ ⊂^{n} the half-harmonic map heat ﬂow may be viewed as an alternative gradient ﬂow for the Plateau problem of disc-type minimal surfaces.

**Karl-Theodor Sturm ***University of Bonn***Conformally Invariant Random Geometry on Riemannian Manifolds of Even Dimension**

We construct and analyze conformally invariant, log-correlated Gaussian random fields on compact Riemannian manifolds of general even dimension defined through its covariance kernel given as inverse of the Graham-Jenne-Mason-Sparkling (GJMS) operator. The corresponding Gaussian multiplicative chaos is a generalization to the n-dimensional case of the celebrated Liouville quantum gravity measure in dimension two. Finally, we study the Polyakov-Liouville measure on the space of distributions on M induced by the copolyharmonic Gaussian field, provide explicit conditions for its finiteness and compute the conformal anomaly.

**Mina Teicher ***Bar-Ilan University***Synchronization in brain activity and application**

In this talk we will present the first proof that Neurons activations is subject to synchronization and not determined by the firing rate . We show it on behaving animals and up to 3 mili seconds. We will also mention some medical applications .

**Wilderich Tuschmann ***Karlsruhe Institute of Technology (KIT)***Manifolds and Moduli Spaces of Positive Ricci Curvature Metrics**

I will report on joint ongoing work with Philipp Reiser on the construction of new metrics of positive Ricci curvature and its applications to the topology of corresponding moduli spaces of metrics.

**Leo van Hemmen ***Technical University Munich***Geometric Perturbation Theory and Sound Localization**

Geometric perturbation theory is universally needed but not recognized as such yet. A typical example is provided by the three-dimensional wave equation, widely used in acoustics; particularly, in modelling the input to animal hearing and sound localization. We face vibrating eardrums as binaural auditory input stemming from an external sound source. In the setup of internally coupled ears (ICE), which are present in more than half of the land-living vertebrates, the two tympana are coupled by an internal air-filled cavity, whose geometry determines the acoustic properties of the ICE system. The eardrums themselves are described by a two-dimensional, damped, wave equation and are part of the spatial boundary conditions of the three-dimensional Laplacian belonging to the wave equation of the internal cavity that couples and internally drives the eardrums. In animals with ICE the resulting signal is the superposition of external sound arriving at both eardrums and the internal pressure coupling them. This is also the typical setup for geometric perturbation theory. In the context of ICE it boils down to acoustic boundary-condition dynamics (ABCD) for the coupled dynamical system of eardrums and internal cavity. In acoustics, the deviations from equilibrium are extremely small (nm range). Perturbation theory therefore seems natural and is shown to be appropriate. In doing so, we use a time-dependent perturbation theory à la Dirac in the context of Duhamel’s principle. We set the general stage for geometric perturbation theory where (d − 1)-dimensional manifolds as subsets of the boundary of a d-dimensional domain are driven by their own dynamics with e.g. the domain pressure p and an external source term as input, at the same time constituting time-dependent boundary conditions for p.

**Kang Zuo ***Johannes Gutenberg University Mainz***Higgs bundles in arithmetic geometry**

I shall report the recent development of Higgs bundles in arithmetic geometry, in particular, constructing rank-2 motivic local systems on the projective line with 4 punctures via p-adic nonabelian Hodge theory and Drinfeld’ work on global Langlands correspondence over function fields of characteristic p. This is a joint program with Raju Krishnamoorthy, Mao Sheng, and Jinbang Yang.

## Date and Location

**May 16 - 25, 2022**

Max Planck Institute for Mathematics in the Sciences

E1 05 (Leibniz-Saal)

Inselstr. 22

04103 Leipzig

## Scientific Organizers

**Nihat Ay**, Hamburg University of Technology (Germany) & Santa Fe Institute**Eckehard Olbrich**, MPI for Mathematics in the Sciences (Germany)**Felix Otto**, MPI for Mathematics in the Sciences (Germany)**Bernd Sturmfels**, MPI for Mathematics in the Sciences (Germany)

## Administrative Contact

**Katharina Matschke**

MPI for Mathematics in the Sciences (Germany)

Contact by Email