# Special Sessions at NCM29

### Organisers of the special sessions invite speakers and take care of the course of the session they are

responsible for.

**Algebraic coding theory and cryptography**

**Matteo Bonini**, Aalborg University, Denmark

**Martino Borello**, University of Paris VIII, France

Coding theory and cryptography are two strands of what we might call information protection, which is fundamental to modern communication. The main focus of coding theory is to protect information from errors: when a message passes through a “noisy” channel, an error may arise, and with coding and decoding an attempt is made to correct it. On the other hand, cryptography aims to protect the message from the presence of adversarial behaviour, constructing and analysing protocols that prevent third parties from reading private messages. While originally these two disciplines began as separate theories (and at very different times, cryptography being much older than coding theory), they are increasingly interacting, bringing ever-new solutions and problems. A significant example of this is code-based cryptography, which is receiving particular attention in recent years as a possible candidate for protecting our data in the quantum era, where quantum computers could jeopardize the cryptosystems used so far.

The session will be devoted to exploring the more mathematical aspects of these two theories. Actually, to construct efficient error-correcting codes and secure cryptosystems, various mathematical tools from abstract algebra, finite geometry, combinatorics, and many more, have proven extremely useful. Conversely, many interesting mathematical problems find their origin or correspondence in a language proper to these two disciplines. Our invited speakers have contributed remarkably to the mathematical development of such theories and will be able to illustrate to us how a rigorous, sound, and formal approach can lead to surprising results.

**Algebraic geometry**

**Gergely Bérczi**, Aarhus University, Denmark

**Jørgen Rennemo**, University of Oslo, Norway

**Sofia Tirabassi**, Stockholm University, Sweden

We aim to bring together early career researchers and established mathematicians working in algebraic geometry and related areas in nordic countries. We have chosen purposely speakers with different expertise in order to foster fruitful discussion and progress through diversity. Here are some of the main topics of the special session:

– geometry of moduli spaces of curves,

– tropical geometry,

– logarithmic geometry,

– Gromov–Witten invariants,

– stacks and derived algebraic geometry

Our primary goal is to improve collaboration, and strengthen the geometry network across the nordic countries

**Complex analysis and geometry**

**Séverine Biard**, Université Polytechnique Hauts-de-France, France

**Barbara Drinovec Drnovšek**, University of Ljubljana, Slovenia

**Zakarias Sjöström Dyrefelt**, Aarhus University, Denmark

**Arkadiusz Lewandowski**, Jagiellonian University, Poland

**Benedikt Magnússon**, University of Iceland, Iceland

Complex analysis and complex geometry are two interconnected branches of mathematics that study the properties of functions of complex variables and their applications to various mathematical and physical phenomena.

Complex analysis focuses on the analysis of functions that are defined on sets in affine complex space of one or many dimensions. Often these functions are assumed to be holomorphic, which means they satisfy a certain partial differential equation (the d-bar of them is zero). These functions are quite rigid, which means that often we have good connections between local and global properties. Existence and properties of these functions on a specific set can give us geometrical and topological information about that set. In the setting of function theory of several complex variables, complex dynamics and the pluripotential theory that has deep applications in the theory of holomorphic foliations and the construction of Kähler-Einstein metrics are important fields.

Complex geometry, on the other hand, deals with the geometric properties of complex manifolds or spaces, which locally resemble a complex affine space. It studies the interplay between complex analysis and differential geometry often with the help of algebra. Key topics in complex geometry include complex manifolds, complex algebraic varieties, sheaf theory, and the study of holomorphic vector bundles. Complex geometry has applications in algebraic geometry, algebraic topology, and theoretical physics.

Both complex analysis and complex geometry provide powerful tools for understanding and analyzing mathematical structures, making significant contributions to various areas of mathematics.

**Geometric analysis**

**Wolfram Bauer**, Leibniz University of Hannover, Germany

**Kenro Furutani**, Osaka Central Advanced Mathematical Institute and Osaka Metropolitan University, Japan

**Erlend Grong**, University of Bergen, Norway

**Irina Markina**, University of Bergen, Norway

Geometric Analysis is a beautiful synthesis of two classical areas of mathematics where methods from both fields can be used. One can say that the analytic tools, historically developed for Euclidean space, are used to study smoothly curved spaces. The influence of geometrical properties of manifolds, such as curvature bounds or volume estimates, reflect the properties of differential operators. One is interested, for instance, in the study of soliton-like solutions arising from geometrical flows, in the analysis of nonlinear wave equations via geometric approaches and mathematical relativity, in the revealing of analytic and geometric properties of smooth and non-smooth spaces satisfying Ricci type curvature bounds. Reformulations of geometric problems as variational problems often allow to establish links between their solutions and the underlying geometry.

Another interesting interplay between geometry and analysis arises from a control theoretical approach to geometry with non-holonomic constrains. Sub-Riemannian manifolds, originated as an example of the Carnot-Caratheodory geometry in studies of M. Gromov of limits of metric spaces, got a boost when the main questions were reformulated in a control theoretical language. Thus existence, regularity of geodesics, minimal surfaces and other extremal problems get satisfactory answers. The study of suitable notions of curvatures on sub-Riemannian manifolds, where the unique Levi-Civita connection is absent, attracted new ideas from Cartan geometry, allowing to use a Lie theoretical approach.

An analysis on metric spaces requires refined instruments and rises such questions as, for instance, how to define the differential structures on a space where any kind of natural linear or coordinate system is absent. What is the “natural” measure and it properties. What is the relation between this “natural” measure and other structures, such as group multiplication or manifold structure, if present. Study of mappings, such as isometries, conformal, symplectic maps and their numerous generalizations on geometric structures is also in the fields of interest of the session on Geometric Analysis.

The main purpose of this session is an exchange of ideas and the promotion of recent results. We aim to provide a platform for enriching the classical areas Geometry and Analysis by new approaches and tools that could lead to novel concepts and breakthroughs in open problems.

**Geometric and topological methods in computer science**

**Dmitry Feichtner-Kozlov**, Universität Bremen, Germany

**Éric Goubault**, Ecole Polytechnique, Institut Polytechnique de Paris, France

**Uli Fahrenberg,**, EPITA, France

**Martin Raussen**, Aalborg University, Denmark

The last three decades have witnessed a fruitful interplay between certain qualitative ideas and methods from geometry and topology on one side and aspects of theoretical computer science on the other.

In this approach, classical questions from the theory of computing, such as feasibility and complexity, could be reformulated using novel topological models. Simplicial and cubical models in distributed computing and in concurrency theory are amenable to investigations using methods from algebraic topology, however, oftentimes with a twist!

Similar developments have emerged originating in logic and rewriting, mainly through the lens of higher category theory and/or connected to homotopy type theory.

This session aims to foster the interaction between several research communities, that otherwise lack a natural venue for exchange, with the goal of letting various approaches stimulate each other.

**Geometric measure theory and nonlinear PDE**

**Katrin Fässler**, University of Jyväskylä, Finland

**Vesa Julin**, University of Jyväskylä, Finland

**Andrea Pinamonti**, Università di Trento, Italy

**Giorgio Saracco**, Università di Trento, Italy

The session aims at bringing together experts and young researchers working in geometric measure theory and nonlinear partial differential equations. The focus lies on connections between the two subjects and other areas prominently represented in the Nordic mathematical community, such as harmonic analysis and analysis in metric spaces. A non-exhaustive list of topics includes questions related to (uniform) rectifiability, free boundary and free interface problems.

The theory of rectifiability is a central topic in geometric measure theory that has been substantially advanced in the recent years. Rectifiable sets exhibit properties similar to those of smooth curves and surfaces, which makes them well suited for the study of geometric variational problems thanks to De Giorgi’s celebrated structure theorem for sets of locally finite perimeter. A quantitative version of rectifiability, known as ‘uniform rectifiability’, involves a multi-scale analysis of the regularity properties of a given set. This theory of uniform rectifiability is closely related with singular integrals and harmonic analysis. It has been quintessential in the study of quasi- and almost minimizers, as well as for the Dirichlet problem of different PDEs in non-smooth domains with rough boundary data. The passage from properties of a Green’s function to uniform rectifiability of the boundary can be seen as a free boundary problem. The session will focus on this and other links between geometric measure theory and partial differential equations.

**Geometry**

**Francisco Martín**, University of Granada, Spain

**Steen Markvorsen**, Danish Technical University, Lyngby, Denmark

**Niels Martin Møller**, University of Copenhagen, Denmark

Differential geometry studies shapes and objects combining calculus and geometry. It focuses on how objects change and behave as they are stretched, bent, and twisted.

Imagine you have a rubber sheet that you can stretch and bend in different ways. Differential geometry studies how points on the sheet move and change relative to each other as the sheet is deformed. It also looks at how the curvature and shape of the sheet are affected by these changes.

In more technical terms, differential geometry uses mathematical tools like vectors, tensors, and differential equations to describe and analyze geometric objects such as curves, surfaces, and higher-dimensional shapes. It involves studying properties like curvature, length, area, volume, and topology, as well as understanding how these properties change as the object is deformed or transformed.

Differential geometry has applications in various fields, including physics, computer graphics, robotics, and even in our everyday lives. For example, it is used in designing car engines, modeling the behavior of light in lenses, and understanding the shape of Earth’s surface. It provides a powerful framework for understanding the geometry of the world we live in and how it changes under different conditions.

**Graph theory**

**Michael Stiebitz**, Technical University of Ilmenau, Germany

**Bjarne Toft**, University of Southern Denmark, Denmark

Graph theory emerged as a subject in its own right at the end of the 1800s, with Danish Julius Petersen’s paper Die Theorie der regulären graphs in Acta Mathematica 1891 as the first graph theory paper dealing with problems motivated by mathematics itself, rather than by outside applications. After a slow start the subject took off in the 1900s, with Dénes König’s monograph in 1936, contributions by at first in particular Hungarian and North-American mathematicians (Erdös, Turan, Gallai, Renyi, and Birkhoff, Whitney, Tutte, Fulkerson, Edmonds), and lateras an important part of theoretical computer science. Today graph theory is a fully fledged part of modern mathematics worldwide, with strong links to topology, algebra, probability theory, game theory, algorithms and complexity theory, and with well-developed theories in areas like coloring, cycles, matching, factorization, extremal theory and graph minors. Some mathematicians working in the area havereceived the Abel Price (Szemerédi, Lovász, Wigderson), the Turing Award (Karp, Tarjan, Hopcroft) and the Nobel Prize in Economics (Shapley).

**Harmonic analysis and representation theory**

**Jan Frahm**, Aarhus University, Denmark

**Michael Pevzner**, Université de Reims Champagne–Ardenne, France

**Genkai Zhang**, Chalmers University of Technology and University of Gothenburg, Sweden.

This session combines topics from non-commutative harmonic analysis and representation theory of reductive groups over local and global fields.

One of its key themes will be the decomposition of spaces of functions on a geometric object under the action of a group, generalizing the classical theory of Fourier series for the circle group and the Fourier transform for the additive group of real numbers. The groups under consideration are mostly reductive groups such as GL(n) defined over the fields of real, complex or p-adic numbers. Their representation theory is intimately related to the harmonic analysis on spaces on which the groups act, and their irreducible representations can be thought of as building blocks in the decomposition of functions.

Several talks will focus on the harmonic analysis on (locally) symmetric spaces and spherical varieties. These topics are related to various integral transformations such as Fourier, Poisson or Penrose transformations, as well as differential operators invariant under the action of the relevant groups. The case of locally symmetric spaces builds a bridge to analytic number theory through the study of automorphic forms and representations.

Another theme concerns branching laws, i.e. the study of restrictions of irreducible representations to subgroups. In the case of representations realized on function spaces, this problem is often intimately related to harmonic analysis. Particular cases of branching problems that have attracted a lot of attention over the last few decades are the Gan-Gross-Prasad conjectures, Plancherel formulæ, the dual pair correspondence as well as analysis of symmetry breaking operators.

The session will also feature more abstract representation theoretic topics such as Dirac cohomology and applications to geometric quantization.

**History of mathematics with a focus on Scandinavian mathematics**

**Jesper Lützen**, University of Copenhagen, Denmark

**Reinhard Siegmund-Schultze**, University of Agder, Kristiansand, Norway

This session presents results of recent research into the history of mathematics, overwhelmingly by historians with connection to Scandinavian countries, and with one focus on the history of Scandinavian mathematics.

Topics range from early Renaissance mathematics (Fibonacci) through 17^{th} century’s construction of conic sections and 19^{th} century mathematical education of a Russian woman (Kovalevskaya). Another focus of the historical session is the 20^{th} century. This includes social and disciplinary history of mathematics, applications of mathematics, and political conditions for mathematics. Among this figure the so-called geometry of reality (Hjelmslev) mathematical biology (Nicolas Rashevsky), mathematical congresses (both Scandinavian, 1913, and the first ICM on Scandinavian soil in Oslo 1936), Scandinavian mathematical journals, emigration of Jewish Italian mathematicians under Mussolini, and international planning of Post-World War II mathematics.

**Homological algebra**

**Martin Herschend**, Uppsala University, Sweden

**Gustavo Jasso**, Lund University, Sweden

**Peter Jørgensen**, Aarhus University, Denmark

**Pierre-Guy Plamondon**, Université Paris-Saclay, France

Homological algebra grew out of algebraic topology in the 1950s through the work of Cartan and Eilenberg. It is an independent branch of pure mathematics which has had a profound impact on numerous other areas, not least on geometry through the work of Grothendieck and Serre.

Work by Auslander and Reiten and later by Iyama showed that homological algebra has an extensive interface with the representation theory of algebras, in particular the resulting combinatorics. This has led to cluster tilting theory and higher dimensional Auslander-Reiten theory.

Parallel developments have seen homological algebra intertwined with the notion of categorification as found, for instance, in Kontsevich’s programme for homological mirror symmetry. All these are key areas of contemporary, international research in homological algebra. This session will draw on the expertise of Nordic and international researchers to promote new interactions between these areas and to introduce the next generation to the powerful methods of homology.

**Invariants of manifolds from quantum field theory and string theory**

**Nezhla Aghaei**, University of Southern Denmark and University of Geneva, Switzerland

**Jørgen Ellegaard Andersen**, University of Southern Denmark, Denmark

**Sergei Gukov**, California Institute of Technology, Pasadena, Californina, USA and Dublin Institute for

Advanced Studies, Ireland

**Du Pei**, University of Southern Denmark, Denmark

Quantum field theory and string theory are theoretical frameworks invented to describe the elementary forces in nature, but the rich mathematical structures that they encodes have made a significant impact on modern mathematics. They are especially useful for uncovering hidden algebraic structures in geometry and topology. The study of them on topologically non-trivial manifolds has led to the discovery of many interesting new topological invariants such as the Witten–Reshetikhin–Turaev (WRT) invariants for 3-manifolds and Seiberg–Witten invariant for 4-manifolds. These new invariants and the algebraic structures behind them are not only interesting from a purely mathematical point of view, but can also be used to study quantum phases of matter and have potential applications to quantum computation. Recent developments indicate that 6-dimensional quantum field theories can be used to unify many previously known invariants, generalizing them as well as discovering new ones. For instance, it was proposed that the “Z-hat invariant” for 3-manifolds “categorifies” the WRT invariant and that there are invariants for 4-manifolds that take values in the ring of “topological modular forms (TMF)” refining the Seiberg–Witten invariants. This session brings together mathematicians and physicists working on this subject to discuss recent progress in this rapidly evolving field.

**Inverse problems**

**Henrik Garde**, Aarhus University, Denmark

**Kim Knudsen**, Technical University of Denmark, Lyngby, Denmark

**Michael Vogelius**, Rutgers University, New Jersey, USA

This session is on the latest advances in the mathematics of inverse problems, e.g. related to inverse coefficient problems from partial differential equations and in image analysis. These problems typically originate from applications in mathematical physics, medical imaging, or industrial tomography. The main topics include unique identifiability, stability estimates, and reconstruction algorithms using techniques coming from applied mathematical analysis, differential geometry, and computational mathematics.

**Mathematics applied to solid state physics**

**Horia Cornean**, Aalborg University, Denmark

**Massimo Moscolari**, University of Tübingen, Germany

**Jacob Schach Møller**, Aarhus University, Denmark

In recent years there have been several advances in the description of topologically ordered phases of matter. In particular, symmetry-protected, disordered, (non)interacting systems have been intensively studied. The definition of new topological indices, their classification and the transport properties of these systems have involved the development of new techniques from functional analysis, spectral and scattering theory for Schrödinger and Dirac operators, to K-theory passing by quantum field theory. Another important mathematical challenge is the rigorous description of the interaction between particles and radiation fields or phonons in crystals. Last but not least, the good old spectral analysis of resonances and bounded states at thresholds is a continuous source of challenging questions.

**Mathematics of communications**

**Oliver W. Gnilke**, Aalborg University, Denmark

**Marcus Greferath**, University College Dublin, Ireland

**Mario Pavčević**, University of Zagreb, Croatia

**Jens Zumbrägel**, Universität Passau, Germany

Problems in communications have initiated and inspired several areas in mathematics. Amongst others coding theory, cryptology, information theory, combinatorics, especially design theory, and number theory. The unifying thread are the applications to reliability, security, privacy or similar aspects of communications. The invited speakers of this session cover a variety of topics related to Mathematics of Communications. We hope that the different backgrounds and areas of expertise will enable a fruitful discussion among the participants.

**Multiscale analysis of multi-physics systems**

**Anastasiia Krushynska**, Groningen University, the Netherlands

**Matthias Liero**, Weierstrass Institute, Berlin, Germany

**Adrian Muntean**, Karlstad University, Sweden

**Grigor Nika**, Karlstad University, Sweden

The mini-symposium is about discussing recent developments in mathematical methods for solving partial differential equations and accurately approximating real-world problems involving multiple scales and multiple physics. These systems are important because they are used in engineering applications as sensors and actuators. Specifically, researchers are interested in developing materials with controlled microstructures like magnetorheological elastomers, electro-active polymers, and others. The session will focus on analyzing these complex systems, including questions related to their solvability, as well as techniques for simplifying them without sacrificing important information. The goal is to bring together mathematicians and mechanics experts who are interested in multiscale, multi-physics systems.

**Nonlinear dispersive equations**

**Mats Ehrnström**, Norwegian University of Science and Technology, Norway

**Fredrik Hildrum**, Norwegian University of Science and Technology, Norway

**Dag Nilsson**, Universität des Saarlandes, Germany

The theory of evolutionary nonlinear dispersive differential equations and fluid flows is of fundamental importance in the analysis of mathematical physics. Both the 18th century-old hydrodynamics’ equations by Euler and many associated modelling equations for water waves and other phenomena have been subject to intense research, and has had a profound effect on the development of mathematics itself. Today, several centuries later, the properties of solutions to dispersive equations are still largely unsettled, but recent breakthroughs have made it possible to address difficult questions. In this regard, both the study of travelling waves, that is, solutions travelling with fixed shape and speed, and solutions influenced by vorticity have attracted great attention. The modern focus also consider the intrinsic nonlocal features in the equations, often paving the way for singularities and break-downs of the solutions. To accomplish this, the mathematical techniques blend a rich array of tools, ranging from optimisation and variational principles to bifurcation analysis and topological arguments. By bringing together experts and young researchers alike, this session seeks to provide fruitful discussions on advances in the theory of nonlinear dispersive equations and fluid flows. In particular, topics related to singularity formations and blow-up phenomena in travelling waves, along with existence theory, regularity issues, and stability and modelling properties of dispersive equations will be addressed.

**Number theory**

**Sigrid Grepstadt**, Norwegian University of Science and Technology, Norway

**Simon Kristensen**, Aarhus University, Denmark

**Ulf Kühn**, Universität Hamburg, Germany

**Pär Kurlberg**, KTH, Stockholm, Sweden

**Tapani Matala-aho**, Aalto University, Finland

This session is devoted to all aspects of number theory.

**Progress and challenges of quantum computing in its NISQ era**

**Jørgen Ellegaard Andersen**, University of Southern Denmark, Denmark

**Peter van Loock**, Mainz University, Germany

**Damian Markham**, CNRS, Sorbonne Université, France

**Shan Shan**, University of Southern Denmark, Denmark

Quantum computing (QC) was first proposed in 1982 by Richard Feynman to simulate quantum physical systems. Interests in quantum computers have spurred in the 90s ever since Schor’s discovery of the quantum factoring algorithm; it was the first algorithm to give an exponential speedup over classical methods for a well-known and interesting problem. Other noteworthy works during that period include Grover’s search algorithm, Quantum Fourier Transform, Hidden subgroup problem, and the HHL algorithm for linear algebra. The first quantum error correction codes (Calderbank-Shor-Steane codes) were also developed during that time (mid 90s), which dramatically lowered the effective error rates of quantum computers.

However, without available hardware, QC remained a theoretical interest until recently. In the “noisy intermediate-scale quantum” (NISQ) era, we began to see the rapid development of trapped ions, superconducting circuits, photonic approaches and quantum annealing technologies for the creation of quantum computers. New challenges have arisen in the NISQ era, such as error correction, scalability, quantum algorithms and integration with classical computers. This opens up a vast space of research opportunities and challenges to mathematicians of diverse fields, such as algebraic topology, algebraic geometry, coding theory, differential geometry, differential topology, statistics, group theory, number theory, numerical analysis, mathematical physics, complexity theory, etc. This proposal aims to demonstrate recent progress in the mathematics of quantum computing and encourage collaboration among quantum researchers in the Nordic region.

**Qualitative analysis of dynamical system**

**Peter Giesl**, University of Sussex, UK

**Sigurdur Freyr Hafstein**, University of Iceland, Iceland

**Stefan Suhr**, Ruhr University Bochum, Germany

Dynamical Systems are of essential importance in science and engineering and are a very active research topic in both applications and theory. In the last decades a plethora of numerical methods have been designed for their qualitative analysis (what is going to happen) together with theoretical tools for the development of such methods. We will call for talks concerned with the qualitative analysis of continuous- and discrete-time dynamical systems, both deterministic and stochastic.

**Quantum Chern-Simons theory**

**Jørgen Ellegaard Andersen**, University of Southern Denmark, Denmark

**François Costantino**, Université de Toulouse III Paul Sabatier, France

**William Elbæk Mistegård**, University of Southern Denmark, Denmark

**Konstantin Wernli**, University of Southern Denmark, Denmark

Using ideas from physics, Witten envisioned that quantum Chern-Simons theory form a three-dimensional topological quantum field theory (TQFT). Mathematically, a TQFT was defined by Atiyah as a monoidal functor from the category of three-dimensional cobordisms to the category of vector spaces. Subsequently Reshetikhin and Turaev used surgery presentations of three-manifolds and representation theory of quantum groups (or more generally: modular tensor categories) to mathematically construct a three-dimensional TQFT. This is now called the Witten-Reshetikhin-Turaev (WRT) TQFT, and is a mathematical model of quantum Chern-Simons theory. Equivalently, the two-dimensional part of the TQFT can be realized by quantization of moduli spaces of flat connections on two-manifolds, or by means of conformal field theory. The WRTTQFT gives topological invariants of three-manifolds containing a link, called quantum invariants. These extend the famous Jones polynomial of links in the three-sphere. On the other hand, studying the perturbative formulation of quantum Chern-Simons theory, one also obtains various three-manifold invariants, e.g. as the Le-Murakami-Ohtsuki, Kontsevich-Kuperber-Thurston-Lescop and Axelrod-Singer invariants, and knot invariants such the Bott-Taubes invariants and the Kontsevich integral.

The field of quantum Chern-Simons theory is a broad and active field of research with connections to low-dimensional topology, knot theory, higher category theory, mathematical physics, algebraic geometry and representation theory of quantum groups.

**Representation theory and combinatorics**

**Karin Baur**, University of Leeds, UK

**Sira Gratz**, Aarhus University, Denmark

**Karin Jacobsen**, Norwegian University of Science and Technology, Norway

**Peter Jørgensen**, Aarhus University, Denmark

This session is inspired by the rich interplay between combinatorics and representation theory, with a focus on finite dimensional algebras. Combinatorics and representation theory have always been deeply intertwined: Combinatorial problems naturally arise in the context of classifying representations and, on the other hand, finding representation theoretic analogues of combinatorial problems often leads to elegant solutions. This connection has been even more emphasised in recent years through cluster theory, the appearance of gentle algebras in mirror symmetry, higher algebra, and the study of wide and thick subcategories. Each of these topics is strongly represented in the Nordic countries.

This session will serve two purposes. First, to showcase recent advances in the area, with an emphasis on topics of relevance to the study of finite dimensional algebras. And secondly, to introduce state of the art advances in combinatorics to the representation theoretical audience, with a view towards future applications.

**Statistics and applications in the pharmaceutical industry**

**Claus Dethlefsen**, Statistical director, Novo Nordisk, and adjunct professor, Aalborg University, Denmark

**Luca La Rocca**, Associate professor, University of Modena and Reggio Emilia, Italy

Statistics is a discipline that involves analysis and interpretation of empirical data. The session will cover both general statistical topics and topics with closer links to the pharmaceutical industry.

The pharmaceutical industry relies on statistics for proving efficacy and safety of new treatments, during the drug development. There is a continuous interest in optimizing trial designs in order to make valid inference while exposing fewer subjects and to bring effective new treatments faster to the market. In recent years, the concept of estimands has been developed in order to make it transparent how observations are used in the statistical analysis following intercurrent events such as treatment discontinuation and change in background medication during a clinical trial.

Data types such as gene expression data where the number of variables is large compared to the number of samples may be computational demanding to analyze. The session will also present faster algorithms that may facilitate data analysis.

**Stochastic processes on random networks**

**Stein Andreas Bethuelsen**, University of Bergen, Norway

**Christian Hirsch**, Aarhus University, Denmark

**Daniel Valesin**, University of Warwick, UK

The theory of stochastic processes on networks has become one of the cornerstones of modern probability theory. This is because the prototypical models such as random walks, Ising/voter models, contact/percolation processes exhibit intimate connections to a variety of different disciplines such as mathematical physics, mathematical biology, epidemiology, materials science, and sociology. The importance of stochastic processes on networks within mathematics is highlighted by three Fields medals in the recent past (Wendelin Werner: 2006; Stanislav Smirnov: 2010; Hugo Duminil-Copin: 2022).

While working on a fixed network such as a regular lattice means that the mathematical derivations often are particularly elegant, this model assumption is often violated massively in problems in application areas. Indeed, when modeling the flow of particles through a porous medium, the spread of an infection in a social network, or the dissemination of data in a wireless ad hoc network, then it does not seem reasonable to rely on a grid for representing such complex phenomena. The need to understand the behavior of stochastic processes on inherently random networks has served as the motivation to develop new mathematical models such as random walks in random environments, or continuum percolation that are today the subject of vigorous international research activities.

**Symplectic and contact geometry and connections to physics**

**Thomas Kragh**, Uppsala University, Sweden

**Georgios Dimitroglou Rizell**, Uppsala University, Sweden

**Vivek Shende**, University of Southern Denmark, Denmark

Modern pseudoholomorphic curve invariants of symplectic and contact geometry have many deep connections to modern theoretical physics. Such invariants include symplectic field theory, open and closed Gromov-Witten invariants, Lagrangian Floer theory and Fukaya categories, and more. Understanding the algebraic structures organizing these theories has often borrowed from and sometimes motivated the latest developments of homological algebra; or more recently spectral algebra as in Floer homotopy theory. In general the subject has been developed in dialogue with ideas from theoretical physics (specially, topological string theory). Applications beyond symplectic topology and theoretical physics include invariants of smooth spaces, such as knot theory and string topology, and invariants in algebraic geometry.

At our session specifically, we have talks around:

* Computations of open and closed Gromov-Witten invariants and connections to symplectic field theory and Floer theory.

* Refinements of Gromov-Witten invariants, e.g. BPS states and skein-valued counts.

* Relations between symplectic field theory, primary and secondary algebraic operations in Floer theory, and corresponding operations in string topology.

* Relations between open and closed string theories and Floer-theoretic formulations.

* Lifts of Floer-theoretic invariants of symplectic and smooth spaces to theories that take values in stable homotopy types.

**Theory of data driven safe machine-decisions**

**Eitan Altman**, INRIA, Sophia Antipolis, France

**Maryam Kamgarpour**, EPFL, Switzerland

**Mihaly Petreczky**, Centre de Recherche en Informatique, Signal et Automatique de Lille, France

**Rafal Wisniewski**, Aalborg University, Denmark

Inquiries about the nature and structure of concepts, data- vs. model-based knowledge, have become central in theoretical control research. It is not surprising that the philosophical rivals, empiricism and rationalism have been battling for many hundred years. Despite advances in learning and success stories in social and mass media, learning is still very challenging. To name one of the unresolved problems is safe learning. Safe learning means to learn a stochastic system (the transition probability kernel, the infinitesimal generator, or the occupation measure) from data such that unsafe states are reached with a small probability. The challenge is that to learn the transition probability of all the states including the forbidden states shall be recursive, but for them to be safe they should be visit them only very seldom.

This workshop is a consequence of the transition of the CPS community from the model-based to data-based methods. From the position of having only a few high-quality sensors to the case where data is omnipresent, but often of varying quality. We strive to learn the machines and infrastructures, to work autonomously, work together and collaborate with people. This invited session is the first step of joining research forces in formalizing and subsequently solving learning for autonomous decisions, safety assessment and verification problems, stochastic control with constraints.

**The ubiquity of quivers and their representations**

**Fabian Haiden**, University of Southern Denmark, Denmark

**Bernhard Keller**, Université Paris Cité, France

**Vidit Nanda**, University of Oxford, UK

Quivers (directed multigraphs) and their linear representations were first introduced to provide a suitable language in which to formulate “linear algebra problems”, vastly generalizing the Jordan normal form. Each quiver poses a challenge: to classify all representations of that quiver up to isomorphism. The difficulty of that problem depends on the underlying graphs of the quiver and exhibits a fundamental finite/tame/wild trichotomy, with the wild case being the general one.

In modern approaches, one studies moduli spaces of quiver representations via geometric methods, in particular using algebraic geometry and geometric invariant theory, which leads to a notion of stability for quiver representations. One particularly fruitful idea has been to apply methods of Donaldson-Thomas theory to extract numerical invariants from moduli of quiver representations. This can then be used to prove certain power series identities known as wall-crossing formulas, which play a key role in quantum field theory. A related phenomenon is that of quiver mutation, a type of local modification of quivers. These occupy a central role in the theory of cluster algebras, but also appear naturally when studying quiver representations from the point of view of homological algebra. The categorification of cluster algebras unifies the two perspectives.

One more recent and very exciting application of the theory of quiver representations has been to the rapidly developing subject of topological data analysis. In order to extract meaningful topological data from real-world datasets, powerful methods of algebraic topology and representation theory are combined in a theory of persistent homology.

Quivers and their representations have proven to be a fundamental structure and a powerful tool in a wide variety of areas of math and will certainly continue to play an important role in the future. This section will bring together world experts with unique perspectives and insights on the subject.

**Topological data analysis. Theory and applications**

**Christophe Biscio**, Aalborg University, Denmark

**Wojciech Chachólski**, KTH, Stockholm, Sweden

**Lisbeth Fajstrup**, Aalborg University, Denmark

**Martina Scolamiero**, KTH, Stockholm, Sweden

In Topological Data Analysis, TDA, data is analyzed using methods developed with algebraic topology and geometry as a point of departure. A by now well-known example is the study of a set of points in Euclidean space by studying the union of balls with center in these point, letting the radius increase and keeping track of topological invariants – persistent homology is one such method.

In other examples, the variation is not through increasing radii but e.g. sublevel sets of functions, and combinations. In multidimensional persistence, several parameters are varied. Periodicity questions are addressed by looking for homology classes of dimension 1 – cycles in data.

The questions in the area are both theoretical, computational and about interpretation of results in the application of the tools; many questions are a mix of theory and application: Where in the data “is” a given homology class can be a natural question in application and leads to both theoretical questions about representations with certain properties and computational questions about how to find them. Stability is another such question: What happens if the data varies – to the combinatorial models, to the computations, and in theory – there are stability questions requiring theoretical and also practical answers.

A broad range of mathematics and statistics is involved.

**Topology**

**Alexander Berglund**, Stockholm University, Sweden

**Rune Hauseng,** Norwegian University of Science and Technology, Norway

**Nathalie Wahl**, University of Copenhagen

Topology originates in the study of properties of geometric objects that stay invariant under appropriate deformations. Nowadays, the word “geometric object” could probably be replaced by any type of mathematical object, and topology, or topological methods, have indeed made their way to many areas of mathematics, from algebraic geometry to data analysis.

The session covers topics of current research in Topology in a broad sense, spanning from geometric topology, with the study of manifolds and their symmetries, to abstract homotopy theory. The session will include neighboring areas such as geometric group theory, where groups are studies using topological methods.