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.