The class of surreal numbers was invented and first described by Conway [1]. This class has the property that it contains both the set of real numbers and the class of ordinal numbers. The operations of addition and multiplication as well as a total order relation arise naturally on the surreal numbers, making them a (linearly) ordered field (of class size). Another way of describing surreal numbers is by expressing them as generalised power series (i.e., Hahn series) via their Conway normal form. This leads to a valuation theoretic approach towards the study of surreal numbers: the ordered field of surreal numbers can be regarded as a Hahn field whose residue field is given by the real numbers and its value group by the surreal numbers. In my talk, I will first outline Conway's surprisingly simple but powerful construction of surreal numbers from first principles. Moreover, I will explain what Hahn fields are and how surreal numbers can be regarded as generalised power series. If time permits, I will report on joint work in progress with E. Kaplan and M. Serra building on the study of automorphisms of Hahn fields initiated in by Kuhlmann and Serra [2]. [1] J. H. Conway, On numbers and games, Lond. Math. Soc. Monogr. 6 (Academic Press, London, 1976). [2] S. Kuhlmann and M. Serra, The automorphism group of a valued field of generalised power series, J. Algebra 605 (2022) 339–376.

We consider a thin elastic sheet with two disclinations in a variational framework, in the so-called von-Karman approximation. Under the non-physical assumption that the out-of-plane deplacement is a convex function, we prove that minimizers display a sharp ridge between the two disclinations. Furthermore we prove the associated energy scaling law (with upper and lower bounds that match up to logarithmic factors in the thickness of the sheet). We note that our result can also be read as a stability result for certain degenerate solutions of the two-dimensional Monge-Ampère equation. Joint work with Peter Gladbach (Bonn)

Die Durchführung von inklusivem Mathematikunterricht in heterogenen Lerngruppen, in denen Kinder mit dem Förderschwerpunkt Geistige Entwicklung unterrichtet werden, stellt Lehrkräfte vor Herausforderungen. Die Lernenden zeigen große Unterschiede im mathematischen Vorwissen und den allgemeinen Lernvoraussetzungen, wodurch gemeinsame Lerngelegenheiten kaum realisierbar scheinen. Im Vortrag wird zunächst auf die Besonderheiten und Meilensteine der mathematischen Entwicklung von Lernenden mit intellektueller Beeinträchtigung eingegangen. Anschließend wird eine Konzeption für den inklusiven Mathematikunterricht vorgestellt, die sowohl gemeinsames Lernen als auch individuelle Förderung ermöglichen soll. Die Konzeption wird anhand eines exemplarischen Verlaufs einer inklusiven Mathematikstunde verdeutlicht und Ergebnisse der Studie, in der die Konzeption erprobt wurde, werdendiskutiert.

Ternärkörper sind als Koordinatenstrukturen projektiver Ebenen eingeführt worden. Sie stellen somit eine Verallgemeinerung des Körperbegriffes dar (welche bekanntermaßen die desarguesschen bzw. im kommutativen Fall die pappusschen projektiven Ebenen koordinatisieren), bringen aber deutlich abgeschwächte algebraische Eigenschaften mit sich. Für Ternärkörper ist von Kalhoff ein Bewertungsbegriff eingeführt worden, welcher viele bekannte Aussagen über bewertete Körper auf solche uniform bewertete Ternärkörper verallgemeinert; so induzieren diese etwa in gewohnter Weise eine Topologie auf dem Ternärkörper, mit welcher dieser einen topologischen Ternärkörper bildet. In diesem Vortrag soll ein Ergebnis vorgestellt werden, welche diese von uniformen Bewertungen stammenden Topologien auf Ternärkörpern charakterisiert und die bekannte Klassifizierung der Topologien bewerteter (kommutativer) Körper von Kowalsky und Dürbaum erweitert.

We develop a novel trust-region method to minimize the sum of a smooth nonconvex function and a nonsmooth convex function. This class of problems that is ubiquitous in data science, learning, optimal control, and inverse problems. Our method is unique in that it permits and systematically controls the use of inexact objective function and derivative evaluations inherent in large-scale system solves and compression techniques, e.g. randomized sketching. When using a quadratic Taylor model for the trust-region subproblem, our algorithm is an inexact, matrix-free proximal Newton-type method that permits indefinite Hessians. We prove global convergence of our method in Hilbert space and elaborate on potential nonsmooth subproblem solvers based on ideas taken from their smooth counter-parts. Under additional assumptions, we can also prove superlinear, or even quadratic convergence to local minima. We demonstrate its efficacy on examples from data science and PDE-constrained optimization.

A mapping of nonzero topological degree from the boundary of a disk to the circle cannot be extended by a continuous mapping defined on the whole disk to the circle as homotopy theory asserts. However, this extension is possible if one allows the extended mapping to have discontinuous points, also called singularities. Geometrical and physical situations that we will describe motivate the problem of finding the 'best' extension possible in the sense of minimization of the p-Dirichlet energy when p=2. However, this is only possible when p<2 for reasons we will explain. We will describe the limiting behavior when p goes to 2 of the minimizers as well as the convergence of the singularities they carry. We will also take the opportunity to raise the question for the 3D variant as well as other manifolds than the circle.

The celebrated 15 theorem of Conway and Schneeberger says that a classical positive definite quadratic form over Z is universal if it represents each element of the set {1,2,3,5,6,7,10,14,15}. Moreover, this is the minimal set with this property. In 2005, B.M. Kim, M.-H. Kim, B.-K. Oh showed that such a finite criterion set exists in a much general setting, but the uniqueness of the criterion set is lost – and the question of uniqueness for some particular situation has been studied by several authors since. We will discuss the analogous questions for totally positive definite quadratic forms over totally real number fields. There, the existence of criterion sets for universality has been known, and Lee determined the set for Q(√5). We will show the uniqueness and a strong connection with indecomposable integers. A part of our uniqueness result is (to our best knowledge) new even over Z. This is joint work with G. Romeo and V. Kala.

In biomechanics, local phenomena, such as tissue perfusion, are strictly related to the global features of the whole blood circulation. We propose a heterogeneous model where a local, accurate, 3D description of tissue perfusion by means of poroelastic equations is coupled with a systemic 0D lumped model of the remainder of the circulation. This represents a multiscale strategy, which couples an initial boundary value problem to be used in a specific tissue region with an initial value problem in the rest of the circulatory system. We discuss well-posedness analysis for this multiscale model, as well as solution methods focused on a detailed comparison between functional iterations and an energy-based operator splitting method and how they handle the interface conditions.

Poroelasticity is the modeling of coupled mechanics and flow in porous media. The Biot equations are one of the most popular poroelastic models and their approximation has attracted a growing attention in recent years. The talk aims at introducing the model and pointing out the main challenges posed by its approximation, with an emphasis on the interplay of analysis and numerics. Then, some new results and open issues will be discussed. This is a joint work with C. Kreuzer.

In recent years, the main challenges in subsurface energy systems (e.g., enhanced geothermal systems and CO2 sequestration) have included issues arising from the multi-physical and multi-scale nature of the problem, as well as uncertainty quantification. Multi-physics involves coupling solid mechanics, fluid mechanics, thermal energy, and chemical reactions, while multi-scale considerations involve relating pore-scale problems to field-scale problems. These problems are complex and require interdisciplinary efforts to achieve meaningful outcomes in research. In this talk, we will focus on the challenges of the multi-physical formulations for solving subsurface applications, and discuss new enriched Galerkin (EG) finite element methods for coupled flow and transport and poro-elasticity systems. The primary goal of the study is to develop computationally efficient and robust numerical methods that are free from oscillations due to a lack of local conservation, maximum principle violations, int-sup issues, and lockingeffects. The locally conservative enriched Galerkin (LC-EG) method, which will be used to solve the flow problem, is constructed by adding a constant function to each element based on the classical continuous Galerkin methods (CG). The locking-free enriched Galerkin (LF-EG) method adds a piecewise linear vector to the displacement space. These EG methods incorporate well-known discontinuous Galerkin (DG) techniques but use approximation spaces with fewer degrees of freedom than typical DG methods, offering an efficient alternative. We will present a priori error estimates of optimal order and demonstrate, through numerical examples, that the new method is free from oscillations and locking.

The generalized Serre question asks whether algebraic vector bundles over topologically contractible smooth affine complex varieties are always trivial or not. This talk gives an overview of the classical results on vector bundles over topologically contractible smooth affine complex varieties and discusses the recent developments on the generalized Serre question.

Die Entwicklung flexiblen Rechnens gilt seit vielen Jahren als zentrales Ziel des Arithmetikunterrichts der Grundschule. Empirische Untersuchungen belegen allerdings, dass sich Flexibilität nicht automatisch zu entwickeln scheint, sondern gezielt gefördert werden muss. Deshalb haben wir im Rahmen meines Dissertationsprojekts – aufbauend auf den Ergebnissen anderer Studien – eine Unterrichtskonzeption zur kontinuierlichen Förderung von Flexibilität vom ersten bis zum vierten Schuljahr entwickelt und umgesetzt. Parallel wurden die Vorgehensweisen der Lernenden beim Addieren und Subtrahieren sieben Mal im Projektzeitraum in leitfadengestützten Einzelinterviews erfasst. Mittels qualitativer Inhaltsanalyse mit darauf folgender Fallkontrastierung und Typenbildung ließen sich dann charakteristische Entwicklungen im Grundschulverlauf rekonstruieren. Im Vortrag werden zentrale Ergebnisse dieser Studie vorgestellt und diskutiert.

Im Kontrast zur derzeitig dominanten Wahrnehmung der Mathematik als 'Schlüsseltechnologie' wird der Vortrag nach dem bildenden Wert der Mathematik jenseits ihrer direkten Anwendbarkeit fragen. Dabei werden u.a. drei Positionen skizziert: Platon als Ausgangspunkt der Überlegungen zur Mathematik im Bildungsgang und zugleich als Höhepunkt der antiken Sicht, Nikolaus Cusanus als Protagonist des christlichen Mittelalters und zugleich als Türöffner der Moderne und schließlich als klassische Repräsentanten der Modernen Mathematik, Karl Weierstraß, David Hilbert und Hermann Weyl.

Erklärvideos gewinnen für den Fachunterricht zunehmend an Bedeutung: Lehrkräfte verwenden Videos für Formate wie den „Flipped Classroom“, aber Schülerinnen und Schüler schauen sich Erklärvideos auch aus eigenem Antrieb an, z.B. mit dem Ziel der Prüfungsvorbereitung auf partizipativen Videoplattformen. Es gibt mittlerweile auch verschiedene Studien, die sich mit Merkmalen, die bei Erklärvideos die Instruktionsqualität fördern, auseinandersetzen und zunehmend steht die Einbettung von Erklärvideos in Lernprozesse im Fokus. In diesem Vortrag soll ein Überblick gegeben werden über Arbeiten aus der Physikdidaktik, die sich mit Erklärvideos befasst haben. Die fachübergreifenden Erkenntnisse sollen dabei herausgearbeitet werden. Im Fokus stehen dabei, wie Lernende nachhaltig Wissen aus Erklärvideos erwerben können und wie Verstehensillusionen als unintendierte Begleitvariable von Erklärvideos minimiert werden können.

Explicit constructions of models of the theory of a valued field are useful tools for understanding its model theory. Since Kaplansky’s work, it has been a topic of interest to characterize value fields in terms of fields of power series. In particular, Kaplansky proved that, under certain assumptions, an equicharacteristic valued field is isomorphic to a Hahn field. In this talk, we show that in the mixed characteristic case, assuming the Continuum Hypothesis, we can provide a characterization, in terms of power series, of pseudo-complete finitely ramified valued fields with a fixed residue field k and valued in a Z-group G, using a Hahn-like construction with coefficients in a finite extension of the Cohen field C(k) of k. In this construction, the elements of the field are “twisted” power series, i.e. powers series whose product is defined by having an extra factor, given by the cross-section and a 2-cocycle determined via the value group. This generalizes a result by Ax and Kochen, who characterize pseudo-complete valued fields elementarily equivalent to the field of p-adic numbers Q_p. If time permits, we will see some consequences of this characterization regarding the problem of lifting automorphisms of the residue field and the value group to automorphisms of the valued field in the mixed characteristic case.

The subject of my currently open doctoral thesis is Witt groups of orders, i.e. Noetherian rings of dimension 1, which are subrings of the Dedekind ring. The Dedekind ring is called the maximal order and for it the Witt ring have been well defined. Under the supervision of Prof. Paweł Gładki, we are working on orders that are not maximal - but we are trying to combine the methods used for the Dedekind ring with those used for example by Paul Balmer or Stefan Gille. The short lecture will be about recent results that deal with the natural homomorphism of Witt groups. More or less known examples of injective and non-injective natural homomorphisms will be shown, as well as recent results that were presented, for example, a year ago in Sopron (Hungary) at the 25th Central European Number Theory Conference. Proposed literature: 1. Kazimierz Szymiczek - Witt ring of a commutative ring 2. Marzena Ciemała, Kazimierz Szymiczek - On injectivity of natural homomorphisms of Witt rings 3. Beata Rothkegel - Witt functor of a quadratic order 4. Paweł Gładki, Mateusz Pulikowski - Natural homomorphism of Witt rings of a certain cubic order

Im Zentrum des Vortrags steht die kooperative Unterrichtsentwicklung von Lehrkräften. Hierfür wird zunächst auf Forschungsbefunde und Ansätze zur kooperativen Unterrichtsentwicklung eingegangen. Die Zusammenarbeit in Form einer Professionellen Lerngemeinschaft stellt dabei einen Schwerpunkt dar. Anschließend wird das Projekt POLKA vorgestellt, das darauf abzielt, Lehrkräfte darin zu unterstützen, ihren Unterricht in Professionellen Lerngemeinschaften kriterial zu analysieren und weiterzuentwickeln. Im POLKA-Projekt kooperieren die Lehrkräfte nicht zu beliebigen Themen, sondern es werden spezifische Merkmale der Unterrichtsqualität fokussiert, für die die Unterrichtsforschung positive Wirkungen auf das Lernen von Kindern und Jugendlichen nachweisen konnte. Obwohl POLKA bisher nur in einem kleinen Rahmen untersucht wurde, sind die bislang vorliegenden Befunde ermutigend. Nach dem Vortrag wird ein Gespräch darüber eröffnet, welche Anregungen sich für die kooperative, fachspezifische Unterrichtsentwicklung ergeben.

Sei R ein semilokaler regulären Ring und U des punktierte Spektrum von R, d.h. des Spektrum ohne die maximalen Ideale. In einer gemeinsamen Arbeit mit Ivan Panin haben wir kürzlich die Wittgruppe W(U) berechnet. Darüber soll im zweiten Teil des Vortrags berichtet werden, nach einer kurzen Einführung in Knebusch's Theorie der Wittgruppen von Schemata.

The u-invariant of a field is the largest dimension of an anisotropic quadratic torsion form over the field. This field invariant was introduced by I. Kaplansky in 1953 and addapted to the study of real fields by R. Elman and T.Y. Lam in 1973. In 2009, D. Harbater, J. Hartmann, and D. Krashen obtained a bound on the u-invariant for nonreal function fields in one variable over a complete discretely valued field. In joint work with N. Daans and V. Mehmeti, we obtain a more general version of this result, namely for function fields of curves over a henselian valued field with arbitrary value group.

Eine gemeinsame Arbeit mit Ivan Soprunov, https://arxiv.org/abs/2112.13128

We address an error analysis of an Eulerian finite element method employed to solve a linearized Navier-Stokes problem posed in a time-dependent domain. In this study, we assume the known evolution of the domain, independent of the solution to the problem at hand. The numerical method under consideration combines a standard BDF-type time-stepping procedure with a geometrically unfitted finite element discretization technique. Additionally, we employ Nitsche's method to enforce the boundary conditions.
The main focus of this presentation is a convergence analysis of the method for various inf-sup stable velocity--pressure elements.
The analysis provides an optimal order convergence estimate in the energy norm for the velocity component and a scaled $L^2(H^1)$-type norm for the pressure component.

Als Fortsetzung an die Charakterisierung topologischer Ternärkörper, deren Topologie von einer uniformen Bewertung induziert wird, werden in diesem Vortrag 'uniforme Absolutwerte' auf Ternärkörpern (in Analogie zum klassischen Fall der Absolutwerte kommutativer Körper) eingeführt und ihre induzierten Topologien charakterisiert. Die relativ ertraglose Suche nach Beispielen echter Ternärkörper mit einem archimedischen, uniformen Absolutwert lässt allerdings wenig Hoffnung, hierbei 'neue' topologische Ternärkörper abgedeckt zu haben, sodass im Anschluss einige Kriterien über die Nicht-Existenz solcher Ternärkörper behandelt werden.

Quantum computing (QC) is an emerging compute technology that has the potential to radically change the way we will be solving computational problems in the future. The potential power of QC stems from the exploitation of quantum mechanical principles, namely, superposition of states, quantum entanglement and quantum parallelism, which makes it possible to solve certain types of computational problems up to exponentially faster or with exponentially less memory storage. In this talk we will briefly discuss the basic concepts of QC and shed some light on the commonalities and differences of the two different paradigms - analog and digital quantum computing. We will give a brief overview of applications that might benefit from using the one or the other paradigm. One of these applications that we will discuss in more detail comes from the field of computational fluid dynamics (CFD). With today's tools and technologies, the fully-resolved direct numerical simulation of large-scale problems such as entire aircrafts or wind-turbine farms is impossible due to excessive compute times and prohibitively high memory requirements. This is the main motivation for our ongoing research on quantum-CFD algorithms. We will outline the working principles of our quantum lattice Boltzmann method and show numerical results that we obtained with high-performance QC simulators.

In this talk I will present a method that enforces bound-preservation (at the degrees of freedom) of the discrete solution. The method is built by first defining an algebraic projection onto the convex closed set of finite element functions that satisfy the bounds given by the solution of the PDE. Then, this projection is hardwired into the definition of the method by writing a discrete problem posed for this projected part of the solution. Since this process is done independently of the shape of the basis functions, and no result on the resulting finite element matrix is used, this process guarantees bound-preservation independently of the underlying mesh. The core of the talk will be devoted to explaining the main idea in the context of linear (and nonlinear) reaction-diffusion equations. Then, I will explain the main difficulties encountered when extending this method to convection-diffusion equations, both in the steady-state and time-dependent cases. The results in this talk have been carried out in collaboration with Abdolreza Amiri (Strathclyde, UK), Emmanuil Geourgoulis (Heriot-Watt, UK and Athens, Greece), Tristan Pryer (Bath, UK), and Andreas Veeser (Milan, Italy).

We describe the construction of frames where the coefficients are based on Fourier coefficients of a function, and characterize local smoothness of the function on different intervals. The ideas are used for blind source signal separation problems, chirp detection in particular. We present some results treating the classification problems in machine learning as a generalization of the signal separation problem.

Interesting combinatorial structures can often be characterised as special subsets of association schemes. In his PhD thesis, Philippe Delsarte developed powerful linear programming techniques to prove non-existence and uniqueness results for such structures. Ideas of this type were fundamental in the work for which Maryna Viazovska was awarded the Fields medal in 2022. This talk will begin with an overview of Delsarte theory. The association scheme of the symmetric group is well known. We will introduce it, and then study generalised permutations. They act on the set {1,2,...,n}, whose elements are coloured with one of r possible colours. We consider different notions of transitivity and interpret these algebraically in the appropriate association scheme . We will give existence results showing that there exist transitive sets of generalised permutations that are small compared to the size of the group. Many of these results extend results previously known for the symmetric group. No particular knowledge of association schemes will be required to appreciate this talk.

Broadly speaking, representation theory focuses on classifying modules over associative algebras up to isomorphism. The representation type of an algebra reflects the complexity of this classification problem. For algebras over an algebraically closed field, a famous theorem by Y. Drozd divides finite-dimensional algebras into three representation types: finite, tame, and wild. A more rough notion of g-representation type is currently being envisioned. The aim is then to classify the so-called $\tau$-rigid modules, which have very nice combinatorial properties. These modules are key in defining the g-vector fan, a polyhedral fan in Euclidean space. The g-representation type of an algebra is linked to the properties of this fan. In this talk, we investigate when traditional representation types and g-representation types coincide. Our main result focuses on incidence algebras of posets. We outline a proof that, for any finite poset, its incidence algebra is of finite representation type if and only if it is of g-finite representation type. If the poset is also simply connected, the incidence algebra is of tame representation type if and only if it is of g-tame representation type. Our original results are joint with J. F. Grevstad and E. S. Rundsveen [arXiv:2407.17965].

Open Q&A session for all topics related to a doctoral thesis

We are pleased to welcome you to the website of the GAMM Junior Research Group! This page is dedicated to our doctoral consultation hour, where we want to give all interested parties the opportunity to discuss questions about a doctoral thesis in a direct exchange with doctoral researchers. It is our goal to deal with topics related to doctoral studies in a transparent manner for everyone. Through a direct exchange with students in the form of an open consultation hour, we also wish to offer the opportunity to ask and clarify very individual questions.

Shape-memory alloys are materials which display a striking thermodynamic behaviour. They undergo a solid-solid phase transition in which symmetry is reduced upon the passage from the high to the low temperature phase. This gives rise to a 'memory' for these materials and to fascinating microstructures determined by a rich energy landscape. In this talk we will approach these structures through the calculus of variations and scaling laws. This allows us to thus rigorously capture some of the complexity of the underlying microstructures. A particular emphasis will be on the analysis of a striking dichotomy between extreme rigidity and flexibility in the mathematical description of these materials.

This work is concerned with the topology optimization of architectured materials, also known as lattice materials, which are two-scale porous structures. These two-scale structures can be represented as periodically perforated materials, where the microscopic periodic cell vary on a macroscopic scale in terms of its parameters (e.g. hole size) and orientation. Lattice materials are becoming increasingly popular since they can be built by additive manufacturing techniques. The main idea is to optimize the homogenized formulation of this problem, which is an easy task of parametric optimization, then to project the optimal microstructure at a desired length-scale, which is a delicate issue, albeit computationally cheap. The main novelty of our work is, in a plane setting, the conformal treatment of the optimal orientation of the microstructure. In other words, although the periodicity cell has varying parameters and orientation throughout the computational domain, the angles between its members or bars are conserved. Several numerical examples are presented for compliance and stress minimization in 2-d. Extension to the 3-d case will also be discussed. This is a joint work with Perle Geoffroy-Donders, Alex Ferrer and Olivier Pantz.

Lehramtsstudierende zeigen häufig Schwierigkeiten, Verbindungen zwischen der akademischen Mathematik, die sie an der Hochschule lernen, und der Schulmathematik, die sie später unterrichten sollen, herzustellen. Viele Hochschulen versuchen diesem bereits von Klein (1908) beschriebenen Problem der doppelten Diskontinuität durch den Einsatz professionsspezifischer Übungsaufgaben zu begegnen. Diese „Lehramtsaufgaben“ adressieren explizit Verbindungen zwischen Schul- und Hochschulmathematik und sollen so den Aufbau eines schulbezogenen Fachwissens unterstützen. In dem Vortrag wird anhand einer Fragebogenstudie erörtert, inwiefern Lehramtsaufgaben einer wahrgenommenen doppelten Diskontinuität entgegenwirken können. Darauf aufbauend wird diskutiert, welche Ansatzpunkte sich für weiterführende Forschung sowie die Gestaltung professionsspezifischer Lerngelegenheiten zur Adressierung der doppelten Diskontinuität ableiten lassen.

Elliptische Kurven gewinnen in der Kryptographie aufgrund eines sehr hohen Sicherheitsniveaus bei vergleichsweise kleiner Schlüssellänge zunehmend an Bedeutung. Der Vortrag soll – beispielhaft am Diffie-Hellman-Schlüsselaustausch – eine Einführung in Elliptische Kurven und deren Einsatz in der Kryptographie (ECC) geben. Exemplarisch wird die Verwundbarkeit von fehlerhaften ECC-Implementierungen anhand des in [1] ausgeführten Angriffs dargestellt. In diesem Rahmen wird auf die Gruppenstruktur elliptischer Kurven über endlichen Körpern eingegangen; insbesondere wird mit dem Algorithmus von Schoof ein Verfahren zur Bestimmung der Gruppenordnung vorgestellt. [1] Tibor Jager, Jörg Schwenk und Juraj Somorovsky. ``Practical Invalid Curve Attacks on TLS-ECDH``. In: Computer Security – ESORICS 2015.

Bei der Untersuchung von algebraischen Varietäten spielt das Verständnis ihrer Symmetriengruppen eine grosse Rolle. In dem Vortrag werde ich das Thema für K3 Flächen und für ihre Verallgemeinerung in höheren Dimensionen untersuchen. K3 Flächen kann man schon als Verallgemeinerung von elliptischen Kurven sehen. Das einfachste Beispiel einer K3 Fläche ist die Nullstellenmenge eines Polynomes vierten Grades im 3-dimensionalen komplexen projectiven Raum, wie zum Beispiel die Fermatsche Quartik : x^4+y^4+z^4+t^4=0. In höheren Dimensionen spricht man von irreduziblen holomorphen symplektischen Mannigfaltigkeiten. Ich werde verschiedene Beispiele angeben und konkrete Probleme diskutieren, wie das Problem der Klassifikation der Symmetrien oder die Untersuchung der Mannigfaltigkeiten die man als Quotienten erhält.

Spin glasses are models of statistical mechanics in which a large number of elementary units, called spins, interact in a disordered manner. They can serve as simplified models for a broad class of problems encountered in theoretical computer science, combinatorics, or high-dimensional statistics. In the simplest case, called the SK model, there are direct interactions between all pairs of spins. The first part of the talk will present the main ideas underpinning the analysis of this model. However, some models that are very similar remain less well-understood. For instance, one can consider a bipartite version of the SK model, in which the spins are divided into two groups, and direct interactions are only present between spins of different groups. I will highlight some of the challenges posed by models of this type, and discuss some partial progress.

Die kommerzielle Lernplattform ANTON hat v.a. seit der Corona-Pandemie Einzug in den Mathematikunterricht an Grundschulen gehalten. Ob im Unterricht selbst, als Teil von Lernplänen oder auch freiwillig zu Hause, ANTON wird von vielen Kindern regelmäßig genutzt. Dabei stellt sich aus fachdidaktischer Perspektive die Frage, inwiefern ANTON Grundschulkindern ein qualitatives Mathematiklernen ermöglicht und wie diese Plattform in den Unterricht eingebunden werden sollte. In meinem Vortrag möchte ich mich dieser Frage annähern, indem drei kleinere Studien zu ANTON vorgestellt, die Ergebnisse miteinander vernetzt und diskutiert werden: Als Erstes wurden Lernende bzgl. ihrer Einstellung zu ANTON befragt und diese als Zweites mit ihren Fähigkeiten im selbstregulierten Lernen verknüpft. Da die Lernenden in den Fragebögen häufig die Qualität der Rückmeldungen in ANTON bemängelten, wurden als Drittes Einzelinterviews mit Kindern zu ihrem Umgang mit den verschiedenen Feedback-Elementen geführt.