Joint Seminar with the universities of Basel and Strasbourg around the topic of transcendence. The next edition will take place in Strasbourg. The speakers are Rosa Winter, Alessio Cangini and Riccardo Tosi.

Zeit und Ort

Freitag, 12.6.26, 10:00–12:00, Seminarraum 125

Zusammenfassung

In this talk, I will explain our idea to establish positive mass theorem (PMT) up to dimension 19, which was later developed by Brendle and Wang to show PMT in all dimensions. This is a joint work with Yuchen Bi, Tianze Hao, Shihang He and Yuguang Shi.

Zeit und Ort

Montag, 15.6.26, 16:15–17:45, Seminarraum 404

Zusammenfassung

Morse theory serves as a way of studying the topology of manifolds using critical points of smooth functions. For example, Morse homology - which is defined by counting gradient flow lines between critical points of index difference 1 - recovers the singular homology of the manifold. However, it is known that the collection of moduli spaces of gradient flow lines of arbitrary index difference captures even the homotopy type of the manifold. It is therefore a natural question how a given invariant of the manifold can be understood through Morse theoretical constructions. In this talk we consider transport functions as a way of studying the isomorphism classes of principal bundles over a manifold. We will further see that a transport function with values in a topological group G and a right G-space F gives rise to a chain complex. The homology of this chain complex is isomorphic to the singular homology of an associated bundle. If G is a Lie group then a transport function can be obtained from actual parallel transport and the induced chain complex is isomorphic to one defined in the style of Barraud-Damian-Humilière and Oancea.

Zeit und Ort

Dienstag, 16.6.26, 14:15–15:45, Seminarraum 226, HH10

Zeit und Ort

Dienstag, 16.6.26, 14:30–16:00, SR 404

Zusammenfassung

Let be the theory of beautiful pairs of algebraically closed fields of fixed characteristic. It is known that for real tuples in models of , SU-rank coincides with Morley rank and can be computed effectively. Building on Pillay's geometric description (2007) of imaginaries in , we define an additive rank on imaginaries of , called the geometric rank. It takes values in and coincides with SU-rank on real tuples. It refines SU-rank and characterizes forking in , from which we derive an explicit criterion for determining forking independence.

Zeit und Ort

Dienstag, 16.6.26, 18:30–20:00, Hörsaal 2

Zusammenfassung

„Lernende sollen Mathematik verstehen“ – diese Forderung findet in Forschung und Unterrichtspraxis breite Zustimmung. Doch was bedeutet es ganz konkret, Verständnis für einen bestimmten mathematischen Inhalt aufzubauen? Jahrzehntelange Unterrichtsforschung hat bereits gezeigt, dass diese Frage für jeden mathematischen Inhalt unterschiedliche Herausforderungen mit sich bringt. Im Vortrag wird diese Frage am Beispiel des Einstiegs in die Binomialverteilung aufgegriffen. Ausgehend von typischen Unterrichtssituationen wird diskutiert, welches Verständnis Lernende entwickeln sollen, welche Vorstellungen sie bereits mitbringen und welche Schwierigkeiten häufig auftreten. Daraus werden Impulse für einen verstehensorientierten Einstieg in die Binomialverteilung entwickelt und an ausgewählten Beispielen aus der Unterrichtspraxis veranschaulicht.

Zeit und Ort

Mittwoch, 17.6.26, 12:15–13:15, Weismannhaus

Zeit und Ort

Donnerstag, 18.6.26, 15:00–16:30, Hörsaal 2

Zusammenfassung

Consider an unknown random vector X, taking values in R^d. Is it possible to"guess" its mean accurately if the only information one is given consists of N independent copies of X? More accurately, given an arbitrary norm on R^d, the goal is to find a mean estimation procedure: upon receiving a wanted confidence parameter \delta and N independent copies X1,...,XN of an unknown random vector X - that has a finite mean and covariance -, the procedure returns \hat{\mu} for which the error | \hat{\mu} - E X| is as small as possible with probability at least 1-\delta (with respect to the product measure). The mean estimation problem has been studied extensively over the years and I will present some of the ideas that have led to its solution (and to the solution of other questions of a similar flavour that I will outline). Two surprising facts are that in all these problems the obvious choices fail miserably (for mean estimation, that choice is N^{-1}\sum{i=1}^N Xi); and, that the solution behaves as if the (arbitrary) random vector X were gaussian.

Zeit und Ort

Montag, 22.6.26, 16:15–17:45, Seminarraum 404

Zusammenfassung

A waveguide is a spatial region \(\Omega \subset \mathbb{R}^{n}\) of tubular form, obtained as a local perturbation of a straight tube \(\Omega_{0} = \mathbb{R} \times B_{1}(0)\). Classically, a particle moving inside such a waveguide and undergoing regular reflections at the boundary will, for almost all initial conditions, eventually leave every bounded region in finite time. In contrast, the quantum-mechanical behaviour can be drastically different. One of the central and perhaps surprising phenomena in the theory of quantum waveguides is that the existence of bound states is closely related to the geometry of the underlying tube.

In the non-relativistic setting, the system is typically described by the Dirichlet Laplacian, and the relation between geometry and discrete spectrum has been studied extensively since the 1980s. However, the study of the relativistic case has just began in recent years.

In this talk, we consider non-uniform relativistic quantum waveguides, modelled by a Dirac operator subject to infinite mass boundary conditions. Under certain assumptions on a localised geometric deformation of the waveguide, we prove the existence of at least one discrete eigenvalue in the spectral gap of the straight waveguide. This eigenvalue corresponds to a geometrically induced bound state, giving a relativistic analogue of a well-known phenomenon from the non-relativistic theory.

Zeit und Ort

Donnerstag, 25.6.26, 15:00–16:30, Hörsaal 2

Zusammenfassung

Partial Differential Equations (PDEs) are often described as the language of Physics as they describe a wide array of physical phenomena over a vast range of scales. Despite their remarkable success over many decades, numerical methods for approximating PDEs can incur a very high computational cost. This limitation has provided the impetus for the design of fast and accurate Machine Learning/AI based neural PDE surrogates which can learn the PDE solution operator from data. In this talk, we review some latest developments in the field of Neural Operators, which are widely used as an ML paradigm for PDEs and discuss state of the art neural operators based on convolutions or attention. We will discuss graph and transformer based architectures for PDEs on arbitrary domains and conditional Diffusion models for PDEs with chaotic multiscale solutions. Finally, the issue of sample complexity is addressed by the design of general purpose Foundation models for PDEs.

Zeit und Ort

Montag, 29.6.26, 16:15–17:45, Seminarraum 404

Zusammenfassung

An irreducible \(G_2\)-manifold is a Riemannian 7-manifold \(M\) with holonomy group equal to the exceptional Lie group \(G_2\). When \(M\) is closed, the Teichmüller space \(T(M)\) of \(G_2\) metrics on \(M\) divided by diffeomorphisms isotopic to the identity is a smooth, finite-dimensional manifold by a result of Joyce. Yet its topology, and that of its quotient by the smooth mapping class group, remains elusive. Using ideas of Crowley, Goette, and Hertl, we exhibit the first known example of a \(G_2\)-manifold \(M\) together with infinitely many diffeomorphisms that both act freely on \(T(M)\) and preserve a connected component. The diffeomorphisms are 7-dimensional analogs of diffeomorphisms of K3 surfaces constructed recently by Farb and Looijenga, and much like the Farb-Looijenga examples, these diffeomorphisms minimize topological entropy among their isotopy class.

Zeit und Ort

Dienstag, 30.6.26, 14:30–16:00, SR 125

Zeit und Ort

Montag, 13.7.26, 16:15–17:45, Seminarraum 404

Zeit und Ort

Dienstag, 14.7.26, 16:15–17:45, Seminarraum 125

Zeit und Ort

Donnerstag, 23.7.26, 15:00–16:30, Hörsaal 2