Abstract: We discuss the spectral properties of three-dimensional Dirac operators with critical combinations of electrostatic and Lorentz scalar shell interactions supported by a compact smooth surface. It turns out that the criticality of the interaction may result in a new interval of essential spectrum. The position and the length of the interval are explicitly controlled by the coupling constants and the principal curvatures of the surface. This effect is completely new compared to lower dimensional critical situations or special geometries considered up to now, in which only a single new point in the essential spectrum was observed. Based on joint work with Konstantin Pankrashkin (Oldenburg).

Location: Endenicher Allee 60, seminar room 0.008

Abstract: We’ll explore the problem of finding effective models for the classifying spaces of certain quotients of fundamental groups of non-positively curved cube complexes. We’ll discuss the framework -- cubical small-cancellation theory -- that provides the necessary tools to do so, and, time permitting, we’ll explain how this viewpoint allows us to compute the homology and cohomology of various examples.

Location: Room 0.003, Endenicher Allee 60

Abstract: A smooth projective variety X is said to be Calabi-Yau if its canonical bundle is trivial. I will discuss joint work with Lukas Brantner, in which we use derived algebraic geometry to study deformations of Calabi-Yau varieties in characteristic p. We prove a positive characteristic analogue of the Bogomolov-Tian-Todorov theorem (which states that deformations of Calabi-Yau varieties in characteristic 0 are unobstructed), and show that 'ordinary' Calabi-Yau varieties admit canonical lifts to characteristic zero (generalising earlier results of Serre-Tate for abelian varieties, and Deligne and Nygaard for K3 surfaces). In this talk, no prior knowledge of derived algebraic geometry will be assumed.

Location: Max Planck Institut – Hörsaal

Abstract: We consider scalar field theories on the line with Ginzburg-Landau (double-well) self-interaction potentials. Prime examples include the \phi^4 model and the sine-Gordon model. These models feature simple examples of topological solitons called kinks. The study of their asymptotic stability leads to a rich class of problems owing to the combination of weak dispersion in one space dimension, low power nonlinearities, and intriguing spectral features of the linearized operators such as threshold resonances or internal modes.

We present a perturbative proof of the full asymptotic stability of the sine-Gordon kink outside symmetry under small perturbations in weighted Sobolev norms. The strategy of our proof combines a space-time resonances approach based on the distorted Fourier transform to capture modified scattering effects with modulation techniques to take into account the invariance under Lorentz transformations and under spatial translations. A major difficulty is the slow local decay of the radiation term caused by the threshold resonances of the non-selfadjoint linearized matrix operator around the modulated kink. Our analysis hinges on two remarkable null structures that we uncover in the quadratic nonlinearities of the evolution equation for the radiation term as well as of the modulation equations.

The entire framework of our proof, including the systematic development of the distorted Fourier theory, is general and not specific to the sine-Gordon model. We conclude with a discussion of potential applications in the generic setting (no threshold resonances) and with a discussion of the outstanding challenges posed by internal modes such as in the well-known \phi^4 model.

This is forthcoming joint work with Gong Chen (GeorgiaTech).

Location: Mathematisches Institut, Endenicher Allee 60, 53115 Bonn, Raum 0.011

Abstract: Let g be a semi-simple complex finite dimensional Lie algebra and M a g-module. Kostant's problem for M asks whether the unviersal enveloping algebra of g surjections onto the algebra of all linear endomorphisms of M that are locally finite with respect to the adjoint action of g. In general, the answer to this question is not known even for simple highest weight modules. In this talk I will survey what is known and present several new developments and results on this problem for various classes of modules in the BGG category O for sl_n.

Location: Mathematisches Institut, Endenicher Allee 60, 53115 Bonn, Raum 1.008

Abstract: In his paper Quantum harmonic analysis on phase space from 1984 (J. Math. Phys.), Reinhard Werner developed a new phase space formalism which allowed for a joint harmonic analysis of functions and operators. Since his reasoning was mostly guided by motivations from the physical side of quantum mechanics, mathematicians ignored this highly interesting contribution for almost 35 years. Only in the last few years, interest in Werner’s approach grew and actually yielded a number of interesting and relevant results in time-frequency analysis as well as in operator theory. The speaker, who has been working mostly on the operator theory side of quantum harmonic analysis (QHA), will try to describe the basic features of QHA and how they relate to problems in operator theory. After presenting some basics of the formalism of QHA, we will discuss one application of the audience’s choice: Either a result in Fredholm theory, results in commutative operator algebras, or a characterization problem of a certain important algebra appearing in QHA.

Location: Seminar room 0.008, Endenicher Allee 60

Abstract: A metric space $X$ satisfies a Euclidean isoperimetric inequality for $n$-spheres, if every $n$-sphere $S\subset X$ bounds a ball $B\subset X$ with $\operatorname{vol}_{n+1}(B)\leq C\cdot \operatorname{vol}_n(S)^\frac{n+1}{n}$. Every CAT(0) space $X$ satisfies Euclidean isoperimetric inequalities for $1$-spheres with the sharp constant $C=1/4\pi$. Moreover, if such inequalities hold with a constant strictly smaller than $1/4\pi$, then $X$ has to be Gromov hyperbolic. In particular, a sharp isoperimetric gap appears. In the talk I will focus on the case $n=2$, namely fillings of 2-spheres by 3-balls. This is based on joint work with Drutu, Lang and Papasoglu.

Location: Endenicher Allee 60, Room 2.040

Abstract: We construct a new type of counterexamples to the integral Tate conjecture over finite fields, where a geometric cycle map is surjective but an arithmetic cycle map is not. We also discuss the relation of this problem with two coniveau filtrations, and show some positive results toward a conjecture of Colliot-Thélène and Kahn. This is joint work with Federico Scavia.

Location: Max Planck Institut - Hörsaal

Abstract: In the last decades, since the first experimental realizations of Bose-Einstein condensates, the study of large bosonic systems has been a very active field of research both in physics and in mathematics. In experiments, Bose gases are often very dilute and can be well described in the Gross-Pitaevskii limit, i.e. as quantum systems of N confined particles, interacting through a potential with scattering length of order 1/N where N tends to infinity. In this talk, we discuss a result on a hard-sphere Bose gas in the Gross-Pitaevskii regime. Namely, we prove a second order upper bound on the ground state energy matching the known expression of the energy for integrable potentials. We also discuss a new upper bound for hard-spheres in the thermodynamic limit where the number of particles and the size of the box are sent to infinity, keeping the density fixed. Our result resolves the ground state energy up to an error of the order of the so-called Lee-Huang-Yang correction. Based on joint works with S. Cenatiempo, A. Giuliani, A. Olgiati, G. Pasqualetti, and B. Schlein.

Location: Lipschitz Saal

Abstract: Recall that a Hecke algebra of a Coxeter group is a deformation of the group algebra, with quantized quadratic relations. Roughly speaking, a quantum wreath product is a deformation of the group algebra of the wreath product G≀S_d of a (possibly infinite) group G by a symmetric group S_d, with quantized wreath relations and quantized quadratic relations in the sense that coefficients are in certain (not necessarily commutative) tensor algebra. A prototype example is the Hecke subalgebra (which we call the Hu algebra) appearing in a Morita equivalence theorem due to Jun Hu. The Hu algebras have highly nontrivial coefficients in their quadratic relations and can be thought of as the Hecke algebras of the wreath product S_m≀S_2, which is not a Coxeter group in general. Quantum wreath products also include important variants of the Hecke algebras, such as affine Hecke algebras, their degenerate versions, and others. In this talk, I will focus on the Clifford theory and the initial steps of Springer theory, motivated by applications to the Ginzburg-Guay-Opdam-Rouquier problem.

Location: Room 1.008, Mathematisches Institut

Abstract: The curve graph of a surface is a combinatorial object that encodes geometric properties of a surface and is key in linking geometric properties with algebraic properties in low-dimensional topology. In this talk, I will present an analogue of the curve graph for the class of CAT(0) spaces and discuss some developments. This is joint work with Harry Petyt and Abdul Zalloum.

Location: Room 0.003, Endenicher Allee 60

Location: MPIM Lecture Hall

Note: This is a talk in the GNOSC-Seminar, https://www.gnosc.net/

Abstract: Telomeres are repetitive sequences situated at both ends of the chromosomes of eukaryotic cells. At each cell division, they are eroded until they reach a critical length that triggers a state in which the cell stops to divide: the senescent state. In this work, we are interested in the link between the initial distribution of telomere lengths and the distribution of senescence times. We propose a method to retrieve the initial distribution of telomere lengths, using only measurements of senescence times. Our approach relies on approximating our model with a transport equation, which provides a natural estimator for the initial telomere length distribution. We investigate this method from a theoretical point of view by providing bounds on the estimation error, pointwise and in all Lebesgue spaces. We also illustrate it with estimations on simulations, and discuss its limitations related to the curse of dimensionality.

Location: Lipschitz Hall

Abstract: Bridgeland stability conditions have been constructed on curves, surfaces, and in some higher dimensional examples. In several cases, there are only so-called "geometric" stability conditions, which are constructed using slope stability for sheaves, whereas in other cases, there are more (for example if there is an equivalence with quiver representations). Lie Fu, Chunyi Li, and Xiaolei Zhao were the first to provide a general result explaining this phenomenon. In particular, they showed that if a variety has a finite map to an abelian variety, then all stability conditions are geometric. In this talk, we test the converse in two ways on surfaces that arise as free quotients by finite groups. One method is via Le Potier functions which characterize the existence of slope-semistable sheaves. The second method uses equivariant categories. This is joint work with Edmund Heng and Anthony Licata, based on arxiv:2307.00815 and arxiv:2311.06857.

Location: Max Planck Institut - Hörsaal

Abstract: Arising in various contexts, such as e.g. quantum symmetric pairs and categorical actions on category O, module categories over monoidal categories are a natural part of the general setting of categorical representation theory. A common way to study, construct and classify module categories over a rigid monoidal category C is by "reconstructing" them as categories of modules for an algebra object in C. In absence of rigidity, e.g. for categories of crystals, monoid representations, and, more generally, modules over non-Hopf bialgebras, it is easy to provide examples where reconstruction fails. In this talk, I will present reconstruction results for module categories over non-rigid categories, where algebra objects are replaced by certain monads on C. I will also explain how these results can be used to obtain an algebraic description of module categories for categories of modules over non-Hopf bialgebras via certain Hopf modules (so-called Hopf trimodules), and present further applications.

Location: Mathematisches Institut, Endenicher Allee 60, 53115 Bonn, Raum 1.008.

Abstract: Here is a motivating question, which is a special case of a more general problem: The moduli space of K3 surfaces in characteristic p is stratified by the height of the formal Brauer group, and the smallest stratum (the supersingular locus) is further stratified by the Artin invariant. If we give ourselves an explicit K3 surface, e.g. a quartic surface in P^3, how can we calculate in which stratum it lies? In my talk, I will explain what an F-zip is (I will not assume you already know this), and how this relates to the above problem. Under mild technical assumptions, we can associate an F-zip to every smooth projective variety in characteristic p, and such F-zips have been classified. I will explain some new techniques that allow us to calculate the F-zips of some accessible types of varieties, such as projective hypersurfaces.

Location: Lipschitzsaal, Endenicher Allee 60

Location: Endenicher Allee 60, Lipschitz Lecture Hall

Abstract: We present a general shape optimization framework based on the method of mappings in the Lipschitz topology. We propose and numerically analyse steepest descent and Newton-like minimisation algorithms for the numerical solution of the respective shape optimization problems. Our work is built upon previous work of the authors in (Deckelnick, Herbert, and Hinze, ESAIM: COCV 28 (2022)), where a Lipschitz framework for star-shaped domains is proposed. To illustrate our approach we present a selection of PDE constrained shape optimization problems and compare our findings to results from so far classical Hilbert space methods and recent p-approximations. This is joint work with Klaus Deckelnick from Magdeburg and Philip Herbert from Sussex.