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

Donnerstag, 13.8.26, 14:00–15:30, SR 404

Zusammenfassung

A \(\lambda\)-self expander is a stationary point for isoperimetric problem in (R^n, e^{\frac{|x|^2}{4}}dx). We show that a closed embedded \(\lambda\)-self expander must be a round sphere centered at the origin. The proof is based on a sharp Heintze-Karcher inequality. We also discuss a more general result for certain warped product manifolds with density satisfying some structural condition. This is joint work with Jinchuan Bai.

Zeit und Ort

Freitag, 24.7.26, 12:00–12:30, Seminarraum 404

Zeit und Ort

Donnerstag, 23.7.26, 16:00–17:30, Seminarraum 125

Zeit und Ort

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

Zeit und Ort

Mittwoch, 22.7.26, 14:15–15:45, SR 125/126 (E1)

Zeit und Ort

Dienstag, 21.7.26, 17:00–18:00, Seminarraum 125

Zusammenfassung

In this talk, based on joint works with X. Cabré, G. Catino, P. Mastrolia and A. Roncoroni, we describe some new tools to study the behaviour of complete, \(2\)-sided stable minimal hypersurfaces (SMH) in non-negatively curved ambient spaces. The guideline is that such SMH are an example of manifolds with spectral Ricci lower bounds, a class of spaces for which we prove a generalization of Cheeger-Gromoll's splitting theorem and sharp pointwise gradient estimates for the Green kernel. Among the applications, we get a short proof of the fact that any SMH \(M^3 \to \R^4\) must be a hyperplane (the stable Bernstein theorem in \(\mathbb{R}^4\)), and a characterization of the 3D-catenoid in \(\R^4\)

Zeit und Ort

Dienstag, 21.7.26, 15:00–17:00, Seminarraum 125

Zusammenfassung

TBD

From 15:00 to 16:00, it is discussion time which takes place at SR 127, please feel free to join. The talk starts from 16:00 at SR404.

Zeit und Ort

Dienstag, 21.7.26, 14:15–15:15, Seminarraum 226, HH10

Zusammenfassung

In this talk we summarize the results of the author’s master thesis, enti- tled ’Space-Time ResNet Approximation of ODEs’. First, we analyze multiple operations regarding feedforward neural networks and residual neural networks (ResNets). Then, using those operations, we show that ResNets are able to ap- proximate solutions to ordinary differential equations (ODEs) with a parameter growth rate of O(| ln(ε)|8).

Zeit und Ort

Montag, 20.7.26, 16:15–17:45, Seminarraum 404

Zusammenfassung

Prime geodesic theorems describe the asymptotic behavior of the distribution of prime closed geodesics. In particular, for a hyperbolic surface, the prime geodesic counting function has the same leading asymptotic as the prime number counting function. The optimal bound on the remainder term remains, however, an open conjecture. In this talk, we will discuss the so-called Gallagherian PGT; that means, we allow the asymptotics to fail on a set of finite logarithmic measure. We improve and extend recent results of this type established by Gušić for compact locally symmetric spaces to new settings. This is a joint work with Jørgen Olsen Lye.

Zeit und Ort

Mittwoch, 15.7.26, 15:30–17:00, online, mailen Sie an Frau Mildenberger bei Interesse

Zusammenfassung

The derived functors \(lim^n\) of the inverse limit measure n-dimensional obstructions to the inverse limit functor preserving exact sequences. Whether or not certain derived limits vanish carries surprisingly deep set theoretic content involving higher dimensional infinitary combinatorics. The vanishing of certain derived limits \(lim^nA\) has been the focus of considerable set theoretic focus and has implications in both strong homology and condensed mathematics. This talk will focus on the combinatorial principles behind the vanishing of \(lim^nA\) and recent developments regarding when they do and do not vanish.

Zeit und Ort

Mittwoch, 15.7.26, 14:15–15:45, SR 125/126 (E1)

Zeit und Ort

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

Zusammenfassung

A surface of least area that intersects a given support surface at a constant angle is called a minimal capillary surface. Minimal capillary surfaces arise naturally in the context of fluid dynamics as the idealized interface between two fluids in a container bounded by the support surface. In recent years, minimal capillary surfaces have been shown to capture aspects of the geometry of the support surface that appear to be beyond the reach of established tools in geometric analysis. In this talk, I will describe some of these developments, including my contributions in joint work with M. Eichmair such as the resolution of the Penrose inequality in extrinsic geometry conjectured by G. Huisken and the characterization of stable minimal capillary surfaces in the so-called tangential limit.

Zeit und Ort

Dienstag, 14.7.26, 14:30–16:00, online, mailen Sie an Frau Mildenberger bei Interesse

Zusammenfassung

Forcing axioms are principles occurring naturally in set theory, which can be seen as generalizations of the Baire Category Theorem. Classical strong forcing axioms, such as PFA or MM, offer a uniformly "saturated" picture of a very low initial segment of the universe. Namely the sets that are hereditarily of cardinality less than the second uncountable cardinal. Forcing axioms have proven to be extremely successful in obtaining consistency results, not only in set theory, but also in other areas of mathematics such as measure theory, topology, group theory, Boolean algebras, and Banach space theory. Most notable is Shelah's proof of the independence of the Whitehead problem in group theory. These two reasons have motivated the search for consistent high analogs of the known classical forcing axioms and the developement of new forcing iteration techniques.

In this talk, I will give an overview of the technique of forcing with side conditions and present some consistency results that contribute towards the search of consistent high forcing axioms.

Zeit und Ort

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

Zusammenfassung

Given a Morse function on a closed smooth manifold and a Morse-Smale pseudo-gradient vector field adapted to it, one can construct a topological category called the flow category associated with this data. Its objects are the critical points of the function and its morphism spaces are the spaces of possibly broken gradient trajectories connecting critical points. This topological category models an infinity category, which can be thought of as providing a homotopy coherent notion of composition of unbroken trajectories connecting critical points. On the other hand, such trajectories determine paths on the manifold, that have the property to be exit paths with respect to the stratification by the stable manifolds of the gradient. A construction of Lurie associates an infinity category to this stratification, which can be thought of as providing a notion of homotopy coherent composition of exit paths. I will define these objects more precisely and then present a result I obtained asserting that these two infinity categories are equivalent.

Zeit und Ort

Mittwoch, 8.7.26, 14:15–15:45, SR 125/126 (E1)

Zeit und Ort

Dienstag, 7.7.26, 14:30–16:00, online, ask Heike Mildenberger for the link

Zeit und Ort

Montag, 6.7.26, 16:15–17:45, Seminarraum 404

Zusammenfassung

For a function \(f\) on a topological space \(X\), Cohen-Steiner--Edelsbrunner--Harer introduced the notion of extended barcode of \(f\). It is a combinatorial gadget that compactly stores information about, in particular, homology of any interlevel set \(f^{-1}([a,b])\) (over a fixed field \(\mathbb F\)). When \(X\) is a manifold (possibly with boundary) and \(f\) is Morse without multiple critical values, we propose an equivalent, geometrical definition of extended barcode. It relies on a Bruhat decomposition, which describes the relative position of two complete flags in a vector space. This approach allows us to answer the following question. Given two manifolds each equipped with a Morse function, choose a diffeomorphism identifiying (parts of) their boundaries. How to obtain the barcode for the function on the glued manifold in terms of two initial barcodes and a given diffeomorphism?

To make the talk (more) accessible to non-topologists, a fraction of the time will be devoted to Bruhat decomposition, which only requires linear algebra.

Zeit und Ort

Dienstag, 30.6.26, 16:15–17:45, Seminarraum 125

Zusammenfassung

I will discuss some recently developed dimension descent methods in connection with the high-dimensional positive mass theorem. The focus will be on weighted scalar curvature and effective dimension, stable minimal hypersurface descent, and conformal blow-up near singular sets.

Zeit und Ort

Dienstag, 30.6.26, 14:30–16:00, SR 404

Zusammenfassung

An amenable group is a group with a finite, additive measure which is invariant under translations; for example, the integers form an amenable group. A fundamental question in additive combinatorics is to determine what kind of structure subsets of positive measure inside a fixed amenable group exhibit. Examples are arithmetic progressions and sumsets.

In this talk, we will see how model theory can be applied to address these problems. We will adapt Hrushovski's stabilizer theorem to show that sets of positive density have the so-called 1-sided productset property, and that dense sets must contain, up to a translate, abundant solutions to many linear equations,

Zeit und Ort

Dienstag, 30.6.26, 14:15–15:15, Seminarraum 226, HH10

Zusammenfassung

Deep neural networks have achieved remarkable performance across numerous domains, including computer vision, natural language processing, and speech recognition. How- ever, the computational and memory demands of these models present significant chal- lenges for deployment on resource-constrained devices such as mobile phones, embed- ded systems, and edge computing platforms. Model quantization has emerged as a critical technique to address these limitations by reducing the numerical precision of network parameters and activations from high-precision floating-point representations to lower-bit fixed-point or integer formats. This thesis investigates the effects of quantization on neural network performance, with particular emphasis on memory effciency and numerical robustness. We provide a comprehensive analysis of quantization techniques, including uniform and non-uniform quantization, symmetric and asymmetric schemes, and various calibration methods. The theoretical foundations of quantization are examined through the lens of informa- tion theory and numerical analysis, establishing a rigorous framework for understanding precision-accuracy trade-offs [11]. Our work explores post-training quantization (PTQ) methodologies, analyzing their respective advantages and limitations. We examine the impact of reduced precision on model accuracy, inference latency, and memory footprint across different neural net- work architectures. Furthermore, we investigate the numerical stability of quantized networks and analyze error propagation through network layers under various quanti- zation configurations. The findings of this research contribute to a deeper understanding of how quantiza- tion affects neural network behavior and provide practical guidelines for implementing effcient quantized models without significant performance degradation. This work has implications for enabling widespread deployment of deep learning on edge devices and reducing the environmental footprint of large-scale machine learning systems.

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

Montag, 29.6.26, 11:00–12:30, Seminarraum 232

Zeit und Ort

Freitag, 26.6.26, 10:30–12:00, Seminarraum 404

Zusammenfassung

The modular Zilber-Pink conjecture is a major open problem in arithmetic geometry governing the behavior of "atypical intersections" between affine complex algebraic varieties and so-called special varieties. In this talk I will introduce the conjecture and present recent joint work with Vahagn Aslanyan and Guy Fowler where we prove that the conjecture holds for "generic" varieties. The proof relies on techniques from the model theory of differential fields.

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

Dienstag, 23.6.26, 16:15–17:45, Seminarraum 125

Zusammenfassung

We establish general monotone quantities and sharp mass-capacity inequalities related to p-capacitary functions in 3-dimensional asymptotically flat half-spaces of simple topology with nonnegative scalar curvature and nonnegative boundary mean curvature. These inequalities attain equality on a Schwarzschild half-space outside a rotationally symmetric half sphere. This is based on a joint work with Prof. Chao Xia and Prof. Jiabin Yin.

Zeit und Ort

Dienstag, 23.6.26, 14:15–15:15, Seminarraum 226, HH10

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

Freitag, 19.6.26, 10:00–11:30, Seminarraum 404

Zusammenfassung

We show that any connected algebraic group \(G\) over a field admits a nilpotent normal subgroup \(Z_\infty(G)\) such that the quotient \(G/Z_\infty(G)\) has trivial center. We construct \(Z_\infty(G)\) as the final term of the transfinitely extended upper central series of \(G\); accordingly, we call it the hypercenter of \(G\). We establish several related results about the upper central series of \(G\), along with an analogue for algebraic groups of a well-known theorem of Fitting's from 1938.

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

Mittwoch, 17.6.26, 16:15–17:00, SR 226, Hermann-Herder-Str. 10

Zusammenfassung

n this presentation, we discuss new tools for comparison principles for viscosity solutions. The discussed comparison principle is based on a test function framework, that allows for the simultaneous treatment of generators of diffusion as well as jump processes. Additionally incorporating Lyapunov functions, we are able to show comparison in the strict rather than the usual sup-norm topology. We then apply the theory to a variety of parabolic equations and elliptic problems arising from, for instance, optimal control or martingale problems.

Zeit und Ort

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

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

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, 14:15–15:15, Seminarraum 226, HH10

Zusammenfassung

Elastic rods arise in numerous applications ranging from engineering structures to biological filaments. In this talk, I present a variational framework for the numerical approximation of inextensible elastic rods governed by bending energy. The dynamics are modeled by an \(H^2\)-gradient flow, which is discretized in space using cubic finite elements and in time by an implicit scheme with linearized constraints. I establish the convergence of the discrete energy functionals via \(\Gamma\)-convergence and analyze the resulting minimizing movements using the theory of curves of maximal slope and the Sandier-Serfaty framework. These results provide a rigorous justification of the numerical scheme and clarify the interplay between spatial and temporal discretization. Finally, I discuss the physical interpretation of the model as an approximation of the motion of elastic rods in viscous fluids.

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

Freitag, 12.6.26, 10:15–12:00, 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

Dienstag, 9.6.26, 16:15–17:45, Seminarraum 125

Zusammenfassung

In this talk I will present some recent results about the quantitative stability of Sobolev-type inequalities on the \(d\)-dimensional round sphere, each of which can be viewed as a generalization of the seminal stability result by Bianchi and Egnell. The first part of my talk concerns the quantitative stability of the reverse fractional Sobolev inequality. Implementing the classical proof strategy by Bianchi and Egnell is non-trivial here because the underlying operator \(A_{2s}\) is not positive definite when \(s > d/2\). Remarkably, the case \(s - d/2 \in (1,2)\) constitutes the first example of a Sobolev-type stability inequality (i) whose best constant is explicit and (ii) which does not admit an optimizer. The second part concerns a fully nonlinear functional inequality for the \(\sigma_2\)-curvature. We prove its stability in a reverse setting occurring in three dimensions. As a geometric application, this implies a quantitative refinement of the almost-Schur lemma of De Lellis and Topping in the special case of the round 3-sphere. This second result is joint work with Jonas Peteranderl (LMU München).

Zeit und Ort

Dienstag, 9.6.26, 14:30–16:00, SR 404

Zusammenfassung

Higher order amalgamation is a family of properties introduced by Shelah, which generalize several important model-theoretic notions. In particular, all simple theories have 3-existence over models. Moreover De Piro, Kim and Millar proved that stable theories with elimination of imaginaries have n-existence over models.

We will explain how to transfer n-existence from a stable theory to a simple theory, provided that the latter is sufficiently "controled" by the former. The motivating example are bounded PAC substructures of stable theories, which were first introduced by Hrushovski to study pseudo algebraically closed fields from a model-theoretic point of view. Our result yields in particular that bounded (possibly non-perfect) pseudo-algebraically closed fields have high amalgamation over elementary substructures.

Zeit und Ort

Dienstag, 9.6.26, 14:15–15:15, Seminarraum 226, HH10

Zusammenfassung

We establish a pointwise limit theorem for a broad class of parameter-dependent BMO-type seminorms as the parameter tends to zero. By introducing novel BMO-type seminorms, we provide a unified framework that extends several existing results and yields non-distributional characterizations of Sobolev-type spaces, both in the scalar and in the vector-valued setting. More precisely, for any open set \(\Omega\subset \mathbb{R}^n\) and any \(p\in (1, \infty)\), we provide a characterization of the Sobolev space \(W^{1,p}(\Omega; \mathbb{R}^m)\). In addition, we characterize the space \(E^{1,p}(\Omega;\mathbb{R}^n)\) of \(L^p\) maps with \(p\)-integrable distributional symmetric gradient. Finally, for all \(p\in [1, \infty)\), we show that these seminorms converge to integral functionals with convex, \(p\)-homogeneous integrands associated with the distributional gradient and the symmetric gradient.

Zeit und Ort

Mittwoch, 3.6.26, 14:15–15:45, SR 125/126 (E1)

Zeit und Ort

Dienstag, 2.6.26, 17:00–18:00, Seminarraum 125

Zusammenfassung

In this talk, we firstly introduce a capillary John ellipsoid theorem for capillary convex bodies in the Euclidean half-space \(\overline{\mathbb{R}^{n+1}_{+}}\). This theorem yields a non-collapsing estimate for capillary hypersurfaces, which provides a new approach to obtaining \(C^{0}\) estimates for solutions to some capillary curvature problems (including the capillary \(L_{p}\) Christoffel–Minkowski problem and the capillary \(L_{p}\) curvature problem). Then, as a further application, we study the capillary \(L_{p}\) dual Minkowski problem, establish existence for \(1<p\leq q\leq 3\), and improve existing results for the case \(p>q\) in \(\overline{\mathbb{R}^{3}_{+}}\).

Zeit und Ort

Dienstag, 2.6.26, 16:00–17:00, Seminarraum 125

Zusammenfassung

In this talk, I will introduce a new notion of convexity in the unit sphere called horo-convexity, inspired by its analogue in hyperbolic space. For horo-convex hypersurfaces, we prove the smooth convergence of the Guan/Li inverse curvature flow and, as a consequence, establish the full set of quermassintegral inequalities on the sphere. If time permits, I will also discuss parallel results for the hypersurface with capillary boundary lying in the Euclidean unit ball. This talk is based on joint works with Julian Scheuer.

Zeit und Ort

Dienstag, 2.6.26, 14:15–15:15, Seminarraum 226, HH10

Zusammenfassung

TBA

Zeit und Ort

Dienstag, 19.5.26, 16:15–17:45, Seminarraum 125

Zusammenfassung

We investigate the validity of the optimal higher-order Sobolev inequality \(H_k^2(M^n)=\{u\in L^2\hbox{ s.t. }\nabla^iu\in L^2\hbox{ for all }i\leq k\}\hookrightarrow L^{\frac{2n}{n-2k}}(M^n)\) on a closed Riemannian manifold when the remainder term is the L^2-norm. Unlike the case k=1, the optimal inequality does not hold in general for k>1. We prove conditions for the validity and non-validity that depend on the geometry of the manifold. Our conditions are sharp when k=2 and in small dimensions. This is joint work with Lorenzo Carletti (Bruxelles).

Zeit und Ort

Dienstag, 19.5.26, 14:15–15:15, Seminarraum 226, HH10

Zusammenfassung

Holographic X-ray phase contrast imaging enables nanometer-scale resolution of large volumes of biological tissues, opening promising perspectives for digital histopathology. Conventional approaches to phase contrast imaging typically assume a perfectly coherent incident beam, an approximation that is well justi- fied for synchrotron radiation sources. In clinical and high-throughput settings, however, one must rely on laboratory-based table-top sources with substantially lower coherence, resulting in a significant loss of image resolution. To address this limitation, we investigate the use of time-resolved intensity measurements, which provide access to intensity correlations. In this talk, we explore and exploit the information contained in such correlations, particularly in comparison with standard mean-intensity data. While in the fully coherent setting intensity correlations carry no additional information compared to mean intensities at all, our numerical experiments with synthetic data demonstrate a substantial improvement under partial coherence. On the theoretical side, we investigate the information content of these corre- lations and analytically demonstrate that, under a symmetry-breaking condition on the sample and the illumination area, both phase and absorption contrast can be uniquely determined in both the full nonlinear and the linearized models. On the algorithmic side we propose an algorithm avoiding the explicit computa- tions of intensity correlations from raw data in a preprocessing step while using the full information content of these correlations. Although the size of intensity correlation data scales quadratically with the number of detector pixels, the computational complexity of the proposed al- gorithm scales only linearly with the number of pixels and quadratically with the rank of the covariance matrix of the incident beam. Since realistic imaging setups typically involve millions of pixels, this near-linear scaling is essential for practical applications to experimental data.

Zeit und Ort

Montag, 18.5.26, 16:15–17:45, Seminarraum 404

Zusammenfassung

Llarull proved that the round sphere is extremal, meaning that one cannot simultaneously increase both its scalar curvature and its metric. Goette and Semmelmann generalized this result and proved that scalar-rigid maps are Riemannian submersions. In this talk, I present a recent generalization showing that a scalar-rigid f: M → N is not only a Riemannian submersion, but that M is essentially a Riemannian products of the base manifold with a Ricci-flat fiber. The proof is based on spin geometry for Dirac operators and an analysis connecting Clifford multiplication with the representation theory of the curvature operator. This is joint work with Oskar Riedler.

Zeit und Ort

Freitag, 15.5.26, 12:00–13:00, Seminarraum 404

Zusammenfassung

Bayesian and related methodologies have been very popular for estimation, prediction and inference in complex statistical models, for instance arising from differential equations (PDEs/SDEs). We present recent statistical and computational guarantees which underpin those methodologies. On the one hand, we will discuss statistical convergence theory towards the ‘ground truth’ as the statistical sample size increases. Secondly, we address recent progress in studying the computational complexity of the numerical algorithms required. This includes polynomial-time mixing results for high-dimensional Markov Chain Monte Carlo (MCMC) methods as well as recent “generalized M-estimators” introduced in here.