Titel: A boundary operator characterization of the recurrence of link graphs

Abstract: In this talk we will observe a phenomenon about boundary operators \partial on infinite weighted simplicial complexes with unbounded geometry. Here, in contrast to the finite setting,  the characteristic property \partial \partial = 0 no longer holds in full generality. We will then characterize this property by showing its equivalence to the recurrence of link graphs, which is based on the idea of localization in the spirit of Garland's method.

Titel: Projective Geometries over finite local rings: Rigidity and Isospectrality

Abstract: We study the simplicial order complexes obtained from free modules over finite local rings, which arise naturally as geodesic spheres in Bruhat-Tits buildings over non-archimedean local fields. We establish powerful connections between the algebra and the geometry. First, we prove two forms of rigidity, showing that the automorphism group of the complex is entirely determined by the underlying algebraic group structure, even when studying sparse induced subgraphs. Next, we analyze these sparse subgraphs and compute their spectra. Our main finding is that these graphs are excellent expanders, demonstrating that these geometric structures possess optimal connectivity properties and providing a direct proof of expansion for geodesic powers of the buildings.
Combining the two results we show that the graphs arising from n-spaces over Z/p^r and (F_p)[t]/t^r are isospectral and non-isomorphic.

Titel: Calculus of variations for nonlocal Sobolev–Bregman forms

Abstract: The Sobolev–Bregman integral forms are an $L^p$ version of quadratic forms defining the fractional Sobolev spaces $H^s$, that emerged in a natural way from the probabilistic potential theory in $L^p$. The forms are strongly nonlinear, in the sense that their natural domain is not linear. Despite that, we prove existence of minimizers for the exterior value problem, and using a special choice of curves we establish an Euler–Lagrange equation for the minimizers. We also prove a Green-type formula and investigate the domain of the polarized form, which is surprisingly challenging. Based on joint work with Krzysztof Bogdan, Katarzyna Pietruska-Pałuba and Christian Rose.

Titel: On some arithmetic problems arising in classical problems of Spectral Geometry

Abstract: It has been known since the 19th century that classical problems in Spectral Geometry in very symmetric circumstances lead to interesting arithmetic problems. In particular, the problems of the vibrating round membrane, vibrating round plate and vibrating sphere lead to questions about common roots of special functions (Bessel, Legendre, etc.). I'll describe the works of Porter, Siegel and an open conjecture by Stieltjes, in addition to recent conjectures and results in the subject.

The talk is based on joint works with Yuri Lvovsky, Adi Weller-Weiser, Daniel Rosenblatt and Borys Kadets.

Titel: An optimal fractional Hardy inequality on the discrete half-line

Abstract: In this talk, we will discuss the discrete fractional Hardy inequality on the discrete half-line ℕ with optimal weight function. We will construct an optimal Hardy-weight W_σ^op of the fractional Laplacian (−Δ_ℕ)^σ on ℕ for exponents σ∈(0,1]. As a consequence, we obtain an estimate of the sharp constant in the fractional Hardy inequality with the classical Hardy-weight n^−2σ on ℕ. Certain unique continuation results will also be derived as an application.
This talk is based on a joint work with Rubén de la Fuente-Fernández.

References

[1] U. Das, R. Fuente-Fernández. An optimal fractional Hardy inequality on the discrete half-line. Arxiv:2507.06716, 2025.

Titel: Packing Dimension of Spectral Measures for Quasiperiodic Operators with Monotone Potentials

Abstract: Quasiperiodic Schrödinger operators exhibit a sharp transition of the spectral type governed by the interplay between the Lyapunov exponent L(E) and the arithmetic properties of the frequency α. These are quantified by β(α), which measures how well α is approximated by rationals. Roughly speaking, L(E)=0 implies absolutely continuous spectrum, L(E)>β(α) implies pure point spectrum, and the intermediate regime 0<L(E)<β(α) produces singular continuous spectrum. A natural question is how singular the spectrum is in this transition region. Since positive Lyapunov exponent forces the Hausdorff dimension of spectral measures to be zero, one instead studies the packing dimension. Recent work of Jitomirskaya–Liu–Tcheremchantsev for the AMO provides new lower-bound techniques via quantitative singularity analysis.

In this talk, we discuss quasiperiodic operators with monotone potentials, present the analytic and arithmetic tools used to study this regime, and show how to obtain sharp packing-dimension bounds in the range 0<L(E)<β(α), with explicit dependence on the difference β(α)−L(E).

Titel: On the spectra of periodic graph operators

Abstract: Periodic Schrödinger operators on graphs play an important role in mathematical physics, as they furnish models of crystals and photonic metamaterials. I will describe some recent results that connect combinatorial and topological aspects of the graph with the measure of the spectrum and spectral gaps.

Title: Diamater and connectivity of amply regular graphs via Ricci curvature

Abstract: A d-regular graph is called amply regular if there are constants α,β such that any two adjacent vertices have α commom neighbors and any two vertices at distance 2 have β commom neighbors.  Through calculations of Lin-Lu-Yau curvature and Bakry-Emery curvature, we obtain new diameter bounds for amply regular graphs, which make a progress on several related conjectures.   Moreover, we explore the interaction between connectivity and Lin-Lu-Yau curvature.  Consequently, we estimate or determine the connectivity and edge-connectivity of amply regular graphs.

Title: Stability of Higher-order Hardy Inequalities for Baouendi–Grushin Vector Fields

Abstract:  In this talk, I will present a unified framework for deriving and improving Hardy, Rellich, and Hardy–Rellich inequalities associated with the Baouendi–Grushin operator, emphasizing both identity-based techniques and stability analysis. In the first part, we explore sharp inequalities and identities for the Grushin operator using spherical harmonics and the Bessel pair approach. In the second part, we analyze the extremizer stability of higher-order Hardy–Rellich inequalities in the radial Grushin setting. By connecting subcritical and critical inequalities through explicit identities, we derive sharp remainder terms and establish quantitative stability estimates via extremal profile approximations. These findings contribute both to the structural understanding of Grushin-type operators and to the development of refined functional inequalities in degenerate settings.

Title: Relating the integrated density of states to spectral flow in aperiodic chains.

Abstract: Aperiodic insulators have typically richer topological invariants than periodic ones. 

In particular the gap labelling group has more generators. We ask, how we can “measure” the gap labels via spectral flows, like, for instance, the flow of eigenvalues of edge states in one dimensional models.  The simple bulk edge correspondence will not do it, because the most interesting models have a discrete hull which is a Cantor set. This was already observed in our earlier work with Emil Prodan on Kohmoto models based on Sturmian sequences. In that work we proposed to augment the hull to make certain flip processes continuous and thereby visible as spectral flow. In this talk I will discuss a generalisation of the above model which has a gap labelling group with one more generator. This generator can be related to a spectral flow density, not of edge states, but of bulk states. When numerically solving the Schroedinger equation to visualise the spectral flow (for a periodic approximation) we face a difficulty: the spectral flow associated with the extra generator obstructs the detection of the edge spectral flow. The solution to this problem is suggested by K-theory: stacking our model with a “topologically trivial” model. 

Title: A gradient flow that is none: Heat flow with Wentzell boundary conditions

Abstract: We establish a representation of the heat flow with Wentzell boundary conditions on smooth domains as gradient descent dynamics for the entropy in a suitably extended Otto manifold of probability measures with additional boundary parts. Yet it is shown that for weak boundary diffusion, the associated Fokker–Planck dynamics cannot be recovered from any entropy-driven metric JKO-Wasserstein scheme, at least if the underlying point metric satisfies certain natural regularity assumptions. The talk is based on joint work with Léonard Monsaingeon, Michiel Renger and Max von Renesse.
 

Titel: On the Eldan–Gross Inequality

Abstract: Recently, Eldan and Gross developed a stochastic analysis approach to proving functional inequalities on discrete hypercubes. Motivated by a conjecture of Talagrand, one of their main results relates to the sensitivity, variance, and influences of Boolean functions. In this talk, I will discuss an alternative proof of this inequality based on hypercontractivity and an isoperimetric-type inequality of the heat semigroup. Our proof also extends to biased hypercubes and continuous spaces with positive Ricci curvature lower bounds in the sense of Bakry and Émery. This is based on joint work with Paata Ivanisvili. 

Titel: Every Salami has two ends

Abstract: In this talk, we prove that every salami has exactly two ends. As is well known to experts, a salami is a weighted graph with non-negative Ollivier Ricci curvature with at least two ends of infinite mass. As an application we show a discrete positive mass theorem. The key idea is based on a non-linear heat flow for which we provide long-term convergence results in a general framework. The talk is based on joint works with Bobo Hua, Ruowei Li and Haohang Zhang.

Titel: Uniqueness for the heat equation on infinite graphs

Abstract: The talk is concerned with uniqueness of solutions to a class of heat equations with positive density posed on infinite weighted graphs. We separately consider the case when the density is bounded from below by a positive constant and the case of possibly vanishing density, showing that these two scenarios lead to two different classes of uniqueness.

Titel: On nonlinear Dirichlet forms

Abstract: The subject of nonlinear Dirichlet forms is rather new and as such, still has many simple open questions. In this talk, we will establish a new characterization that more closely resembles the well-known bilinear case. Furthermore, we show applications of function-valued analogues of the capacity set function and the resistance metric. The former answers an open problem posed by Burkhard Claus in his PhD thesis, while the latter generalizes the various known results on homogeneous forms.

Titel: Classical transport in a maximally chaotic chain

Abstract: CAT maps (for Continuous Automorphisms on the Torus) are known as one family of chaotic systems. They have been intensively used to study the quantum/classical correspondence. In the last decade the focus has shifted towards coupling them along a chain or a lattice. They have become one central example in "quantum many-body chaos". In this talk I will use a more recently introduced type of coupling for the cat maps, based on symmetry. I will describe how to establish ergodicity and count the periodic orbits for this coupling. It further allows for a fully explicit treatment of the transport problem. The diffusion coefficient is evaluated using a saddle point approximation and agrees well with our numerical experiments.

Titel: Dynamics of the Sturmian Trace Skew Product

Abstract: In this talk, we focus on the spectrum of the discrete Schrödinger equation with a quasi-periodic potential called Sturmian potential. Eigenvector problem with a Sturmian potential is associated to a dynamics of the Markov surface, together with a control variable of "rotation angle," leading us to a study of a skew product system.
Our understanding is that this skew product system exhibits a sort of hyperbolicity. As a first step to establish it, we have shown that there exists a cone field on the Markov surface that contracts by the dynamics, which is independently defined by the angle variable. The discovery is more or less elementary, initiated by some geometric observations of the Markov dynamics. After sketching the tricks, we will announce some prospectives after having a cone field, including the regularity of the Sturmian spectra.
This talk is based on a joint work with Anton Gorodetski and Victor Kleptsyn.

Titel: Zero measure Cantor spectrum for Sturmian metric graphs

Abstract: A classical result by Belissard et al states that the spectrum of a 1-dimensional Hamiltonian with a Sturmian potential is a singular continuous Cantor set of Lebesgue measure zero. More recently, the same result was proven by Damanik-Fang-Sukhtaiev for a class of aperiodic metric antitrees equipped with the standard Laplacian.

In this talk, we present an analogous result for a large family of metric graphs whose local geometric structure is determined by Sturmian sequences. We prove that almost surely, the spectrum of these metric graphs is a zero measure and purely singular continuous generalized Cantor set. The proof is based on a mixture of Kotani theory with tools from the world of quantum graphs.

Based on joint work with Ram Band.

Titel: Quantum information scrambling and chaos induced by a Hermitian matrix (joint work with Uzy Smilansky)

Abstract: Given an arbitrary Hermitian matrix, considered as a finite discrete quantum Hamiltonian, we use methods from graph and ergodic theories to construct a corresponding unitary scattering approach by defining a quantum Poincaré map and a corresponding stochastic classical Poincaré-Markov map at the same energy on an appropriate discrete phase space. The correspondence between quantum Poincaré map and classical Poincaré-Markov map is an alternative to the standard quantum-classical correspondence based on a classical limit. Most importantly it can be constructed where no such limit exists. Using standard methods from ergodic theory we then proceed to define an expression for the mean Lyapunov  exponent of the classical map. It measures the rate of loss of classical information in the dynamics and relates it to the separation of stochastic classical trajectories in the phase space. We suggest that loss of information in the underlying classical dynamics is an indicator for quantum information scrambling.

Titel: Essential self-adjointness of the Laplacian on weighted graphs: harmonic functions, stability, characterizations and capacity