Weekly Bulletin

The Hofer–Zehnder capacity is a numerical symplectic invariant defined via Hamiltonian dynamics and closely related to the existence of periodic orbits. In this talk, I survey known and still missing results on the Hofer–Zehnder capacity of subsets of (co-)tangent bundles, with an emphasis on explicit computations for disk bundles. Lower bounds are typically obtained via Riemannian billiards, while upper bounds rely on Floer theoretic methods. So far, all explicit computations are restricted to highly degenerate settings, such as constant curvature metrics or more generally symmetric spaces. I will report on work in progress aimed at extending these techniques to less degenerate Riemannian metrics and relates its value to certain diastolic quantities.

We use generalized minimal or capillary hypersurfaces to study comparison theorems of certain class of positively curved manifolds (with boundary); this is also called Gromov's μ-bubble method. We apply these results to obtain useful geometric bounds such as Urysohn width, bandwidth estimates, and study rigidity of (free boundary) minimal hypersurfaces.

<p>In 1997, Benjamin Weiss constructed a non-trivial translation-invariant probability measure on entire functions. In this talk, we give several constructions of Aut(D)-invariant probability measures on holomorphic self-maps of the unit disk by looking at conformally-natural shapes of deterministic and random infinite hyperbolic trees. The construction uses an infinite version of Grothendieck’s dessin d'enfant. No prior knowledge is assumed.<br><br>This is joint work with O. Ivrii.</p>

<p style="margin-top: 0px; margin-bottom: 0px; caret-color: #000000; color: #000000; font-family: Calibri, Helvetica, sans-serif; font-size: 18.666666px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-line: none; text-decoration-thickness: auto; text-decoration-style: solid;" data-start="64" data-end="188">I will begin with a brief survey of the history of the subject. I will then present results from two of my recent articles. The first, joint work with Athreya and Hubert, investigates complexity in rational lattice polygons. The second, with Krueger and Nogueira, studies complexity in generic polygons, as well as the metric complexity of arbitrary polygons.</p>

I will discuss how the strong expansion properties of residually finite groups with Kazhdan's property (T) are incompatible with three dimensional Poincaré duality in the sense of Wall.

The study of the limiting absorption principle for elliptic equations with periodic structures is very challenging when the dimension is greater than one . The fundamental obstacle lies in the mismatch between directional physical reality and the direction-independent classic spectral analysis, resulting in intricate geometry of band structures and Fermi surfaces in high dimensions. In this paper, we introduce the novel Sliced Spectral Analysis (SSA) framework which introduce the direction into classic spectral analysis. It dismantles the geometrical complexities by reducing the problems to its analytical essence on the one-dimensional slices. With the new SSA framework, the solution can be formulated in a semi-analytic form, which not only gives an explicit representation, but also reflects the phenomenon in physics. The new SSA approach resolves the mismatch between mathematics and physics, and also breaks the dimensional barriers. It also opens a door to a lot of further possibilities, ranging from the analysis of solutions and numerical simulations for the solutions.

Empirical research in the social, health, and economic sciences is often based on categorical variables, such as questionnaire responses, self-reported health, or counting processes. Yet, just like in continuous variables, contamination might be present in such data, for instance (but not limited to) careless responses, which can cause severe biases in the commonly employed maximum likelihood estimation. However, robustifying estimation against contamination is challenging because categorical variables, by their very nature, cannot take arbitrarily large values and may not even admit a numerical interpretation in the first place. Consequently, the extensive literature on outlier-robust M-estimation may not be applicable. As a remedy, we propose a general framework for robust estimation and inference of models for categorical data, called C-estimation ("C" for categorical; Welz, 2024). In addition to offering enhanced robustness, we show that C-estimators are asymptotically consistent, normally distributed, and fully efficient. The latter property starkly contrasts M-estimation, which is characterized by a fundamental tradeoff between robustness and efficiency. C-estimators avoid this tradeoff by exploiting the categorical nature of the data. Furthermore, C-estimators do not incur any additional computational cost and are therefore also attractive from a practical perspective. This talk aims to strike a balance between theoretical aspects of C-estimators and an application thereof to psychometric structural equation models (SEMs) with ordinal measurements. We show that using a robustly estimated polychoric correlation Matrix (Welz, Mair & Alfons, 2025+) for SEM estimation can substantially improve SEM fit, enhance the accuracy of parameter estimates, and help identify low-quality responses (such as careless responses). Furthermore, the proposed approach is very general because it does not necessitate any adjustments to the SEM itself and it can be used in conjunction with any method for SEM fitting, such as maximum likelihood, least-squares-based approaches like Diagonally Weighted Least Squares (DWLS), or Bayesian techniques. Our proposed procedure is implemented in the free open-source R package "robcat" (https://CRAN.R-project.org/package=robcat), whose source is written in fast and efficient C++ code. REFERENCES: - Welz, M. (2024). Robust estimation and inference for categorical data [arXiv:2403.11954]. https://doi.org/10.48550/arXiv.2403.11954 - Welz, M., Mair, P., & Alfons, A. (2025+). Robust estimation of polychoric correlation. Psychometrika, https://doi.org/10.1017/psy.2025.10066

Relying solely on carbon pricing to meet Paris Agreement targets imposes prohibitive economic costs. We show that achieving the 2 ˝C stabilization goal through taxation alone requires carbon prices reaching approximately $474/tCO 2 , a level that triggers widespread capital divestment. To resolve this dilemma, we develop a dynamic stochastic integrated assessment model that optimizes a portfolio of carbon taxation, clean-capital subsidies, adaptation investment, and carbon dioxide removal (CDR). Our analysis identifies CDR as a necessary condition for stabilization rather than a supplementary measure. In the optimal portfolio, net carbon removal scales from 0.04 to 3.7 GtCO 2 /year by 2050 to maintain temperature targets at a feasible cost. These instruments act as economic complements: carbon pricing and subsidies target new emissions, CDR reduces the legacy atmospheric stock, and adaptation protects the economic base from immediate damages. Consequently, the welfare gains from the integrated portfolio significantly exceed the sum of individual instrument effects. We conclude that optimal climate policy requires shifting from a price-centric framework to a diversified approach in which carbon removal and adaptation serve as core pillars of decarbonization.

In 1973, Shimura discovered a way to associate a holomorphic half-integral weight modular form h with a classical cusp form f. Subsequently, in the 1980s, Waldspurger proved a remarkable formula relating squares of the Fourier coefficients of h and quadratic twists of L-values of f. In the spirit of these results, we prove that one can associate "quaternionic" modular forms on the group G2 with dihedral cusp forms f, whose Fourier coefficients are explicitly related to cubic twists of L-values of f. This gives the first examples where a conjecture of Gross from 2000 has been fully verified. (Joint work with Petar Bakić, Siyan Daniel Li-Huerta, and Naomi Sweeting.)