2025

Scalar Curvature in Dimension 4

Deng, Jialong — Mon, 01 Dec 2025 00:00:00 GMT

Scalar Curvature in Dimension 4 Deng, Jialong We prove that every locally conformally flat metric on a closed, oriented hyperbolic $4$-manifold with scalar curvature bounded below by $-12$ satisfies Schoen’s conjecture. We also classify all closed Riemannian $4$-manifolds of positive scalar curvature that arise as total spaces of fibre bundles. For a closed locally conformally flat manifold $(M^4,g)$ with scalar-flat and $\pi_2(M^4) \neq 0$, we show that the universal Riemannian cover $(\widetilde{M},\tilde{g})$ is homothetic to the standard product $\mathbb{H}^2 \times \mathbb{S}^2$. This affirmatively answers a question of N. H. Noronha. The author acknowledges support from the Oberwolfach Leibniz Fellows programme (MFO), the YMSC Overseas Shuimu Scholarship, the Simons Center for Geometry and Physics, and ICMS Edinburgh (workshops on Geometric Measure Theory on Metric Spaces with Applications to Physics and Geometry and Geometric Moduli Spaces, respectively). I thank Gerhard Huisken for discussions on the Ricci flow. This work originates from a broader project initiated during my postdoctoral stay at the Yau Center. During that time, this work was also supported by NSFC 12401063 and partially by NSFC 12271284. I am deeply grateful to Shing-Tung Yau and Akito Futaki for their trust and support, which allowed me to pursue independent research.

Inversion of the Unbounded Finite Hilbert Transform on $L^1$

Curbera, Guillermo P. — Mon, 01 Dec 2025 00:00:00 GMT

Inversion of the Unbounded Finite Hilbert Transform on $L^1$ Curbera, Guillermo P.; Okada, Susumu; Ricker, Werner J. The finite Hilbert transform $T$ is a classical singular integral operator with its roots in aerodynamics, elasticity theory and image reconstruction. The setting has always been to consider $T$ as acting in those rearrangement invariant spaces $X$ over (−1, 1) which $T$ maps boundedly into itself (e.g., $L^p$ for 1 < $p$ < ∞), a setting which excludes $L^1$. Our aim is to go beyond boundedness and to address the case $X$ = $L^1$. For this, we need to consider $T$ as an unbounded operator on $L^1$. Is there a “suitable” domain for $T$? Yes. Remarkably, for $T$ acting on this domain, we prove a full inversion theorem, together with refined versions of both the Parseval and Poincaré-Bertrand formulae, which are crucial results needed for the proof. This domain, a somewhat unusual space, turns out to be a rather extensive subspace of $L^1$, fails to be an ideal and properly contains the Zygmund space $L$log$L$ (which is the largest ideal of functions that $T$ maps boundedly into $L^1$). [MSC 2020] (Primary) 44A15; 46E30; (Secondary) 47A53; 47B34; The first and third authors acknowledge the support of the Mathematisches Forschungsinstitut Oberwolfach via the Oberwolfach Research Fellows program (March, 2025). The first author also acknowledges the support of PID2021-124332NB-C21 (FEDER(EU)/Ministerio de Ciencia e Innovación-Agencia Estatal de Investigación) and FQM-262 (Junta de Andalucía).

The $q$-Deformed Random-to-Random Family in the Hecke Algebra

Brauner, Sarah — Sat, 01 Nov 2025 00:00:00 GMT

The $q$-Deformed Random-to-Random Family in the Hecke Algebra Brauner, Sarah; Commins, Patricia; Grinberg, Darij; Saliola, Franco We generalize Reiner-Saliola-Welker's well-known but mysterious family of $k$-random-to-random shuffles from Markov chains on symmetric groups to Markov chains on the Type-$A$ Iwahori-Hecke algebras. We prove that the family of operators pairwise commutes and has eigenvalues that are polynomials in $q$ with non-negative integer coefficients. Our work generalizes work of Reiner-Saliola-Welker and Lafrenière for the symmetric group, and simplifies all known proofs in this case. Acknowledgments: We thank Pavel Etingof, Nadia Lafrenière, and Vic Reiner for interesting and informative conversations. This paper was started at the Mathematisches Forschungsinstitut Oberwolfach in October 2024, as the four authors were Oberwolfach Research Fellows (2442p), and finished at the ICERM program "Categorification and Computation in Algebraic Combinatorics" in Fall 2025. The first author is partially supported by the NSF MSPRF DMS-2303060 and the second author was partially supported by an NSF GRFP fellowship. The fourth author was supported by NSERC (RGPIN-2023-04476). The SageMath computer algebra system [22] was used to find several of the results.; [MSC 2020] 20C08; 20C30; 60J10; 05E10

Some Notes on Pontryagin Duality of Abelian Topological Groups

Hofmann, Karl Heinrich — Sat, 01 Nov 2025 00:00:00 GMT

Some Notes on Pontryagin Duality of Abelian Topological Groups Hofmann, Karl Heinrich; Kramer, Linus We consider several questions related to Pontryagin duality in the category of abelian pro-Lie groups. This research was supported through the program ”Oberwolfach Research Fellows” by the Mathematisches Forschungsinstitut Oberwolfach in 2025. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2044-390685587, Mathematics M¨unster: Dynamics-Geometry-Structure.

Parabolic Normalizers in Finite Coxeter Groups as Subdirect Products

Douglass, J. Matthew — Mon, 01 Sep 2025 00:00:00 GMT

Parabolic Normalizers in Finite Coxeter Groups as Subdirect Products Douglass, J. Matthew; Pfeiffer, Götz; Röhrle, Gerhard We revisit the structure of the normalizer $N_W(P)$ of a parabolic subgroup $P$ in a finite Coxeter group $W$, originally described by Howlett. Building on Howlett's Lemma, which provides canonical complements for reflection subgroups, and inspired by a recent construction of Serre for involution centralizers, we refine this understanding by interpreting $N_W(P)$ as a subdirect product via Goursat's Lemma. Central to our approach is a Galois connection on the lattice of parabolic subgroups, which leads to a new decomposition \begin{align*} N_W(P) \cong (P \times Q) \rtimes ((A \times B) \rtimes C)\text, \end{align*} where each subgroup reflects a structural feature of the ambient Coxeter system. This perspective yields a more symmetric description of $N_W(P)$, organized around naturally associated reflection subgroups on mutually orthogonal subspaces of the reflection representation of $W$. Our analysis provides new conceptual clarity and includes a case-by-case classification for all irreducible finite Coxeter groups. Acknowledgements: Work on this paper began during a visit to the Mathematisches Forschungsinstitut Oberwolfach under the Oberwolfach Research Fellows Programme; we thank them for their support. J.M. Douglass would like to acknowledge that some of this material is based upon work supported by, and while serving at, the National Science Foundation. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Renormalisation of Singular SPDEs with Correlated Coefficients

Clozeau, Nicolas — Mon, 01 Sep 2025 00:00:00 GMT

Renormalisation of Singular SPDEs with Correlated Coefficients Clozeau, Nicolas; Singh, Harprit We show local well-posedness of the g-PAM and the $\phi^{K+1}_2$-equation for $K\geq 1$ on the two-dimensional torus when the coefficient field is random and correlated to the driving noise. In the setting considered here, even when the model in the sense of [Hai14] is stationary, naive use of renormalisation constants in general leads to variance blow-up. Instead, we prove convergence of renormalised models choosing random renormalisation functions analogous to the deterministic variable coefficient setting. The main technical contribution are stochastic estimates on the model in this correlated setting which are obtained by a combination of heat kernel asymptotics, Gaussian integration by parts formulae and Hairer-Quastel type bounds [HQ18]. Both authors would like to thank the Oberwolfach Research Fellows (OWRF) program for supporting a research stay during which a part of this work was carried out, as well as Rhys Steele and Lucas Broux for valuable discussions during that stay. HS gratefully acknowledges financial support from the Swiss National Science Foundation (SNSF), grant number 225606.

On Angular Momentum Algebras and their Relations

Calvert, Kieran — Fri, 01 Aug 2025 00:00:00 GMT

On Angular Momentum Algebras and their Relations Calvert, Kieran; de Martino, Marcelo; Oste, Roy In this paper, we study the centraliser of $\mathfrak{osp}(1|2)$, denoted the total angular momentum algebra (TAMA), in the Weyl Clifford algebra. The TAMA extends the angular momentum algebra (AMA), which arises as the centraliser of $\mathfrak{sl}(2)$ and admits a diagrammatic presentation via the crossing relation described by Feigin and Hakobyan. Using Young symmetrisers we construct an analogue relation for the even subalgebra of the TAMA. We prove that for rank $4$ and $5$ these relations generate a presentation for the even subalgebra of the TAMA. Acknowledgments. We gratefully acknowledge the hospitality and excellent working conditions provided by the Mathematisches Forschungsinstitut Oberwolfach, where KC and MDM were Oberwolfach Research Fellows during the Spring of 2024. MDM thanks the Mathematics Department at Lancaster University for hospitality during a visit and collaboration with KC. MDM acknowledges support from the special research fund (BOF) of Ghent University [BOF20/PDO/058].

Renormalised Models for Variable Coefficient Singular SPDEs

Broux, Lucas — Tue, 01 Jul 2025 00:00:00 GMT

Renormalised Models for Variable Coefficient Singular SPDEs Broux, Lucas; Singh, Harprit; Steele, Rhys In this work we prove convergence of renormalised models in the framework of regularity structures [Hai14] for a wide class of variable coefficient singular SPDEs in their full subcritical regimes. In particular, we provide for the first time an extension of the main results of [CH16, HS24, BH23] beyond the translation invariant setting. In the non-translation invariant setting, it is necessary to introduce renormalisation functions rather than renormalisation constants. We show that under a very general assumption, which we prove covers the case of second order parabolic operators, these renormalisation functions can be chosen to be local in the sense that their space-time dependence enters only through a finite order jet of the coefficient field of the differential operator at the given space-time point. Furthermore we show that the models we construct depend continuously on the coefficient field.

Generalized Bose-Einstein Condensation in the Kac-Luttinger Model

Boccato, Chiara — Tue, 01 Jul 2025 00:00:00 GMT

Generalized Bose-Einstein Condensation in the Kac-Luttinger Model Boccato, Chiara; Kerner, Joachim; Pechmann, Maximilian; Spitzer, Wolfgang In this article, we prove generalized Bose–Einstein condensation (BEC) at zero temperature in the random Kac–Luttinger model for repulsive two-particle interactions that are scaled suitably in the limit of large volume. Compared to previous works, by proving generalized condensation rather than the macroscopic occupation of finitely many single-particle states (type-I BEC), we can allow for stronger two-particle interactions. We discuss implications of the result which include a possible transition in the type of the condensation. Acknowledgement. This research was supported through the program ”Oberwolfach Research Fellows” by the Mathematisches Forschungsinstitut Oberwolfach in 2024.

Spectral Regularity and Defects for the Kohmoto Model

Beckus, Siegfried — Tue, 01 Apr 2025 00:00:00 GMT

Spectral Regularity and Defects for the Kohmoto Model Beckus, Siegfried; Bellissard, Jean; Thomas, Yannik We study the Kohmoto model including Sturmian Hamiltonians and the associated Kohmoto butterfly. We prove spectral estimates for the operators using Farey numbers. In addition, we determine the impurities at rational rotations leading to the spectral defects in the Kohmoto butterfly. Our results are similar to the ones obtained for the Almost-Mathieu operator and the associated Hofstadter butterfly. Acknowledgment. S.B. and Y.T. are grateful to Ram Band for various interesting discussions on the Kohmoto butterfly and its presentation in this work. S.B. and Y.T. are thankful to Lior Tenenbaum for inspiring discussions on the Lebesgue measure of approximations and for pointing out flaws and typos in an earlier versions. In addition, S.B. would like to thank Daniel Lenz for an interesting discussion on Sturmian systems and their factorization through the torus. This research was supported through the program “Research in Pairs” by the Mathematisches Forschungsinstitut Oberwolfach in 2018 hosting S.B. and J.B.. This work was supported by the Deutsche Forschungsgemeinschaft [BE 6789/1-1 to S.B.].