2026

On Wall-Crossing Invariant Combinations of Welschinger Numbers for Rational Surfaces with a Conic Bundle

Finashin, Sergey — Tue, 01 Sep 2026 00:00:00 GMT

On Wall-Crossing Invariant Combinations of Welschinger Numbers for Rational Surfaces with a Conic Bundle Finashin, Sergey; Kharlamov, Viatcheslav We refine the real enumerative invariants insensitive to changes in the real structure introduced in our previous paper on del Pezzo surfaces, and extend these refined invariants to arbitrary rational surfaces. The construction involves a choice of an auxiliary conic bundle together with a Pinˉ-structure. We prove that the resulting invariants are independent of these auxiliary choices and establish recursive relations for their computation. Acknowledgements. This research was initiated in 2024 during our Research in Residence visits to Centre International de Rencontres Mathématiques, Luminy, and to Institut des Hautes Études Scientifiques, Paris. We completed it during our Research Fellows staying in MFO, Oberwolfach, 2026. We thank these institutions for hospitality and excellent working conditions.

Locally Conformally Flat Manifolds with Positive Scalar Curvature: Kleinian Groups, Moduli Spaces, and Euclidean Rigidity

Deng, Jialong — Tue, 01 Sep 2026 00:00:00 GMT

Locally Conformally Flat Manifolds with Positive Scalar Curvature: Kleinian Groups, Moduli Spaces, and Euclidean Rigidity Deng, Jialong We study locally conformally flat (LCF) Riemannian manifolds with nonnegative or positive scalar curvature (PSC), using the conformal boundary of the developing image. For closed oriented LCF $n$-manifolds with PSC and infinite fundamental group, $n\ge5$, we bound the macroscopic dimension of their Riemannian universal covers by $\lfloor(n-1)/2\rfloor$, establish the existence of a nontrivial homotopy group above the middle dimension, and, under mild additional hypotheses, bound the Hausdorff dimension of the limit sets of their Kleinian groups in the interval $(1,(n-2)/2)$.
In particular, no closed aspherical manifold admits an LCF metric with PSC. If the scalar curvature is at least $n(n-1)$ and the manifold is not isometric to the round sphere, then every smooth nonzero-degree map to the sphere expands somewhere when its Kleinian group is elementary, while the developing map expands somewhere whenever the fundamental group is infinite. We prove that the space of LCF metrics with PSC and its moduli space are contractible in dimension three for finite fundamental group and for $S^2\times S^1$, and that the moduli space is empty or contractible for smooth manifolds homeomorphic to spherical space forms but not diffeomorphic to $S^n$ in dimensions $n\ge4$. For complete open simply connected LCF manifolds of nonnegative scalar curvature, we obtain Euclidean rigidity under additional topological hypotheses at infinity (and, in dimension three, from vanishing second homology alone). We also show that, in dimensions $n\ge4$, the Euclidean conclusion can fail when neither of the two additional topological hypotheses is assumed: we construct complete contractible examples of PSC that are not homeomorphic to $\mathbb{R}^n$. Acknowledgments. This research was supported through the program “Oberwolfach Leibniz Fellows” by the Mathematisches Forschungsinstitut Oberwolfach. This work originates from a broader project initiated during my postdoctoral stay at the Yau Center. I thank Akito Futaki and Shing-Tung Yau for their support during that time, and this work was supported by the YMSC Overseas Shuimu Scholarship and NSFC 12401063, and partially supported by NSFC 12271284. Results were presented in seminars in 2025. I thank Gerhard Huisken, Thomas Schick, and Uwe Semmelmann for helpful discussions in person.

Spherical Subgroups in Reductive Algebraic Groups

Knop, Friedrich — Wed, 01 Jul 2026 00:00:00 GMT

Spherical Subgroups in Reductive Algebraic Groups Knop, Friedrich; Röhrle, Gerhard In [KR15], we determined all spherical affine homogeneous varieties for simple algebraic groups in arbitrary characteristic. The present paper extends this classification to semisimple groups. This generalizes work done independently by Brion and Mikityuk in characteristic zero. Our primary approach relies on a lifting lemma to characteristic zero, enabling us to directly apply Brion’s and Mikityuk’s results. [MSC 2020] 20G15; 14M27; 14M17; 14L30; 20G05; Acknowledgments: Work on this paper began during a visit to the Mathematisches Forschungsinstitut Oberwolfach (MFO) under the "Oberwolfach Research Fellows" program; we thank the MFO for its support.

Transverse Foliations for Two-Degree-of-Freedom Mechanical Systems

de Paulo, Naiara V. — Sun, 01 Mar 2026 00:00:00 GMT

Transverse Foliations for Two-Degree-of-Freedom Mechanical Systems de Paulo, Naiara V.; Kim, Seongchan; Salomão, Pedro A. S.; Schneider, Alexsandro We investigate the dynamics of a two-degree-of-freedom mechanical system for energies slightly above a critical value. The critical set of the potential function is assumed to contain a finite number of saddle points. As the energy increases across the critical value, a disk-like component of the Hill region gets connected to other components precisely at the saddles. Under certain convexity assumptions on the critical set, we show the existence of a weakly convex foliation in the region of the energy surface where the interesting dynamics takes place. The binding of the foliation is formed by the index-2 Lyapunov orbits in the neck region about the rest points and a particular index-3 orbit. Among other dynamical implications, the transverse foliation forces the existence of periodic orbits, homoclinics, and heteroclinics to the Lyapunov orbits. We apply the results to the Hénon-Heiles potential for energies slightly above 1/6. We also discuss the existence of transverse foliations for decoupled mechanical systems, including the frozen Hill's lunar problem with centrifugal force, the Stark problem, the Euler problem of two centers, and the potential of a chemical reaction. NdP was partially supported by CAPES/MATH-AMSUD 88881.878892/2023-01. SK was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No. RS-2025-16070003). A part of this work was done during SK’s visit to the Mathematisches Forschungsinstitut Oberwolfach (MFO) as an Oberwolfach Leibniz Fellow in 2020. SK cordially thanks the MFO for its excellent support and stimulating working atmosphere. PS acknowledges the support of the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai and the 2022 National Foreign Experts Program. PS was partially supported by FAPESP (2016/25053-8) and CNPq (306106/2016-7). PS was partially supported by the National Natural Science Foundation of China (grant number W2431007). PS thanks the support of the Shenzhen International Center for Mathematics - SUSTech. AS thanks the Instituto de Matemática Pura e Aplicada (IMPA) for the post-doc position. Part of this work was conducted during visits to the Southern University of Science and Technology (SUSTech) and the Kongju National University (KNU). AS thanks both institutes for their hospitality.; [MSC 2020] Primary 37J55; Secondary 53D35.

On Constructing Small Subgraphs in the Budget-Constrained Random Graph Process

Antoniuk, Sylwia — Sun, 01 Feb 2026 00:00:00 GMT

On Constructing Small Subgraphs in the Budget-Constrained Random Graph Process Antoniuk, Sylwia; Espuny Díaz, Alberto; Petrova, Kalina; Stojaković, Miloš Consider the budget-constrained random graph process introduced by Frieze, Krivelevich and Michaeli, where each time an edge is offered through the (standard) random graph process we must irrevocably decide whether to "purchase" this edge or not, with our goal being to construct a graph which satisfies some property within a given time $t$ and while purchasing at most $b$ edges. We consider the problem of constructing graphs containing certain fixed small subgraphs. We provide an optimal strategy for building a graph which contains a copy of $K_4$, showing that budget $b=\omega(\max\{n^8/t^5,n^2/t\})$ suffices and that if $b=o(\max\{n^8/t^5,n^2/t\})$ then no strategy can a.a.s. produce a graph containing a copy of $K_4$. This resolves a problem raised by Iľkovič, León and Shu. More generally, we obtain analogously tight results for containing a wheel of any fixed size, or a graph consisting of a tree plus one additional universal vertex. We also tackle the problem of constructing graphs containing a copy of $K_5$, obtaining both lower and upper bounds on the optimal budget, though a gap remains in this case. This research was supported by the Oberwolfach Research Institute for Mathematics through its Oberwolfach Research Fellows (OWRF) program. S. Antoniuk was supported by Narodowe Centrum Nauki, grant 2024/53/B/ST1/00164. A. Espuny Díaz was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through project no. 513704762. K. Petrova was supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 101034413 . M. Stojaković was partly supported by the Science Fund of the Republic of Serbia, Grant #7462: Graphs in Space and Time: Graph Embeddings for Machine Learning in Complex Dynamical Systems (TIGRA), and partly supported by the Ministry of Science, Technological Development and Innovation of the Republic of Serbia (grants 451-03-33/2026-03/200125 & 451-03-34/2026-03/200125).