Nikita Kalinin, Ernesto Lupercio, Mikhail Shkolnikov
We define an $\operatorname{SL}_n(\mathbb{Z})$-invariant tropical zeta function of a convex domain. In dimension 2 it admits boundary Dirichlet-series representation with summands indexed by Farey pairs. For $C^3$ strictly convex domains, it extends meromorphically to $\Re(s)>3/5$, holomorphic there except for a simple pole at $s=2/3$, with residue proportional to equiaffine perimeter. A Tauberian argument yields the $t^{1/3}$ wave-front lattice-perimeter asymptotic for $t\rightarrow 0$.
Joé Brendel, Jean-Philippe Chassé, Laurent Côté
Given a symplectic manifold, can one pack uncountably many Lagrangian submanifolds in a given Hamiltonian isotopy class of this symplectic manifold? We address $C^\infty$ and $C^0$ versions of this question.
Robert Lipshitz, Peter Ozsváth
Fix a 3-manifold $Y$ with boundary $F\amalg F$ and an orientation-preserving involution $τ: Y\to Y$ exchanging the boundary components, with nonempty fixed set. To an appropriate kind of Heegaard diagram for $Y$, we describe how to associate a module over the bordered Heegaard Floer algebra of $F$. These modules satisfy a gluing, or pairing, theorem, and extend the "hat" variant of Guth-Manolescu's real Heegaard Floer homology, $\widehat{HFR}(Y,τ)$. Using these modules, we give a practical algorithm to compute $\widehat{HFR}(Y,τ)$ for real 3-manifolds $(Y,τ)$ with connected fixed set.
Leonardo F. Cavenaghi, Lino Grama, Ludmil Katzarkov, Pedro Antonio Muniz Martins
This paper investigates the geometric and cohomological properties of non-Kähler SYZ mirror symmetry for dual torus fibrations over solvmanifolds in the sense of Lau, Tseng and Yau. We are mainly concerned with three questions: \textbf{(a)} How the Lau-Tseng-Yau notion of non-Kähler SYZ is related to the mapping of supersymmetric branes between symplectic and complex sides; \textbf{(b)} Finding explicit non-Kähler SYZ mirror pairs determined purely by Lie-theoretic data; \textbf{(c)} better understand the cohomological correspondence in the Lau-Tseng-Yau framework (given by a Fourier-Mukai transform), especially concerning the role of Tseng-Yau cohomology. We prove that the Fourier-Mukai transform introduced by Lau-Tseng-Yau exchanges type-A supersymmetric cycles, which are given by special Lagrangian sections equipped with flat $\mathrm{U}(1)$ connections, with type-B cycles, corresponding to line bundles whose connections satisfy the deformed Hermitian-Yang-Mills (dHYM) equation. We provide pure Lie-theoretic criteria for the existence of non-Kähler SYZ mirror pairs whose base manifolds are solvmanifolds. Applying these criteria, we construct new explicit families of mirror pairs from almost abelian and generalized Heisenberg Lie groups, and provide a complete classification of such pairs arising from nilpotent Lie groups. To contextualize the role of the Tseng-Yau cohomology, we link it to noncommutative geometry. We introduce the Tseng-Yau and Bott-Chern mirror bicomplexes. We show that (some of) their enclosed cohomologies reduce to the primitive Tseng-Yau and Bott-Chern cohomologies and that for basic forms they are isomorphic under the Fourier-Mukai transform. As a last contribution, we discuss how to explicitly compute the Tseng-Yau and the Bott-Chern cohomology for the non-Kähler SYZ mirror pairs constructed here.
Hansjörg Geiges, Jakob Hedicke, Murat Sağlam
We show that there is no universal upper bound for the systolic ratio of Bott-integrable contact forms on closed 3-manifolds, thus providing further evidence for the relative flexibility of integrable contact forms. For the proof, we study piecewise linear approximations of Lutz forms and establish integrability of a `plug' constructed by Abbondandolo, Bramham, Hryniewicz and Salomão for pushing up the systolic ratio.
L. Feher, H. R. Dullin
We investigate certain Liouville integrable systems constructed earlier via reduction of the quasi-Hamiltonian double of $\mathrm{SU}(n)$. These systems live on compact connected symplectic manifolds of dimension $2(n-1)$ and can be interpreted as compactified trigonometric Ruijsenaars--Schneider systems. Depending on the value of a parameter $0<y< π$, they arise in two drastically different forms: in type (i) these are toric systems, while in the type (ii) cases they possess globally continuous action variables that generate a Hamiltonian torus action (only) on a dense open subset of the phase space. The principal goal of the paper is to study those fibers of the action map (alias the $\mathbb{T}^{n-1}$ momentum map) which are contained in the complement of the domain of the densely defined torus action occurring in the type (ii) cases. We demonstrate that all such `singular fibers' are smooth connected isotropic submanifolds. We also work out a model of the fibers as quotient spaces of certain subgroups of $\mathrm{SU}(n)$ with respect to an action of another subgroup. The general results are exemplified by determining the vertices of the polytope filled by the action variables in the simplest type (ii) cases that appear for any $n\geq 4$ with $π/(n-1) <y < π/(n-2)$, and proving that the fibers over the `singular vertices' are diffeomorphic to $S^3 \simeq \mathrm{SU}(2)$ in these cases. In this way, our findings enrich the set of examples of Liouville integrable systems with spherical singularities.
Yi Lin
In this article, we introduce a transverse averaging operator for basic forms on a Riemannian foliation equipped with an isometric transverse Lie algebra action, under the assumption that the leaf closure space is compact. Unlike the classical averaging operator in equivariant geometry, which is defined by integration over a compact Lie group, our operator is built purely from infinitesimal transverse data and does not require any global group action. We show that it sends every closed basic form to an invariant basic form representing the same basic cohomology class. As a main application, we compute the diffeological de Rham cohomology of the homogeneous space $G/H$, where $G$ is a connected Lie group, not necessarily compact, and $H$ is a connected Lie subgroup, not necessarily closed. Let $\mathfrak g$ and $\mathfrak h$ be the Lie algebras of $G$ and $H$, respectively. Assuming that $\mathfrak g$ is of compact type and that $G/\overline{H}$ is compact, we prove that \[ H^\bullet_{dR}(G/H)\cong H^\bullet(\mathfrak g,\mathfrak h). \] If, in addition, $\mathfrak h$ is an ideal in $\mathfrak g$, then under the weaker assumption that $G/\overline{H}$ is compact, we obtain \[ H^\bullet_{dR}(G/H)\cong H^\bullet(\mathfrak g/\mathfrak h), \] without requiring $\mathfrak g$ to be of compact type.
Joshua Evan Greene, Andrew Lobb
Suppose that $γ\subset \mathbb{C}$ is a Jordan curve of diameter $2R$ which encloses a region of area $A$. We prove that there exists a subset $I \subset (0,π)$ of measure at least $A/R^2$ such that if $θ\in I$, then there exist four points on $γ$ at the vertices of a rectangle whose diagonals meet at angle $θ$.
Simon Lapointe, Mykola Matviichuk, Brent Pym, Boris Zupancic
We establish existence of functorial orbifold reductions of singularities for Poisson subvarieties in smooth Poisson threefolds. Namely, we show that with enough weighted blowups, one can reduce the singularities of such Poisson subvarieties to certain simple, explicit, local normal forms: Du Val surface singularities where the Poisson structure is locally Jacobian, and plane curves lying in the vanishing locus of a particular linear Poisson structure. The proof combines Abramovich--Temkin--Włodarczyk and McQuillan's recent approach to resolution of singularities for varieties via weighted blowups with some new normal forms for three-dimensional Poisson brackets derived via Poisson cohomology. Along the way, we describe necessary and sufficient conditions for a polyvector field to lift to the weighted blowup of an orbifold along a suborbifold, generalizing criteria of Polishchuk for unweighted blowups of Poisson structures on smooth varieties.
John B. Etnyre
These notes are an expanded version of evening talks at the 2025 Georgia International Topology Conference, and an abbreviated version of talks at Georgia Tech, which were aimed at graduate students. The hope was to indicate a common framework that has been used since the late 1980s to construct homology theories in low-dimensional topology and symplectic and contact geometry. In addition to this, we also try to indicate how the specific nature of the situation being studied dictates the algebraic nature of the chain groups used to define the homology.
Fiammetta Battaglia, Elisa Prato
Symplectic and complex toric quasifolds are a generalization of toric manifolds and orbifolds to the nonrational case. In this paper, we reframe these notions from the viewpoint of algebraic geometry.
Hokuto Konno
This article provides a survey of gauge theory for families, with a particular focus on its applications to diffeomorphism groups of $4$-manifolds that were developed during the period 2021--2025.
Hokuto Konno
This article surveys gauge theory for families and its applications to the comparison between the diffeomorphism group and the homeomorphism group of $4$-manifolds, up to 2021.
Konstantinos Efstathiou, Gabriela Jocelyn Gutierrez-Guillen, Pavao Mardešić, Dominique Sugny
Singular Lagrangian fibrations arising from three-degree-of-freedom integrable Hamiltonian systems remain largely unexplored. While several results describe the global structure of large classes of systems with two degrees of freedom, only a few examples are understood in higher dimensions. We present a three-degree-of-freedom system derived from the two-spin Tavis-Cummings model whose singular Lagrangian fibration exhibits a topology that has not been observed in other physical models. We show that the most degenerate singular fiber is homeomorphic to $\mathbf{S}^2\times\mathbf{S}^1$ with a singularity of $A_2$ type. We further describe the bifurcation diagram and the global topology of the fibration, and we compute its Hamiltonian monodromy.
Marián Poppr
Given a thin torus $T_K$ around a knot $K\subset \mathbb{R}^3$, we construct Morse models of cord algebra $Cord(T_K)$ with $\mathbb{Z}$ and loop space coefficients. Using the Multiple time scale dynamics we identify $Cord(T_K; \mathbb{Z})$ with $Cord(K; \mathbb{Z})$. In combination with the works of Cieliebak-Ekholm-Latschev-Ng and Petrak this indirectly relates $Cord(T_K)$ to $0$-th degree Legendrian contact homology $LCH_0(\mathcal{L}^\ast_+ T_K)$ of one component of the unit conormal bundle over $T_K$. Our definition of $Cord(T_K)$ is motivated by $J$-holomorphic curves with boundary on the Lagrangian submanifold $L^\ast_+ T_K\cup\mathbb{R}^3$ with an arboreal singularity along the torus $T_K$.
Alexander Braverman, Michael Finkelberg, Roman Travkin
Consider a pair of $S$-dual hyperspherical varieties $G\circlearrowright X$ and $G^\vee\circlearrowright X^\vee$ equipped with equivariant quantizations $Q(X)$, $Q(X^\vee)$. Assume that the local conjecture of Ben-Zvi, Sakellaridis and Venkatesh holds for this pair, and also that $X\simeq T^*_ψ(Y)$ is polarized, so that $Q(X)=D_ψ(Y)$. Let $B\subset G$ (resp. $B^\vee\subset G^\vee$) be Borel subgroups. Then using a variant of the $S^1$-equivariant localization of arxiv:0706.0322, we deduce an equivalence between the ${\mathbb Z}/2$-graded $B$-equivariant category $(D_ψ(Y)\operatorname{-mod})^{{\mathbb Z}/2})^B$ and the ${\mathbb Z}/2$-graded unipotent $B^\vee$-monodromic category $(Q(X^\vee)\operatorname{-mod}^{{\mathbb Z}/2})^{B^\vee,\operatorname{mon}}$.
Lina Deschamps, Levin Maier, Tom Stalljohann
This paper develops new links between contact geometry, magnetic dynamics, and symmetry in exact magnetic systems. First, we establish an interpolation property for Killing magnetic systems on contact manifolds under an additional condition. Specifically, we show that the corresponding magnetic geodesic flow interpolates smoothly between the sub-Riemannian geodesic flow on the contact distribution and the flow of the vector field associated with a primitive of the magnetic field. Second, we show that Hamiltonian group actions associated with the magnetomorphism group produce Poisson-commuting integrals of motion for the magnetic flow. Finally, we obtain new structural results on totally magnetic submanifolds, showing that fixed-point sets of magnetomorphisms and intersections of totally magnetic submanifolds are again totally magnetic. The latter two results may be viewed as extensions of classical phenomena from Riemannian geometry to magnetic geometry.
Guangbo Xu
We give a definition of all-genus reduced Gromov-Witten invariants of symplectic manifolds by using effectively supported multivalued perturbations on derived orbifold/Kuranishi charts, which bypasses the hard analytical result of sharp compactification/ghost bubble censorship of Zinger and Ekholm-Shende.
Clément Cren, Jean-Marie Lescure, Omar Mohsen
We use Toeplitz operators to define a star-product on Poisson manifolds whose Poisson structure is induced by a symplectic Lie algebroid. The Toeplitz operators we consider are defined on groupoids whose algebroid can be endowed with a Heisenberg group structure on the fibers. This generalizes an approach due to Guillemin and Melrose in the symplectic case.
C. Evans Hedges
In the variational approach to statistical mechanics, equilibrium states are the rigorous analogues of thermodynamic phases; the question of which invariant measures can arise as equilibrium states is therefore the question of which phases are thermodynamically realizable. We prove that for continuous actions of locally compact amenable groups on compact metrizable spaces with finite topological entropy, an ergodic measure $μ$ is an equilibrium state for some continuous potential if and only if the entropy map $h$ is upper semicontinuous at $μ$; equivalently, the unrealizable phases are exactly those hidden behind the convex envelope of the free energy. More generally, the same criterion applies whenever $(X, T)$ has bounded entropy and embeds as an invariant subsystem of a compact metrizable system. As a canonical case, one-point compactification yields a $C_0$-potential realization theorem for locally compact $σ$-compact systems, with applications to countable-state Markov shifts. We also show that the equilibrium-face realization stated by Jenkinson (2006) omits a necessary continuity hypothesis, exhibiting a counterexample on the full shift, and give the sharp corrected statement: a weak-$*$ closed set $\mathcal{E}$ of ergodic measures determines an equilibrium face if and only if $h|_{\mathcal{E}}$ is continuous and $h$ is upper semicontinuous at each point of $\mathcal{E}$.