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This https://arxiv.org/abs/2505.08752 has been replaced.
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Three Tone Networks and a Tessellation
The Eulerian tonnetz, which associates three minor chords to each major chord and three major chords to each minor chord, can be represented by a bipartite graph with twelve white vertices signifying major chords and twelve black vertices signifying minor chords. This so-called Levi graph uniquely determines the combinatorial geometry of a remarkable configuration of twelve points and twelve lines in the real projective plane with the property that three points lie on each line and three lines …

Closed Estimates of Leray Projected Transport Noise and Strong Solutions of the Stochastic Euler Equations
Daniel Goodair
https://arxiv.org/abs/2507.00787 …

Closed Estimates of Leray Projected Transport Noise and Strong Solutions of the Stochastic Euler Equations
We consider the incompressible Euler and Navier-Stokes equations on the three dimensional torus, in velocity form, perturbed by a transport or transport-stretching Stratonovich noise. Closed control of the noise contributions in energy estimates are demonstrated, for any positive integer ordered Sobolev Space and the equivalent Stokes Space; difficulty arises due to the presence of the Leray Projector disrupting cancellation of the top order derivative. This is particularly pertinent in the cas…

Euler characteristics, higher Kazhdan projections and delocalised $\ell^2$-Betti numbers
Sanaz Pooya, Baiying Ren, Hang Wang
https://arxiv.org/abs/2507.20119 https://

Euler characteristics, higher Kazhdan projections and delocalised $\ell^2$-Betti numbers
For non-amenable finitely generated virtually free groups, we show that the combinatorial Euler characteristic introduced by Emerson and Meyer is the preimage of the K-theory class of higher Kazhdan projections under the Baum-Connes assembly map. This allows to represent the K-theory class of their higher Kazhdan projection as a finite alternating sum of the K-theory classes of certain averaging projections. The latter is associated to the finite subgroups appearing in the fundamental domain of…

Birational Invariants from Hodge Structures and Quantum Multiplication
Ludmil Katzarkov, Maxim Kontsevich, Tony Pantev, Tony Yue YU
https://arxiv.org/abs/2508.05105 https://

Birational Invariants from Hodge Structures and Quantum Multiplication
We introduce new invariants of smooth complex projective varieties, called Hodge atoms. Their construction combines rational Gromov-Witten invariants with classical Hodge theory and relies on the notion of an F-bundle, which is a non-archimedean version of a non-commutative Hodge structure. The Hodge atoms arise from the spectral decomposition of the F-bundle under the Euler vector field action, and behave additively under blowups, in accordance with Iritani's blowup theorem. We compute several…

1/2 order convergence rate of Euler-type methods for time-changed stochastic differential equations with super-linearly growing drift and diffusion coefficients
Yuanling Niu, Shuai Wang, Ying Zhang
https://arxiv.org/abs/2507.14562

1/2 order convergence rate of Euler-type methods for time-changed stochastic differential equations with super-linearly growing drift and diffusion coefficients
This paper investigates the convergence rates of two Euler-type methods for a class of time-changed stochastic differential equations with super-linearly growing drift and diffusion coefficients. Building upon existing research, we adapt the backward Euler method to time-changed stochastic differential equations where both coefficients exhibit super-linear growth and introduce an explicit counterpart, the projected Euler method. It is shown that both methods achieve the optimal strong convergen…

On the Virtual Euler Characteristic of the Moduli Space of Stable Pairs on Surfaces
Ana Pavlakovi\'c
https://arxiv.org/abs/2505.23531 https://

On the Virtual Euler Characteristic of the Moduli Space of Stable Pairs on Surfaces
We study the stable pair theory on toric surfaces and determine the virtual tangent space over the fixed point loci. Further, we present a program to compute the virtual Euler characteristic, illustrated by the case of the projective plane. As an application, conjectures regarding rationality and symmetry are supported by verification of a special case.

Quadratic Segre indices
Felipe Espreafico, Stephen McKean, Sabrina Pauli
https://arxiv.org/abs/2506.01547 https://arxiv.org/pdf/2506.…

Quadratic Segre indices
We prove that the local Euler class of a line on a degree $2n-1$ hypersurface in projective $n+1$ space is given by a product of indices of Segre involutions. Segre involutions and their associated indices were first defined by Finashin and Kharlamov over the reals. Our result is valid over any field of characteristic not 2 and gives an infinite family of problems in enriched enumerative geometry with a shared geometric interpretation for the local type.

Generic vanishing on homogeneous spaces in characteristic $p>0$
Ankit Rai, K. V. Shuddhodan
https://arxiv.org/abs/2507.13452 https://

Generic vanishing on homogeneous spaces in characteristic $p>0$
Let $X$ be a projective variety that is a homogeneous space under the action of a connected algebraic group $G$. Let $W, Z \subset X$ be locally closed, smooth affine subvarieties. In a recent work, Schürmann--Simpson--Wang proved that for a generic $g \in G$, the inequality $(-1)^{\dim(gW \cap Z)} χ(gW \cap Z) \geq 0$ holds, when the base field has characteristic $0$. More precisely, they prove a new generic vanishing result and obtain the assertion about the Euler characteristic as a coroll…