Euler characteristic of crepant resolutions of specific modular quotient singularities

Abstract The compactly supported $\mathbb {A}^1$ -Euler characteristic, introduced by Hoyois and later refined by Levine and others, is an analogue in motivic homotopy theory of the classical Euler characteristic of complex topological manifolds. It is an invariant on the Grothendieck ring of varieties $\mathrm {K}_0(\mathrm {Var}_k)$ taking values in the Grothendieck-Witt ring $\mathrm {GW}(k)$ of the base field k . The former ring has a natural power structure induced by symmetric powers of varieties. In a recent preprint, the first author and Pál construct a power structure on $\mathrm {GW}(k)$ and show that the compactly supported $\mathbb {A}^1$ -Euler characteristic respects these two power structures for $0$ -dimensional varieties, or equivalently étale k -algebras. In this paper, we define the class $\mathrm {Sym}_k$ of symmetrisable varieties to be those varieties for which the compactly supported $\mathbb {A}^1$ -Euler characteristic respects the power structures and study the algebraic properties of the subring $\mathrm {K}_0(\mathrm {Sym}_k)$ of symmetrisable varieties. We show that it includes all cellular varieties, and even linear varieties as introduced by Totaro. Moreover, we show that it includes non-linear varieties such as elliptic curves. As an application of our main result, we compute the compactly supported $\mathbb {A}^1$ -Euler characteristics of symmetric powers of Grassmannians and certain del Pezzo surfaces.

We study the subrack lattices of conjugation racks arising from subsets of a finite group that are closed under conjugation. For connected conjugation racks, we show that the subrack lattice can be associated with a subposet of a partition lattice and a subposet of an integer partition lattice in a canonical way. For racks that are not connected, we reduce the analysis of the homotopy type of the subrack lattice to the study of the subposet of parabolic subracks. We further prove that, for a certain class of p-power racks, the order of a Sylow p-subgroup divides the reduced Euler characteristic of the associated subrack lattice. This statement may be viewed as the rack analogue of a result by Brown concerning the Euler characteristic of the poset of nontrivial p-subgroups of a group.

Nonnegativity of signed Euler characteristics of moduli of curves and abelian varieties

On the ext analog of the Euler characteristic

Abstract This paper studies rings of integral piecewise‐exponential functions on rational fans. Motivated by lattice‐point counting in polytopes, we introduce a special class of unimodular fans called Ehrhart fans, whose rings of integral piecewise‐exponential functions admit a canonical linear functional that behaves like a lattice‐point count. In particular, we verify that all complete unimodular fans are Ehrhart and that the Ehrhart functional agrees with lattice‐point counting in corresponding polytopes, which can otherwise be interpreted as holomorphic Euler characteristics of vector bundles on smooth toric varieties. We also prove that all Bergman fans of matroids are Ehrhart and that the Ehrhart functional in this case agrees with the Euler characteristic of matroids, introduced recently by Larson, Li, Payne, and Proudfoot. A key property that we prove about the Ehrharticity of fans is that it only depends on the support of the fan, not on the fan structure, thus providing a uniform framework for studying ‐rings and Euler characteristics of complete fans and Bergman fans simultaneously.

Nonpositively curved 4-manifolds with zero Euler characteristic

This paper presents some nontrivial computational results on derived category and Fourier–Mukai technique in algebraic geometry. In particular, it aims at presenting calculations involving spherical twists as a certain class of Fourier–Mukai functors and its cohomological descent on the singular rational cohomology of smooth projective variety. The purpose of this investigation is to present a new perspective, based upon Fourier–Mukai technique, on solving classical problems involving characteristic classes: in particular, the Chern and the Euler characteristics.

In algebraic topology, a k-dimensional simplex is defined as a convex polytope consisting of k + 1 vertices. If spatial dimensionality is not considered, it corresponds to the complete graph with k + 1 vertices in graph theory. The alternating sum of the number of simplices across dimensions yields a topological invariant known as the Euler characteristic, which has gained significant attention due to its widespread application in fields such as topology, homology theory, complex systems, and biology. The most common method for calculating the Euler characteristic is through simplicial decomposition and the Euler–Poincaré formula. In this study, we introduce a new “subgraph” polynomial, termed the simplex polynomial, and explore some of its properties. Using those properties, we provide a new method for computing the Euler characteristic and prove the existence of the Euler characteristic as an arbitrary integer by constructing the corresponding simplicial complex structure. When the Euler characteristic is 1, we determined a class of corresponding simplicial complex structures. Moreover, for three common network structures, we present the recurrence relations for their simplex polynomials and their corresponding Euler characteristics. Finally, at the end of this study, three basic questions are raised for the interested readers to study deeply.

Abstract We study Morse–Bott functions on surfaces and compute their cobordism group by defining Morse–Bott maps, which yield an equivalence relation between functions. The generators of the cobordism group correspond to the number of Morse singularities mod two or endowed with signs and to the number of components of Morse–Bott singular loci endowed with signs. We give geometric cobordism invariants and also prove a formula about the relation of the numbers of different singularity types of a Morse–Bott function on a surface. We also get a similar formula about the Euler characteristic of the surface. We show that for $n \ge 3$ the rank of the cobordism group of Morse–Bott functions on $n$-dimensional manifolds is equal to infinity.

Fluctuations of observables provide unique insights into the nature of physical systems, and their study stands as a cornerstone of both theoretical and experimental science. Generalized fluctuations, or cumulants, provide information beyond the mean and variance of an observable. In this paper, we develop a systematic method to determine the asymptotic behavior of cumulants of local observables as the region becomes large. Our analysis reveals that the expansion is closely tied to the geometric characteristics of the region and its boundary, with coefficients given by convex moments of the connected correlation function: the latter is integrated against intrinsic volumes of convex polytopes built from the coordinates, which can be interpreted as average shadows. A particular application of our method shows that, in two dimensions, the leading behavior of odd cumulants of conserved quantities is topological, specifically depending on the Euler characteristic of the region. We illustrate these results with the paradigmatic strongly-interacting system of two-dimensional quantum Hall state at filling fraction 1/2 1 / 2 , by performing Monte-Carlo calculations of the skewness (third cumulant) of particle number in the Laughlin state.

We find an algorithm to compute the quadratic Euler characteristic of a smooth projective complete intersection of hypersurfaces of the same degree. As an example, we compute the quadratic Euler characteristic of a smooth projective complete intersection of two generalized Fermat hypersurfaces. The results presented here also form a chapter in the author's thesis, which was submitted on May 30'th, 2023.

Abstract Building on work of Harer, we construct a spine for the decorated Teichmüller space of a non‐orientable surface with at least one puncture and negative Euler characteristic. We compute its dimension, and show that the deformation retraction onto this spine is equivariant with respect to the pure mapping class group of the non‐orientable surface. As a consequence, we obtain a model for the classifying space for proper actions of the pure mapping class group of a punctured non‐orientable surface, which is of minimal dimension in the case there is a single puncture.

Abstract We introduce a scissors congruence ‐theory spectrum that lifts the equivariant scissors congruence groups for compact ‐manifolds with boundary, and we show that on , this is the source of a spectrum‐level lift of the Burnside ring‐valued equivariant Euler characteristic of a compact ‐manifold. We also show that the equivariant scissors congruence groups for varying subgroups assemble into a Mackey functor, which is a shadow of a conjectural higher genuine equivariant structure.

Convolutional neural networks (CNNs) have emerged to reduce clinical resources and standardize auto-contouring of organs-at-risk (OARs). Although CNNs perform adequately for most patients, understanding when the CNN might fail is critical for effective and safe clinical deployment. However, the limitations of CNNs are poorly understood because of their black-box nature. Explainable artificial intelligence (XAI) can expose CNNs' inner mechanisms for classification. Here, we investigate the inner mechanisms of CNNs for segmentation and explore a novel, computational approach to a-priori flag potentially insufficient parotid gland (PG) contours. First, 3D UNets were trained in three PG segmentation situations using (1) synthetic cases; (2) 1925 clinical computed tomography (CT) scans with typical and (3) more consistent contours curated through a previously validated auto-curation step. Then, we generated attribution maps for seven XAI methods, and qualitatively assessed them for congruency between simulated and clinical contours, and how much XAI agreed with expert reasoning. To objectify observations, we explored persistent homology intensity filtrations to capture essential topological characteristics of XAI attributions. Principal component (PC) eigenvalues of Euler characteristic profiles were correlated with spatial agreement (Dice-Sørensen similarity coefficient; DSC). Evaluation was done using sensitivity, specificity and the area under receiver operating characteristic (AUROC) curve on an external AAPM dataset, where as proof-of-principle, we regard the lowest 15% DSC as insufficient. PatternNet attributions (PNet-A) focused on soft-tissue structures, whereas guided backpropagation (GBP) highlighted both soft-tissue and high-density structures (e.g. mandible bone), which was congruent with synthetic situations. Both methods typically had higher/denser activations in better auto-contoured medial and anterior lobes. Curated models produced "cleaner" gradient class-activation mapping (GCAM) attributions. Quantitative analysis showed that PCλ1 of guided GCAM's (GGCAM) Euler characteristic (EC) profile had good predictive value (sensitivity>0.85, specificity>0.90) of DSC for AAPM cases, with AUROC=0.66, 0.74, 0.94, 0.83 for GBP, GCAM, GGCAM and PNet-A. For for λ1<-1.8e3 of GGCAM's EC-profile, 87% of cases were insufficient. GBP and PNet-A qualitatively agreed most with expert reasoning on directly (structure borders) and indirectly (proxies used for identifying structure borders) important features for PG segmentation. Additionally, this work investigated as proof-of-principle how topological data analysis could be used for quantitative XAI signal analysis to a-priori mark potentially inadequate CNN-segmentations, using only features from inside the predicted PG. This work used PG as a well-understood segmentation paradigm and may extend to target volumes and other organs-at-risk.

Hydrogen (H2) storage in porous geological formations offers a promising means to balance supply and demand in the renewable energy sector, supporting the energy transition. Important unknowns to this technology include the H2 fluid flow dynamics through the porous medium which affect H2 injectivity and recovery. We used time-resolved X-ray computed microtomography to image real-time unsteady and steady state injections of H2 and brine (2M KI) into a Clashach sandstone core at 5MPa and ambient temperature. In steady state injections, H2 entered the brine-saturated rock within seconds, dispersing over several discrete pores. Over time, some H2 ganglia connected, disconnected and then reconnected from each other (intermittent flow), indicating that the current presumption of a constant connected flow pathway during multiphase fluid flow is an oversimplification. Pressure oscillations at the sample outlet were characterized as red noise, supporting observations of intermittent pore-filling. At higher H2 fractional flow the H2 saturation in the pore space increased from 20-22% to 28%. Average Euler characteristics were generally positive over time at all H2 flow fractions, indicating poorly connected H2 clusters and little control of connectivity on the H2 saturation. In unsteady state injections, H2 displaced brine in sudden pore-filling events termed Haines jumps, which are key to understanding fluid dynamics in porous media. Our results suggest a lower H2 storage capacity in sandstone aquifers with higher injection-induced hydrodynamic flow and suggest a low H2 recovery. For more accurate predictions of H2 storage potential and recovery, geological models should incorporate energy-dissipating processes such as Haines jumps.

We show an intrinsic version of Thomason's fixed-point theorem. Then we determine the local structure of the Hilbert scheme of at most $7$ points in $\mathbb{A}^3$. In particular, we show that in these cases, the points with the same extra dimension have the same singularity type. Using these results, we compute the equivariant Hilbert functions at the singularities and verify a conjecture of Zhou on the Euler characteristics of tautological sheaves on Hilbert schemes of points on $\mathbb{P}^3$ for at most $6$ points.An error in Lemma 5.1 is corrected, and the proof of this lemma is improved

This paper establishes a connection between the Hawking temperature of spacetime horizons and global topological invariants, specifically the Euler characteristic of Wick-rotated Euclidean spacetimes. This is demonstrated for both de Sitter and Schwarzschild, where the compactification of the near-horizon geometry allows for a direct application of the Chern–Gauss–Bonnet theorem. For de Sitter, a simple argument connects the Gibbon–Hawking temperature of the Bunch–Davies state to the global thermal de Sitter temperature. This establishes that spacetime thermodynamics are a consequence of the geometrical structure of spacetime itself, therefore suggesting a deep connection between global topology and semi-classical analysis.

Background:As the availability of single-cell RNA sequencing (scRNA-seq) data expands, there is a growing need for robust methods that enable integration and comparison across diverse biological conditions and experimental protocols. Persistent homology (PH), a technique from topological data analysis (TDA), provides a deformation-invariant framework for capturing structural patterns in high-dimensional data.Methods:In this study, PH was applied to a diverse collection of scRNA-seq datasets spanning eight tissue types to investigate how data integration affects the topological features and biological interpretability of the resulting representations. Clustering was performed based on PH-derived pairwise distances and global topological structure was assessed through Betti curves, Euler characteristics, and persistence landscapes. By comparing these summaries across raw, normalized, and integrated datasets, we examined whether integration enhances the detection of biologically meaningful patterns, or, conversely, obscures fine-scale structure.Results:This approach demonstrates that PH can serve as a powerful complementary strategy for evaluating the impact of integration and reveals how topological summaries can help disentangle biological signal from batch-related noise in single-cell data. This work establishes a framework for using topological methods to assess integration quality and highlights new avenues for interpreting complex transcriptomic landscapes beyond conventional clustering.

Abstract Let $$S_n$$ S n denote the symmetric group. We consider $$\begin{aligned} N_{\ell }(n):= \frac{\left| \textrm{Hom}\left( {\mathbb {Z}}^{\ell },S_n\right) \right| }{n!} \end{aligned}$$ N ℓ ( n ) : = Hom Z ℓ , S n n ! which also counts the number of $$\ell $$ ℓ -tuples $$\pi =\left( \pi _1, \ldots , \pi _{\ell }\right) \in S_n^{\ell }$$ π = π 1 , … , π ℓ ∈ S n ℓ with $$\pi _i \pi _j = \pi _j \pi _i$$ π i π j = π j π i for $$1 \le i,j \le \ell $$ 1 ≤ i , j ≤ ℓ scaled by n!. A recursion formula, generating function, and Euler product have been discovered by Dey, Wohlfahrt, Bryan and Fulman, and White. Let $$a,b, \ell \ge 2.$$ a , b , ℓ ≥ 2 . It is known by Bringmann, Franke, and Heim, that the Bessenrodt–Ono inequality $$\begin{aligned} \Delta _{a,b}^{\ell }:= N_{\ell }(a) \, N_{\ell }(b) - N_{\ell }(a+b) &gt;0 \end{aligned}$$ Δ a , b ℓ : = N ℓ ( a ) N ℓ ( b ) - N ℓ ( a + b ) &gt; 0 is valid for $$a,b \gg 1$$ a , b ≫ 1 and by Bessenrodt and Ono that it is valid for $$\ell =2$$ ℓ = 2 and $$a+b &gt;9.$$ a + b &gt; 9 . In this paper, we prove that for each pair (a, b) the sign of $$\{\Delta _{a,b}^{\ell } \}_{\ell }$$ { Δ a , b ℓ } ℓ is getting stable. In each case we provide an explicit bound. The numbers $$N_{\ell }\left( n\right) $$ N ℓ n had been identified by Bryan and Fulman as the nth orbifold characteristics, generalizing work by Macdonald and Hirzebruch–Höfer concerning the ordinary and string-theoretic Euler characteristics of symmetric products, where $$N_2(n)=p(n) $$ N 2 ( n ) = p ( n ) represents the partition function.