We prove that the ($\tau$-weighted, sheaf-theoretic) $\operatorname{SL}(2,\mathbb{C})$ Casson–Lin invariant introduced by Manolescu and the first author is generically independent of the parameter $\tau$ and additive under connected sums of knots in integral homology 3-spheres. This addresses two questions asked by Manolescu and the first author. Our arguments involve a mix of topology and algebraic geometry, and rely crucially on the fact that the $\operatorname{SL}(2,\mathbb{C})$ Casson–Lin invariant admits an alternative interpretation via the theory of Behrend functions.

In 1929, Siegel defined $E$-functions as power series in $\overline{\mathbb{Q}}[[z]]$, with Taylor coefficients satisfying certain growth conditions, and solutions of linear differential equations with coefficients in $\overline{\mathbb{Q}}(z)$. The Siegel–Shidlovskii theorem (1956) generalized to $E$-functions the Diophantine properties of the exponential function. In 2000, André proved that the finite singularities of a differential operator in $\overline{\mathbb{Q}}(z)[d/dz] \setminus \{0\}$ of minimal order for some non-zero $E$-function are apparent, except possibly 0 which is always regular singular. We pursue the classification of such operators and consider those for which 0 is $\eta$-apparent, in the sense that there exists $\eta \in \mathbb C$ such that $L$ has a local basis of solutions at 0 in $z^{\eta} \mathbb C[[z]]$. We prove that they have a $\mathbb{C}$-basis of solutions of the form $Q_{j}(z)z^{\eta} e^{\beta_{j} z}$, where $\eta \in \mathbb{Q}$, the $\beta_{j} \in \overline{\mathbb{Q}}$ are pairwise distinct and the $Q_{j}(z) \in \overline{\mathbb{Q}}[z] \setminus \{0\}$. This generalizes a previous result by Roques and the author concerning $E$-operators with an apparent singularity or no singularity at the origin, of which certain consequences are also given here.

For an abelian group $A$, we study a close connection between braided $A$-crossed tensor categories with a trivialization of the $A$-action and $A$-graded braided tensor categories. Additionally, we prove that the obstruction to the existence of a trivialization of a categorical group action $T$ on a tensor category $\mathcal{C}$ is given by an element $O(T) \in H^2(G, \operatorname{Aut}_{\otimes}(\operatorname{Id}_{\mathcal{C}}))$. In the case that $O(T) = 0$, the set of obstructions forms a torsor over $\operatorname{Hom}(G, \operatorname{Aut}_{\otimes}(\operatorname{Id}_{\mathcal{C}}))$, where $\operatorname{Aut}_{\otimes}(\operatorname{Id}_{\mathcal{C}})$ is the abelian group of tensor natural automorphisms of the identity.

The cohomological interpretation of trivializations, together with the homotopical classification of (faithfully graded) braided $A$-crossed tensor categories developed Etingof et al., allows us to provide a method for the construction of faithfully $A$-graded braided tensor categories. We work out two examples. First, we compute the obstruction to the existence of trivializations for the braided $A$-crossed tensor category associated with a pointed semisimple tensor category. In the second example, we compute explicit formulas for the braided $\mathbb{Z}/2\mathbb{Z}$-crossed structures over Tambara–Yamagami fusion categories and, consequently, a conceptual interpretation of the results by Siehler about the classification of braidings over Tambara–Yamagami categories.

We mainly answer two open questions about finite multiple harmonic $q$-series on 3-2-1 indices at roots of unity, posed recently by Bachmann, Takeyama, and Tasaka. Two conjectures regarding cyclic sums which generalize the given results are also provided.

In this paper we settle the two-dimensional case of a conjecture involving unknown semialgebraic functions with specified smoothness. More precisely, we prove the following result: Let $\mathcal{H}$ be a semialgebraic bundle with respect to $C^m_{loc}(\mathbb{R}^2, \mathbb{R}^{D})$. If $\mathcal{H}$ has a section, then it has a semialgebraic section.

For any compact Riemannian surface of genus three $(\Sigma, ds^2)$ Yang and Yau proved that the product of the first eigenvalue of the Laplacian $\lambda_1(ds^2)$ and the area $Area(ds^2)$ is bounded above by $24\pi$. In this paper we improve the result and we show that $\lambda_1(ds^2) Area(ds^2) \leq 16(4 - \sqrt{7})\hspace{.02cm}\pi \approx 21.668\hspace{.02cm}\pi$. About the sharpness of the bound, for the hyperbolic Klein quartic surface numerical computations give the value $\approx 21.414\hspace{.02cm}\pi$.

(Non-)displaceability of fibers of integrable systems has been an important problem in symplectic geometry. In this paper, for a large class of classical Liouville integrable systems containing the Lagrangian top, the Kovalevskaya top and the C. Neumann problem, we find a non-displaceable fiber for each of them. Moreover, we show that the non-displaceable fiber which we detect is the unique fiber which is non-displaceable from the zero-section. As a special case of this result, we also show the existence of a singular level set of a convex Hamiltonian, which is non-displaceable from the zero-section. To prove these results, we use the notion of superheaviness introduced by Entov and Polterovich.

Let $\bar{Y}$ be a normal surface that is the canonical $\mu_{2}$- or $\alpha_{2}$-covering of a classical or supersingular Enriques surface in characteristic 2. We determine all possible configurations of singularities on $\bar{Y}$, and for each configuration we describe which type of Enriques surfaces (classical or supersingular) appear as quotients of $\bar{Y}$.

We classify outer actions (or $\mathscr{G}$-kernels) of discrete amenable groupoids on injective factors. Our method based on unified approach for classification of discrete amenable groups actions, and cohomology reduction theorem of discrete amenable equivalence relations. We do not use Katayama–Takesaki type resolution group approach.

In this note, we introduce and study a notion of bi-exactness for creation operators acting on full, symmetric and anti-symmetric Fock spaces. This is a generalization of our previous work, in which we studied the case of anti-symmetric Fock spaces. As a result, we obtain new examples of solid actions as well as new proofs for some known solid actions. We also study free wreath product groups in the same context.

Let $p$ be an odd prime number and let $2^{e+1}$ be the highest power of 2 dividing $p - 1$. For $0 \leq n \leq e$, let $k_{n}$ be the real cyclic field of conductor $p$ and degree $2^{n}$. For a certain imaginary quadratic field $L_{0}$, we put $L_{n} = L_{0} k_{n}$. For $0 \leq n \leq e - 1$, let $\mathcal{F}_{n}$ be the imaginary quadratic subextension of the imaginary $(2, 2)$-extension $L_{n+1}/k_{n}$ with $\mathcal{F}_{n} \neq L_{n}$. We study the Galois module structure of the 2-part of the ideal class group of the imaginary cyclic field $\mathcal{F}_{n}$. This generalizes a classical result of Rédei and Reichardt for the case $n = 0$.

We show that the number of lines contained in a supersingular quartic surface is 40 or at most 32, if the characteristic of the field equals 2, and it is 112, 58, or at most 52, if the characteristic equals 3. If the quartic is not supersingular, the number of lines is at most 60 in both cases. We also give a complete classification of large configurations of lines.