In this paper we study a long-standing conjecture of Huneke and Wiegand which is concerned with the torsion submodule of certain tensor products of modules over one-dimensional local domains. We utilize Hochster's theta invariant and show that the conjecture is true for two periodic modules. We also make use of a result of Orlov and formulate a new condition which, if true over hypersurface rings, forces the conjecture of Huneke and Wiegand to be true over complete intersection rings of arbitrary codimension. Along the way we investigate the interaction between the vanishing of Tate (co)homology and torsion in tensor products of modules, and obtain new results that are of independent interest.

Let $G$ be a finite group. If $n \leq 5$ then any $n$-dimensional homotopy sphere never admits a smooth action of $G$ with exactly one fixed point. Let $A_n$ and $S_n$ denote the alternating group and the symmetric group on some $n$ letters. If $n \geq 6$ then the $n$-dimensional sphere possesses a smooth action of $A_5$ with exactly one fixed point. Let $V$ be an $n$-dimensional real $G$-representation with exactly one fixed point. It is interesting to ask whether there exists a smooth $G$-action with exactly one fixed point on the $n$-dimensional sphere such that the associated tangential $G$-representation is isomorphic to $V$. In this paper, we study this problem for nonsolvable groups $G$ and real $G$-representations $V$ satisfying certain hypotheses. Applying a theory developed in this paper, we can prove that the $n$-dimensional sphere has an effective smooth action of $S_5$ with exactly one fixed point if and only if $n = 6$, 10, 11, 12, or $n \geq 14$ and that the $n$-dimensional sphere has an effective smooth action of $A_5 \times Z$ with exactly one fixed point if $n$ satisfies $n \geq 6$ and $n \neq 9$, where $Z$ is a group of order 2.

We shall propose a new proof scheme for Planar Cover Conjecture, focusing on the rotation systems of planar coverings of connected graphs. We shall introduce the notion of “rotation compatible coverings” and show that a rotation compatible covering of $G$ embedded on the sphere can be covered by a regular covering of $G$ embedded on an orientable closed surface on which its covering transformation group acts. The surface may not be homeomorphic to the sphere in general, but its quotient becomes either the sphere or the projective plane which contains $G$. As an application of our theory, we shall prove that if a 3-connected graph $G$ has a 3-connected finite planar covering such that the pre-images of each vertex has sufficiently large distance, then $G$ can be embedded on the projective plane.

In this paper, we give a refinement of a generalized Dedekind's theorem. In addition, we show that all possible values of integer group determinants of any group are also possible values of integer group determinants of its any abelian subgroup. By applying the refinement of a generalized Dedekind's theorem, we determine all possible values of integer group determinants of the direct product group of the cyclic group of order 8 and the cyclic group of order 2.

Time-dependent free surface problem for the incompressible Navier–Stokes equations which describes the motion of viscous incompressible fluid nearly half-space are considered. We obtain global well-posedness of the problem for a small initial data in scale invariant critical Besov spaces. Our proof is based on maximal $L^{1}$-regularity of the corresponding Stokes problem in the half-space and special structures of the quasi-linear term appearing from the Lagrangian transform of the coordinate.

We prove the convergence of (solid) ellipsoids to a Gaussian space in Gromov's concentration/weak topology as the dimension diverges to infinity. This gives the first discovered example of an irreducible nontrivial convergent sequence in the concentration topology, where ‘irreducible nontrivial’ roughly means to be not constructed from Lévy families nor box convergent sequences.

In this paper, we study the parabolic BGG categories for graded Lie superalgebras of Cartan type over the field of complex numbers. The gradation of such a Lie superalgebra $\mathfrak{g}$ naturally arises, with the zero component $\mathfrak{g}_{0}$ being a reductive Lie algebra. We first show that there are only two proper parabolic subalgebras containing Levi subalgebra $\mathfrak{g}_{0}$: the “maximal one” $\mathsf{P}_{\max}$ and the “minimal one” $\mathsf{P}_{\min}$. Furthermore, the parabolic BGG category arising from $\mathsf{P}_{\max}$ essentially turns out to be a subcategory of the one arising from $\mathsf{P}_{\min}$. Such a priority of $\mathsf{P}_{\min}$ in the sense of representation theory reduces the question to the study of the “minimal parabolic” BGG category $\mathcal{O}^{\min}$ associated with $\mathsf{P}_{\min}$. We prove the existence of projective covers of simple objects in these categories, which enables us to establish a satisfactory block theory. Most notably, our main results are as follows.

(1) We classify and obtain a precise description of the blocks of $\mathcal{O}^{\min}$.

(2) We investigate indecomposable tilting and indecomposable projective modules in $\mathcal{O}^{\min}$, and compute their character formulas.

We consider a continuum percolation built over stationary ergodic point processes. Assuming that the occupied region has a unique unbounded cluster and the cluster satisfies volume regularity and isoperimetric condition, we prove a quenched invariance principle for reflecting diffusions on the cluster.

An element of a Weyl group of classical type is skew if it is the left factor in a reduced factorization of a Grassmannian element. The skew Grothendieck polynomials are those which are indexed by skew elements of the Weyl group. We define set-valued tableaux which are fillings of the associated skew Young diagrams and use them to prove tableau formulas for the skew double Grothendieck polynomials in all four classical Lie types. We deduce tableau formulas for the Grassmannian Grothendieck polynomials and the $K$-theoretic analogues of the (double mixed) skew Stanley functions in the respective Lie types.

In this paper, we establish a refined transversality theorem on linear perturbations from a new perspective of Hausdorff measures. Furthermore, we give its applications not only to singularity theory but also to multiobjective optimization.

One of the important research subjects in the study of multiple zeta functions is to clarify the linear relations and functional equations among them. The Schur multiple zeta functions are a generalization of the multiple zeta functions of Euler–Zagier type. Among many relations, the duality formula and its generalization are important families for both Euler–Zagier type and Schur type multiple zeta values. In this paper, following the method of previous works for multiple zeta values of Euler–Zagier type, we give an interpolation of the sums in the generalized duality formula, called Ohno relation, for Schur multiple zeta values. Moreover, we prove that the Ohno relation for Schur multiple zeta values is valid for complex numbers.

For every fixed $h \geq 1$, we construct an infinite family of simply connected symplectic 4-manifolds $X'_{g,h}[i]$, for all $g > h$ and $0 \leq i < 2p - 1$, where $p = \lfloor \frac{g+1}{h+1} \rfloor$. Each manifold $X'_{g,h}[i]$ is the total space of a symplectic genus $g$ Lefschetz pencil constructed by an explicit monodromy factorization. We then show that each $X'_{g,h}[i]$ is diffeomorphic to a complex surface that is a fiber sum formed from two standard examples of hyperelliptic genus $h$ Lefschetz fibrations, here denoted $Z_{h}$ and $H_{h}$. Consequently, we see that $Z_{h}$, $H_{h}$, and all fiber sums of them admit an infinite family of explicitly described Lefschetz pencils, which we observe are different from families formed by the degree doubling procedure.

Let $V_{L}$ be the vertex algebra associated to a non-degenerate even lattice $L$, $\theta$ the automorphism of $V_{L}$ induced from the $-1$ symmetry of $L$, and $V_{L}^{+}$ the fixed point subalgebra of $V_{L}$ under the action of $\theta$. In this series of papers, we classify the irreducible weak $V_{L}^{+}$-modules and show that any irreducible weak $V_{L}^{+}$-module is isomorphic to a weak submodule of some irreducible weak $V_{L}$-module or to a submodule of some irreducible $\theta$-twisted $V_{L}$-module. Let $M(1)^{+}$ be the fixed point subalgebra of the Heisenberg vertex operator algebra $M(1)$ under the action of $\theta$. In this paper (Part 2), we show that there exists an irreducible $M(1)^{+}$-submodule in any non-zero weak $V_{L}^{+}$-module and we compute extension groups for $M(1)^{+}$.

This paper deals with links and braids up to link-homotopy, studied from the viewpoint of Habiro's clasper calculus. More precisely, we use clasper homotopy calculus in two main directions. First, we define and compute a faithful linear representation of the homotopy braid group, by using claspers as geometric commutators. Second, we give a geometric proof of Levine's classification of 4-component links up to link-homotopy, and go further with the classification of 5-component links in the algebraically split case.

Consider a cohomologically hyperbolic birational self-map defined over the algebraic numbers, for example, a birational self-map in dimension two with the first dynamical degree greater than one, or in dimension three with the first and the second dynamical degrees distinct. We give a boundedness result about heights of its periodic points. This is motivated by a conjecture of Silverman for polynomial automorphisms of affine spaces. We also study the Kawaguchi–Silverman conjecture concerning dynamical and arithmetic degrees for certain rational self-maps in dimension two. In particular, we reduce the problem to the dynamical Mordell–Lang conjecture and verify the Kawaguchi–Silverman conjecture for some new cases. As a byproduct of the argument, we show the existence of Zariski dense orbits in these cases.

Let $n$ be an integer such that $n = 5$ or $n \geq 7$. In this article, we introduce a recipe for a certain infinite family of non-singular plane curves of degree $n$ which violate the local-global principle. Moreover, each family contains infinitely many members which are not geometrically isomorphic to each other. Our construction is based on two arithmetic objects; that is, prime numbers of the form $X^{3} + NY^{3}$ due to Heath-Brown and Moroz and the Fermat type equation of the form $x^{3} + Ny^{3} = Lz^{n}$, where $N$ and $L$ are suitably chosen integers. In this sense, our construction is an extension of the family of odd degree $n$ which was previously found by Shimizu and the author. The previous construction works only if the given degree $n$ has a prime divisor $p$ for which the pure cubic fields $\mathbb{Q}(p^{1/3})$ or $\mathbb{Q}((2p)^{1/3})$ satisfy a certain indivisibility conjecture of Ankeny–Artin–Chowla–Mordell type. In this time, we focus on the complementary cases, namely the cases of even degrees and exceptional odd degrees. Consequently, our recipe works well as a whole. This means that we can unconditionally produce infinitely many explicit non-singular plane curves of every degree $n = 5$ or $n \geq 7$ which violate the local-global principle. This gives a conclusion of the classical story of searching explicit ternary forms violating the local-global principle, which was initiated by Selmer (1951) and extended by Fujiwara (1972) and others.

Infinite-dimensional stochastic differential equations (ISDEs) describing systems with an infinite number of particles are considered. Each particle undergoes a Lévy process, and the interaction between particles is determined by the long-range interaction potential. The potential is of Ruelle's class or logarithmic. We discuss the existence and uniqueness of strong solutions of the ISDEs.

In this paper we give an explicit formula of the Fourier coefficients of the Siegel Eisenstein series of degree 3, level $p$ with quadratic character. For the calculation we use the result of the generalized Gauss sum computed by Hiroshi Saito. After the tedious calculation we can get a rather simple result.

We show that a Busemann space $X$ covered by parallel bi-infinite geodesics is homeomorphic to a product of another Busemann space $Y$ and the real line. We also show that a semi-simple isometry on $X$ preserving the foliation by parallel geodesics canonically induces a semi-simple isometry on $Y$.

Let $A$ be a semi-abelian variety with an exponential map $\exp : \mathop{\mathrm{Lie}}(A) \to A$. The purpose of this paper is to explore Nevanlinna theory of the entire curve $\hskip1pt{\widehat{\mathrm{exp}}}\hskip2pt f := (\exp f, f) : \mathbf{C} \to A \times \mathop{\mathrm{Lie}}(A)$ associated with an entire curve $f : \mathbf{C} \to \mathop{\mathrm{Lie}}(A)$. Firstly we give a Nevanlinna theoretic proof to the analytic Ax–Schanuel Theorem for semi-abelian varieties, which was proved by J. Ax 1972 in the case of formal power series (Ax–Schanuel Theorem). We assume some non-degeneracy condition for $f$ such that the elements of the vector-valued function $f(z) - f(0) \in \mathop{\mathrm{Lie}}(A) \cong \mathbf{C}^n$ are $\mathbf{Q}$-linearly independent in the case of $A = (\mathbf{C}^{*})^{n}$. Our proof is based on the Log Bloch–Ochiai Theorem and a key estimate which we show.

Our next aim is to establish a Second Main Theorem for $\hskip1pt{\widehat{\mathrm{exp}}}\hskip2pt f$ and its $k$-jet lifts with truncated counting functions at level one. We give some applications to a problem of a type raised by S. Lang and the unicity. The results clarify a relationship between the problems of Ax–Schanuel type and Nevanlinna theory.

Using Hahn series, one can attach to any linear Mahler equation a basis of solutions at 0 reminiscent of the solutions of linear differential equations at a regular singularity. We show that such a basis of solutions can be produced by using a variant of Frobenius method.

For any $1 < q < p < \infty$, we identify the best constant $K_{p,q}$ with the following property. If $\mathcal{H}$ is the Hilbert transform on the unit circle $\mathbb{T}$ and $A \subset \mathbb{T}$ is an arbitrary measurable set, then $\int_{A} | \mathcal{H}f | \mathrm{d}m \leq K_{p,q} \| f \|_{L^p(\mathbb{T},m)} m(A)^{1-1/q}$. The proof rests on the construction of certain special superharmonic functions on the plane, which are of independent interest.

We show the strong cohomological rigidity of Hirzebruch surface bundles over Bott manifolds. As a corollary, we have that the strong cohomological rigidity conjecture is true for Bott manifolds of dimension 8.

A Hurwitz group is a conformal automorphism group of a compact Riemann surface with precisely $84(g - 1)$ automorphisms, where $g$ is the genus of the surface. Our starting point is a result on the smallest Hurwitz group $\mathrm{PSL}(2,\mathbb{F}_{7})$ which is the automorphism group of the Klein surface. In this paper, we generalize it to various classes of simple Hurwitz groups and discuss a relationship between the surface symmetry and spectral asymmetry for compact Riemann surfaces. To be more precise, we show that the reducibility of an element of a simple Hurwitz group is equivalent to the vanishing of the $\eta$-invariant of the corresponding mapping torus. Several wide classes of simple Hurwitz groups which include the alternating group, the Chevalley group and the Monster, which is the largest sporadic simple group, satisfy our main theorem.

This paper studies the obstructions to deforming a map from a complex variety to another variety which is an immersion of codimension one. We extend the classical notion of semiregularity of subvarieties to maps between varieties, and prove the unobstructedness of the deformation of such a map, even when the image is non-reduced. As an application, we give a simple but effective criterion for the vanishing of the obstructions to equisingular deformations of nodal curves on surfaces.

We give the inversion formula and the Plancherel formula for the hypergeometric Fourier transform associated with a root system of type $BC$, when the multiplicity parameters are not necessarily non-negative.

We study moduli spaces of certain sextic curves with a singularity of multiplicity 3 from both perspectives of Deligne–Mostow theory and periods of $K3$ surfaces. In both ways we can describe the moduli spaces via arithmetic quotients of complex hyperbolic balls. We show in Theorem 7.4 that the two ball-quotient constructions can be unified in a geometric way.