For a line arrangement ${\mathcal{A}}$ in the complex projective plane $\mathbb{P}^{2}$, we investigate the compactification $\overline{F}$ in $\mathbb{P}^{3}$ of the affine Milnor fiber $F$ and its minimal resolution $\tilde{F}$. We compute the Chern numbers of $\tilde{F}$ in terms of the combinatorics of the line arrangement ${\mathcal{A}}$. As applications of the computation of the Chern numbers, we show that the minimal resolution is never a quotient of a ball; in addition, we also prove that $\tilde{F}$ is of general type when the arrangement has only nodes or triple points as singularities. Finally, we compute all the Hodge numbers of some $\tilde{F}$ by using some knowledge about the Milnor fiber monodromy of the arrangement.

We analyse infinitesimal deformations of pairs $(X,{\mathcal{F}})$ with ${\mathcal{F}}$ a coherent sheaf on a smooth projective variety $X$ over an algebraically closed field of characteristic 0. We describe a differential graded Lie algebra controlling the deformation problem, and we prove an analog of a Mukai–Artamkin theorem about the trace map.

In the first part of the paper, we introduce notions of asymptotic continuous orbit equivalence and asymptotic conjugacy in Smale spaces and characterize them in terms of their asymptotic Ruelle algebras with their dual actions. In the second part, we introduce a groupoid $C^{\ast }$-algebra that is an extended version of the asymptotic Ruelle algebra from a Smale space and study the extended Ruelle algebras from the view points of Cuntz–Krieger algebras. As a result, the asymptotic Ruelle algebra is realized as a fixed point algebra of the extended Ruelle algebra under certain circle action.

Let $\unicode[STIX]{x1D701}(s)$ be the Riemann zeta function. In 1929, Hardy and Littlewood proved the approximate functional equation for $\unicode[STIX]{x1D701}^{2}(s)$ with error term $O(x^{1/2-\unicode[STIX]{x1D70E}}((x+y)/|t|)^{1/4}\log |t|)$, where $-1/2<\unicode[STIX]{x1D70E}<3/2,x,y\geqslant 1,xy=(|t|/2\unicode[STIX]{x1D70B})^{2}$. Later, in 1938, Titchmarsh improved the error term by removing the factor $((x+y)/|t|)^{1/4}$. In 1999, Hall showed the approximate functional equations for $\unicode[STIX]{x1D701}^{\prime }(s)^{2},\unicode[STIX]{x1D701}(s)\unicode[STIX]{x1D701}^{\prime \prime }(s)$, and $\unicode[STIX]{x1D701}^{\prime }(s)\unicode[STIX]{x1D701}^{\prime \prime }(s)$ (in the range $0<\unicode[STIX]{x1D70E}<1$) whose error terms contain the factor $((x+y)/|t|)^{1/4}$. In this paper we remove this factor from these three error terms by using the method of Titchmarsh.

Let $Y$ be a complex Enriques surface whose universal cover $X$ is birational to a general quartic Hessian surface. Using the result on the automorphism group of $X$ due to Dolgachev and Keum, we obtain a finite presentation of the automorphism group of $Y$. The list of elliptic fibrations on $Y$ and the list of combinations of rational double points that can appear on a surface birational to $Y$ are presented. As an application, a set of generators of the automorphism group of the generic Enriques surface is calculated explicitly.

We compute cup-product pairings in the integral cohomology ring of the moduli space of rank two stable bundles with odd determinant over a Riemann surface using methods of Zagier. The resulting formula is related to a generating function for certain skew Schur polynomials. As an application, we compute the nilpotency degree of a distinguished degree two generator in the mod two cohomology ring. We then give descriptions of the mod two cohomology rings in low genus, and describe the subrings invariant under the mapping-class group action.

A result of Haglund implies that the $(q,t)$-bigraded Hilbert series of the space of diagonal harmonics is a $(q,t)$-Ehrhart function of the flow polytope of a complete graph with netflow vector $(-n,1,\ldots ,1)$. We study the $(q,t)$-Ehrhart functions of flow polytopes of threshold graphs with arbitrary netflow vectors. Our results generalize previously known specializations of the mentioned bigraded Hilbert series at $t=1$, $0$, and $q^{-1}$. As a corollary to our results, we obtain a proof of a conjecture of Armstrong, Garsia, Haglund, Rhoades, and Sagan about the $(q,q^{-1})$-Ehrhart function of the flow polytope of a complete graph with an arbitrary netflow vector.

Motivated by the definition of a semigroup compactication of a locally compact group and a large collection of examples, we introduce the notion of an (operator) homogeneous left dual Banach algebra (HLDBA) over a (completely contractive) Banach algebra $A$. We prove a Gelfand-type representation theorem showing that every HLDBA over A has a concrete realization as an (operator) homogeneous left Arens product algebra: the dual of a subspace of $A^{\ast }$ with a compatible (matrix) norm and a type of left Arens product $\Box$. Examples include all left Arens product algebras over $A$, but also, when $A$ is the group algebra of a locally compact group, the dual of its Fourier algebra. Beginning with any (completely) contractive (operator) $A$-module action $Q$ on a space $X$, we introduce the (operator) Fourier space $({\mathcal{F}}_{Q}(A^{\ast }),\Vert \cdot \Vert _{Q})$ and prove that $({\mathcal{F}}_{Q}(A^{\ast })^{\ast },\Box )$ is the unique (operator) HLDBA over $A$ for which there is a weak$^{\ast }$-continuous completely isometric representation as completely bounded operators on $X^{\ast }$ extending the dual module representation. Applying our theory to several examples of (completely contractive) Banach algebras $A$ and module operations, we provide new characterizations of familiar HLDBAs over A and we recover, and often extend, some (completely) isometric representation theorems concerning these HLDBAs.

In this paper, we complete the ADE-like classification of simple transitive 2-representations of Soergel bimodules in finite dihedral type, under the assumption of gradeability. In particular, we use bipartite graphs and zigzag algebras of ADE type to give an explicit construction of a graded (non-strict) version of all these 2-representations.

Moreover, we give simple combinatorial criteria for when two such 2-representations are equivalent and for when their Grothendieck groups give rise to isomorphic representations.

Finally, our construction also gives a large class of simple transitive 2-representations in infinite dihedral type for general bipartite graphs.