In this paper we give a classification of discrete series for $SO\left( 2n,\,F \right)$, $F$$p$–adic, similar to that of Mœglin–Tadić for the other classical groups. This is obtained by taking the Mœglin–Tadić classification for $O\left( 2n,\,F \right)$ and studying how the representations restrict to $SO\left( 2n,\,F \right)$. We then extend this to an analysis of how admissible representations of $O\left( 2n,\,F \right)$ restrict.

In the paper, we introduce a new concept, topological orbit dimension of an $n$-tuple of elements in a unital ${{\text{C}}^{*}}$-algebra. Using this concept, we conclude that Voiculescu's topological free entropy dimension of every finite family of self-adjoint generators of a nuclear ${{\text{C}}^{*}}$-algebra is less than or equal to 1. We also show that the Voiculescu's topological free entropy dimension is additive in the full free product of some unital ${{\text{C}}^{*}}$-algebras. We show that the unital full free product of Blackadar and Kirchberg's unital $\text{MF}$ algebras is also an $\text{MF}$ algebra. As an application, we obtain that $\text{Ext(}C_{r}^{*}\text{(}{{F}_{2}}\text{)}{{\text{*}}_{\mathbb{C}}}C_{r}^{*}\text{(}{{F}_{2}}\text{))}$ is not a group.

An abstract polytope of rank $n$ is said to be chiral if its automorphism group has precisely two orbits on the flags, such that adjacent flags belong to distinct orbits. This paper describes a general method for deriving new finite chiral polytopes from old finite chiral polytopes of the same rank. In particular, the technique is used to construct many new examples in ranks 3, 4, and 5.

If ${{\Phi }_{\lambda }}$ is a master function corresponding to a hyperplane arrangement $\mathcal{A}$ and a collection of weights $\lambda $, we investigate the relationship between the critical set of ${{\Phi }_{\lambda }}$, the variety defined by the vanishing of the one-form ${{\omega }_{\lambda }}=\text{d}\log {{\Phi }_{\lambda }}$, and the resonance of $\lambda $. For arrangements satisfying certain conditions, we show that if $\lambda $ is resonant in dimension $p$, then the critical set of ${{\Phi }_{\lambda }}$ has codimension at most $p$. These include all free arrangements and all rank 3 arrangements.

We analyze flat ${{S}_{3}}$-covers of schemes, attempting to create structures parallel to those found in the abelian and triple cover theories. We use an initial local analysis as a guide in finding a global description.

We characterize the compactness of linear combinations of analytic composition operators on the Bloch space. We also study their boundedness and compactness on the little Bloch space.

Let $A$ be an $\text{AI}$ algebra; that is, $A$ is the ${{\text{C}}^{*}}$-algebra inductive limit of a sequence

$${{A}_{1}}\xrightarrow{{{\phi }_{1,2}}}{{A}_{2}}\xrightarrow{{{\phi }_{2,3}}}{{A}_{3}}\to \cdot \cdot \cdot \to {{A}_{n}}\to \cdot \cdot \cdot ,$$

where ${{A}_{n}}=\oplus _{i=1}^{{{k}_{n}}}{{M}_{\left[ n,i \right]}}\left( C\left( X_{n}^{i} \right) \right),X_{n}^{i}$ are [0, 1], ${{k}_{n}}$, and $\left[ n,\,i \right]$ are positive integers. Suppose that $A$ has the ideal property: each closed two-sided ideal of $A$ is generated by the projections inside the ideal, as a closed two-sided ideal. In this article, we give a complete classification of $\text{AI}$ algebras with the ideal property.

Reversibility of the Fleming-Viot process with mutation, selection, and recombination is well understood. In this paper, we study the reversibility of a system of Fleming-Viot processes that live on a countable number of colonies interacting with each other through migrations between the colonies. It is shown that reversibility fails when both migration and mutation are non-trivial.

We calculate reducibility for the representations of metaplectic groups induced from cuspidal representations of maximal parabolic subgroups via theta correspondence, in terms of the analogous representations of the odd orthogonal groups. We also describe the lifts of all relevant subquotients.

Ramanujan famously found congruences like $p(5n\,+\,4)\,\equiv \,0$ mod 5 for the partition function. We provide a method to find all simple congruences of this type in the coefficients of the inverse of a modular form on ${{\Gamma }_{1}}(4)$ that is non-vanishing on the upper half plane. This is applied to answer open questions about the (non)-existence of congruences in the generating functions for overpartitions, crank differences, and 2-colored $F$-partitions.

In this note we search the parameter space of Horrocks–Mumford quintic threefolds and locate a Calabi–Yau threefold that is modular, in the sense that the $L$-function of its middle-dimensional cohomology is associated with a classical modular form of weight 4 and level 55.

A diagonal ind-group is a direct limit of classical affine algebraic groups of growing rank under a class of inclusions that contains the inclusion

$$SL(n)\,\to \,SL(2n),\,\,M\mapsto \,\left( \begin{matrix} M & 0 \\ 0 & M \\ \end{matrix} \right)$$

as a typical special case. If $G$ is a diagonal ind-group and $B\,\subset \,G$ is a Borel ind-subgroup, we consider the ind-variety $G/B$ and compute the cohomology ${{H}^{\ell }}(G/B,\,{{\mathcal{O}}_{-\lambda }})$ of any $G$-equivariant line bundle ${{\mathcal{O}}_{-\lambda }}$ on $G/B$. It has been known that, for a generic $\lambda $, all cohomology groups of ${{\mathcal{O}}_{-\lambda }}$ vanish, and that a non-generic equivariant line bundle ${{\mathcal{O}}_{-\lambda }}$ has at most one nonzero cohomology group. The new result of this paper is a precise description of when ${{H}^{j}}(G/B,\,{{\mathcal{O}}_{-\lambda }})$ is nonzero and the proof of the fact that, whenever nonzero, ${{H}^{j}}(G/B,\,{{\mathcal{O}}_{-\lambda }})$ is a $G$-module dual to a highest weight module. The main difficulty is in defining an appropriate analog ${{W}_{B}}$ of the Weyl group, so that the action of ${{W}_{B}}$ on weights of $G$ is compatible with the analog of the Demazure “action” of the Weyl group on the cohomology of line bundles. The highest weight corresponding to ${{H}^{j}}(G/B,\,{{\mathcal{O}}_{-\lambda }})$ is then computed by a procedure similar to that in the classical Bott–Borel–Weil theorem.

Let ${{A}_{p}}\left( G \right)$ be the Figa-Talamanca, Herz Banach Algebra on $G$; thus ${{A}_{2}}\left( G \right)$ is the Fourier algebra. Strong Ditkin $\left( \text{SD} \right)$ and Extremely Strong Ditkin $\left( \text{ESD} \right)$ sets for the Banach algebras $A_{P}^{r}\left( G \right)$ are investigated for abelian and nonabelian locally compact groups $G$. It is shown that $\text{SD}$ and $\text{ESD}$ sets for ${{A}_{p}}\left( G \right)$ remain $\text{SD}$ and $\text{ESD}$ sets for $A_{P}^{r}\left( G \right)$, with strict inclusion for $\text{ESD}$ sets. The case for the strict inclusion of $\text{SD}$ sets is left open.

A result on the weak sequential completeness of ${{A}_{2}}\left( F \right)$ for $\text{ESD}$ sets $F$ is proved and used to show that Varopoulos, Helson, and Sidon sets are not $\text{ESD}$ sets for ${{A}_{2}}\left( G \right)$, yet they are such for $A_{2}^{r}\left( G \right)$ for discrete groups $G$, for any $1\,\le \,r\,\le \,2$.

A result is given on the equivalence of the sequential and the net definitions of $\text{SD}$ or $\text{ESD}$ sets for $\sigma $-compact groups.

The above results are new even if $G$ is abelian.

Speyer and Sturmfels associated Gröbner toric degenerations $\text{G}{{\text{r}}_{2}}{{({{\mathbb{C}}^{n}})}^{\mathcal{T}}}$ of $\text{G}{{\text{r}}_{2}}{{({{\mathbb{C}}^{n}})}^{{}}}$ with each trivalent tree $\mathcal{T}$ having $n$ leaves. These degenerations induce toric degenerations $M_{r}^{\mathcal{T}}$ of ${{M}_{r}}$, the space of $n$ ordered, weighted (by $\mathbf{r}$) points on the projective line. Our goal in this paper is to give a geometric (Euclidean polygon) description of the toric fibers and describe the action of the compact part of the torus as “bendings of polygons”. We prove the conjecture of Foth and Hu that the toric fibers are homeomorphic to the spaces defined by Kamiyama and Yoshida.

For a maximal torus in a quasi-split semi-simple simply-connected group over a local field of characteristic 0, Langlands and Shelstad constructed a cohomological invariant called the splitting invariant, which is an important component of their endoscopic transfer factors. We study this invariant in the case of a split real group and prove a decomposition theorem which expresses this invariant for a general torus as a product of the corresponding invariants for simple tori. We also show how this reduction formula allows for the comparison of splitting invariants between different tori in the given real group.

Certain polynomials in ${{n}^{2}}$ variables that serve as generating functions for symmetric group characters are sometimes called $\left( {{S}_{n}} \right)$ character immanants. We point out a close connection between the identities of Littlewood–Merris–Watkins and Goulden–Jackson, which relate ${{S}_{n}}$ character immanants to the determinant, the permanent and MacMahon's Master Theorem. From these results we obtain a generalization of Muir's identity. Working with the quantum polynomial ring and the Hecke algebra ${{H}_{n}}\left( q \right)$, we define quantum immanants that are generating functions for Hecke algebra characters. We then prove quantum analogs of the Littlewood–Merris–Watkins identities and selected Goulden–Jackson identities that relate ${{H}_{n}}\left( q \right)$ character immanants to the quantum determinant, quantum permanent, and quantum Master Theorem of Garoufalidis–Lê–Zeilberger. We also obtain a generalization of Zhang's quantization of Muir's identity.

In this paper, we study the non-vanishing and transcendence of special values of a varying class of $L$-functions and their derivatives. This allows us to investigate the transcendence of Petersson norms of certain weight one modular forms.

Let $F$ be a non-separable $\text{LF}$-space homeomorphic to the direct sum ${{\sum }_{n\in \text{N}}}\,{{\ell }_{2}}\left( {{\tau }_{n}} \right)$, where ${{\aleph }_{0}}<{{\tau }_{1}}<{{\tau }_{2}}<\cdot \cdot \cdot $. It is proved that every open subset $U$ of $F$ is homeomorphic to the product $\left| K \right|\,\times \,F$ for some locally finite-dimensional simplicial complex $K$ such that every vertex $v\,\in \,{{K}^{\left( 0 \right)}}$ has the star $\text{St}\left( v,\,K \right)$ with card $\text{St}{{\left( v,K \right)}^{\left( 0 \right)}}<\tau =\sup {{\tau }_{n}}$ (and card ${{K}^{\left( 0 \right)}}\le \tau $), and, conversely, if $K$ is such a simplicial complex, then the product $\left| K \right|\,\times \,F$ can be embedded in $F$ as an open set, where $\left| K \right|$ is the polyhedron of $K$ with the metric topology.

Let ${{S}_{k}}(\Gamma )$ be the space of holomorphic cusp forms of even integral weight $k$ for the full modular group. Let ${{\lambda }_{f}}(n)$ and ${{\lambda }_{g}}(n)$ be the $n$-th normalized Fourier coefficients of two holomorphic Hecke eigencuspforms $f(z),\,g(z)\,\in \,{{S}_{k}}(\Gamma )$, respectively. In this paper we are able to show the following results about higher moments of Fourier coefficients of holomorphic cusp forms.

(i)For any $\varepsilon \,>\,0$, we have

$$\sum\limits_{n\le x}{\lambda _{f}^{5}(n){{\ll }_{f,\varepsilon }}}{{x}^{\frac{15}{16}+\varepsilon }}\text{and}\sum\limits_{n\le x}{\lambda _{f}^{7}(n){{\ll }_{f,\varepsilon }}}{{x}^{\frac{63}{64}+\varepsilon }}.$$

(ii)If $\text{sy}{{\text{m}}^{3\,}}{{\pi }_{f}}\,\ncong \,\text{sy}{{\text{m}}^{3\,}}{{\pi }_{g}}\,$, then for any $\varepsilon \,>\,0$, we have

$$\sum\limits_{n\le x}{\lambda _{f}^{3}(n)\lambda _{g}^{3}(n){{\ll }_{f,\varepsilon }}}{{x}^{\frac{31}{32}+\varepsilon }};$$

If $\text{sy}{{\text{m}}^{2}}\,{{\pi }_{f}}\,\ncong \,\text{sy}{{\text{m}}^{2}}\,{{\pi }_{g}}$, then for any $\varepsilon \,>\,0$, we have

$$\sum\limits_{n\le x}{\lambda _{f}^{4}(n)\lambda _{g}^{2}(n)}=cx\log x+{c}'x+{{O}_{f,\varepsilon }}({{x}^{\frac{31}{32}+\varepsilon }});$$

If $\text{sy}{{\text{m}}^{2}}\,{{\pi }_{f}}\,\ncong \,\text{sy}{{\text{m}}^{2}}\,{{\pi }_{g}}$ and $\text{sy}{{\text{m}}^{4}}{{\pi }_{f}}\,\ncong \,\text{sy}{{\text{m}}^{4}}{{\pi }_{g}}$, then for any $\varepsilon \,>\,0$, we have

$$\sum\limits_{n\le x}{\lambda _{f}^{4}(n)\lambda _{g}^{4}(n)}=xP(\log x)+{{O}_{f,\varepsilon }}({{x}^{\frac{127}{128}+\varepsilon }}),$$

where $P\left( x \right)$ is a polynomial of degree 3.

Let $G$ be the $F$-rational points of the symplectic group $S{{p}_{2n}}$, where $F$ is a non-Archimedean local field of characteristic 0. Cogdell, Kim, Piatetski-Shapiro, and Shahidi constructed local Langlands functorial lifting from irreducible generic representations of $G$ to irreducible representations of $G{{L}_{2n+1}}\left( F \right)$. Jiang and Soudry constructed the descent map from irreducible supercuspidal representations of $G{{L}_{2n+1}}\left( F \right)$ to those of $G$, showing that the local Langlands functorial lifting from the irreducible supercuspidal generic representations is surjective. In this paper, based on above results, using the same descent method of studying $S{{O}_{2n+1}}$ as Jiang and Soudry, we will show the rest of local Langlands functorial lifting is also surjective, and for any local Langlands parameter $\phi \,\in \,\Phi \left( G \right)$, we construct a representation $\sigma $ such that $\phi $ and $\sigma $ have the same twisted local factors. As one application, we prove the $G$-case of a conjecture of Gross-Prasad and Rallis, that is, a local Langlands parameter $\phi \,\in \,\Phi \left( G \right)$ is generic, i.e., the representation attached to $\phi $ is generic, if and only if the adjoint $L$-function of $\phi $ is holomorphic at $s\,=\,1$. As another application, we prove for each Arthur parameter $\psi $, and the corresponding local Langlands parameter ${{\phi }_{\psi }}$, the representation attached to ${{\phi }_{\psi }}$ is generic if and only if ${{\phi }_{\psi }}$ is tempered.