ABSTRACT We develop an affine scheme-theoretic version of Hamiltonian reduction by symplectic groupoids. It works over $\mathbb {k}=\mathbb {R}$ or $\mathbb {k}=\mathbb {C}$, and is formulated for an affine symplectic groupoid $\mathcal {G}\,\,\substack{\longrightarrow \\ \longrightarrow }\,\,X$, an affine Hamiltonian $\mathcal {G}$-scheme $\mu :M\longrightarrow X$, a coisotropic subvariety $S\subseteq X$, and a stabilizer subgroupoid $\mathcal {H}\,\,\substack{\longrightarrow \\\longrightarrow }\,\,S$. Our first main result is that the Poisson bracket on $\mathbb {k}[M]$ induces a Poisson bracket on the subquotient $\mathbb {k}[\mu ^{-1}(S)]^{\mathcal {H}}$. The Poisson scheme $M\mathbin {//}_{{S,\mathcal {H}}}\mathcal {G}:= \mathrm{Spec}(\mathbb {k}[\mu ^{-1}(S)]^{\mathcal {H}})$ is then declared to be a Hamiltonian reduction of M. Other main results include sufficient conditions for $M\mathbin {//}_{{S,\mathcal {H}}}\mathcal {G}$ to inherit a residual Hamiltonian scheme structure. Our main results are best viewed as affine scheme-theoretic counterparts to [6], where we simultaneously generalize several Hamiltonian reduction processes. In this way, the present work yields scheme-theoretic analogues of Marsden–Ratiu reduction [19], Mikami–Weinstein reduction [20], Śniatycki–Weinstein reduction [23], and symplectic reduction along general coisotropic submanifolds [6]. The initial impetus for this work was its utility in formulating and proving generalizations of the Moore–Tachikawa conjecture.

ABSTRACT We investigate the mean value of the inner product of squared $\mathrm{GL}_{n}$ degenerate maximal parabolic Eisenstein series against a smooth compactly supported function lying in a restricted space of incomplete Eisenstein series induced from an $\mathrm{SL}_{2}(\mathbb {Z})$ Hecke–Maaß cusp form $\varphi$. Our result breaks the fundamental threshold with a polynomial power-saving beyond the pointwise implications of the generalized Lindelöf hypothesis for L-functions attached to $\varphi$. Furthermore, we evaluate the Archimedean quantum variance and establish approximate orthogonality, expanding upon Zhang’s work on quantum unique ergodicity for $\mathrm{GL}_{n}$ degenerate maximal parabolic Eisenstein series as well as Huang’s work on quantum variance for $\mathrm{GL}_{2}$ Eisenstein series. Despite the theoretical strength of these manifestations, our argument relies exclusively on the Watson–Ichino-type formula for incomplete Eisenstein series of type $(2, 1, \ldots , 1)$ and Jutila’s asymptotic formula for the second moment of L-functions attached to $\varphi$ in long intervals, supplemented by a standard analytical toolbox.

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.

ABSTRACT We study the areal Mahler measure of the two-variable, $k$-parameter family $x+y+k$ and prove explicit formulas that demonstrate its relation to the standard Mahler measure of these polynomials. The proofs involve interpreting the areal Mahler measure as a random walk in the complex plane and utilizing the areal analogue of the Zeta Mahler function to arrive at the result. Using similar techniques, we also present formulas for a three-variable family $(x+1)(y+1)+kz$ in terms of the standard Mahler measure, along with terms that involve certain hypergeometric functions. For both families, we show that its areal Mahler measure is, up to elementary functions, a linear combination of the normal Mahler measure and the volume of the Deninger cycle of the corresponding family.

ABSTRACT We study the continuity of archimedean zeta integrals, as well as the lower bound of their operator norms, with respect to a natural family of Sobolev norms arising from the functional calculus of harmonic oscillator. The relevant notion and results will be generalized to archimedean Rankin–Selberg integrals of the tempered principal series representations of general linear groups.

ABSTRACT We consider the twistor theory approach to Kronheimer's ALE metrics on the resolution of a Kleinian singularity. The circle action on the 4-manifold induces a C*-action on a compactification of the twistor space and we identify the orbit of a generic twistor line as a nodal rational curve in a particular cohomology class of a projective rational surface. Using the results of N.Honda et. al. we identify this surface with the minitwistor space for the Einstein-Weyl structure on the 3-dimensional quotient of the ALE space by the circle action.

ABSTRACT Moments of moments of the Riemann zeta function, defined by $$ \text{MoM}_T(k,\beta) := \frac{1}{T}\int_T^{2T} \Bigg(\,\int\limits_{ |h|\leq (\log T)^\theta}|\zeta(\frac{1}{2} + i t + ih)|^{2\beta}\ dh\Bigg)^k\ dt, $$ where $k,\beta \geq 0$ and $\theta \gt -1$ were introduced by Fyodorov and Keating, Freezing transitions and extreme values: random matrix theory, and disordered landscapes, Philos. Trans. Roy. Soc. A: 372 no. 2007 (2014), 20120503 A doi:10.1098/rsta.2012.0503 when comparing extreme values of zeta in short intervals to those of characteristic polynomials of random unitary matrices. We study the k = 2 case as $T \rightarrow \infty$ and obtain sharp upper bounds for $\text{MoM}_T(2,\beta)$ for all real $0\leq \beta \leq 1$ as well as lower bounds of the conjectured order for all $\beta \geq 0$. In particular, we show that the second moment of moments undergoes a freezing phase transition with critical exponent $\beta = \frac{1}{\sqrt{2}}$. The main technical ingredient is a new estimate for the correlation of two shifted $2\beta^{\text{th}}$ powers of zeta with $0\leq \beta \leq 1$.

ABSTRACT We show that if a Sylow p-subgroup of a finite group G is nilpotent of class at most p, then the p-part of the Bogomolov multiplier of G is locally controlled.

ABSTRACT A set $A\subset \mathbb{F}_p^n$ is sum-free if A + A does not intersect A. If $p\equiv 2 \;\mathrm{mod}\; 3$, the maximal size of a sum-free set in $\mathbb{F}_p^n$ is known to be $(p^n+p^{n-1})/3$. We show that if a sum-free set $A\subset \mathbb{F}_p^n$ has size at least $p^n/3-p^{n-1}/6+p^{n-2}$, then there exists subspace $V\lt\mathbb{F}_p^n$ of codimension 1 such that A is contained in $(p+1)/3$ cosets of V. For p = 5 specifically, we show the stronger result that every sum-free set of size larger than $1.2\cdot 5^{n-1}$ has this property, thus improving on a recent theorem of Lev.

ABSTRACT We consider the subgroup of points of finite orbit through the action of an endomorphism of a finitely generated virtually free group, with particular emphasis on the subgroup of eventually fixed points, $\text{EvFix}(\varphi)$: points whose orbit contains a fixed point. We provide an algorithm to compute the subgroup of fixed points of an endomorphism of a finitely generated virtually free group and prove that finite orbits have cardinality bounded by a computable constant, which allows us to solve several algorithmic problems: deciding if φ is a finite order element of $\text{End}(G)$, if φ is aperiodic, if $\text{EvFix}(\varphi)$ is finitely generated and if $\text{EvFix}(\varphi)$ is a normal subgroup. In the cases where $\text{EvFix}(\varphi)$ is finitely generated, we also present a bound for its rank and an algorithm to compute a generating set.