Heat kernels arise in a variety of contexts including probability, geometry, and functional analysis; the automorphic heat kernel is particularly important in number theory and string theory. The typical construction of an automorphic heat kernel as a Poincaré series presents analytic difficulties, which can be dealt with in special cases (e.g., hyperbolic spaces) but are often sidestepped in higher rank by restricting to the compact quotient case. In this paper, we present a new approach, using global automorphic Sobolev theory, a robust framework for solving automorphic PDEs that does not require any simplifying assumptions about the rank of the symmetric space or the compactness of the arithmetic quotient. We construct an automorphic heat kernel via its automorphic spectral expansion in terms of cusp forms, Eisenstein series, and residues of Eisenstein series. We then prove uniqueness of the automorphic heat kernel as an application of operator semigroup theory. Finally, we prove the smoothness of the automorphic heat kernel by proving that its automorphic spectral expansion converges in the $C^{\mathrm{\infty }}$-topology.

In this paper, we study a particular model of distorted Brownian motion (dBM) on state spaces with varying dimension. Roughly speaking, the state space of such a process consists of two components: a 3-dimensional component and a 1-dimensional component. These two parts are joined together at the origin. The restriction of dBM on the 3- or 1-dimensional component receives a strong “push” toward the origin. On each component, the “magnitude” of a “push” can be parametrized by a constant $\mathit{\gamma }>0$. In this article, using the probabilistic method, we get the exact expressions for the transition density functions of dBM with varying dimension for any $0<t<\mathrm{\infty }$.

We show that the monodromy of a spherical conical metric g is reducible if and only if the metric g has a real-valued eigenfunction with eigenvalue 2 for the holomorphic extension ${\mathrm{\Delta }}_g^{\mathrm{Hol}}$ of the associated Laplace–Beltrami operator. Such an eigenfunction produces a meromorphic vector field, which is then related to the developing maps of the conical metric. We also give a lower bound of the first nonzero eigenvalue of ${\mathrm{\Delta }}_g^{\mathrm{Hol}}$, together with a complete classification of the dimension of the space of real-valued 2-eigenfunctions for ${\mathrm{\Delta }}_g^{\mathrm{Hol}}$ depending on the monodromy of the metric g. This paper can be seen as a new connection between the complex analysis method and the PDE approach in the study of spherical conical metrics.

In this paper, we completely classify, up to homeomorphism, the orientable and nonorientable orbifolds which have a Heegaard decomposition consisting of orbifold handlebodies with Euler number zero. In addition we compute their fundamental groups.

Let $(R,\mathfrak{m})$ be a complete discrete valuation ring with the finite residue field $R∕\mathfrak{m}={\mathbb{F}}_q$. Given a monic polynomial $P(t)\in R[t]$ whose reduction modulo $\mathfrak{m}$ gives an irreducible polynomial $\bar{P}(t)\in {\mathbb{F}}_q[t]$, we initiate an investigation of the distribution of $\mathrm{coker}(P(A))$, where $A\in {\mathrm{Mat}}_n(R)$ is randomly chosen with respect to the Haar probability measure on the additive group ${\mathrm{Mat}}_n(R)$ of $n\times n$ R-matrices. In particular, we provide a generalization of two results of Friedman and Washington about these random matrices. We use some concrete combinatorial connections between ${\mathrm{Mat}}_n(R)$ and ${\mathrm{Mat}}_n({\mathbb{F}}_q)$ to translate our problems about a Haar-random matrix in ${\mathrm{Mat}}_n(R)$ into problems about a random matrix in ${\mathrm{Mat}}_n({\mathbb{F}}_q)$ with respect to the uniform distribution. Our results over ${\mathbb{F}}_q$ are about the distribution of the $\bar{P}$-part of a random matrix $\bar{A}\in {\mathrm{Mat}}_n({\mathbb{F}}_q)$ with respect to the uniform distribution, and one of them generalizes a result of Fulman. We heuristically relate our results to a celebrated conjecture of Cohen and Lenstra, which predicts that given an odd prime p, any finite abelian p-group (i.e., ${\mathbb{Z}}_p$-module) H occurs as the p-part of the class group of a random imaginary quadratic field extension of $\mathbb{Q}$ with a probability inversely proportional to $|{\mathrm{Aut}}_{\mathbb{Z}}(H)|$. We review three different heuristics for the conjecture of Cohen and Lenstra, and they are all related to special cases of our main conjecture, which we prove as our main theorems.

A meromorphic quadratic differential on a punctured Riemann surface induces horizontal and vertical measured foliations with pole singularities. In a neighborhood of a pole, such a foliation comprises foliated strips and half-planes, and its leaf space determines a metric graph. We introduce the notion of an asymptotic direction at each pole and show that for a punctured surface equipped with a choice of such asymptotic data, any compatible pair of measured foliations uniquely determines a complex structure and a meromorphic quadratic differential realizing that pair. This proves the analogue of a theorem of Gardiner–Masur for meromorphic quadratic differentials. We also prove an analogue of the Hubbard–Masur theorem; namely, for a fixed punctured Riemann surface there exists a meromorphic quadratic differential with any prescribed horizontal foliation, and such a differential is unique provided we prescribe the singular flat geometry at the poles.

Let f be a martingale on an arbitrary atomic probability space equipped with a tree-like structure and let $S(f,q)$ denote the associated q-function. The paper is devoted to weighted $L^p$-estimates,

$$c_{p,q,w}^{-1}{\Vert S(f,q)\Vert }_{L^p(w)}\le \Vert f{\Vert }_{L^p(w)}\le C_{p,q,w}{\Vert S(f,q)\Vert }_{L^p(w)},1\le p<\mathrm{\infty },$$

for Muckenhoupt weights. Using the combination of the theory of sparse operators, extrapolation, and Bellman function method, we identify the optimal dependence of the constants $c_{p,q,w}$ and $C_{p,q,w}$ on the $A_p$ characteristics of the weights involved.

We determine the asymptotic behavior as the degree tends to infinity of the minimal $L^1$ norm $m(n,d)$ of the solution of an optimization problem arising when studying polynomial sphere maps. Here n is the source dimension and d is the degree. We provide upper and lower bounds for $m(n,d)$. We use these bounds to show that the function $d\to m(n,d)$ is monotone increasing in d. We prove that ${lim}_{d\to \mathrm{\infty }}\frac{m(n,d)}{d}=n(n-1)$. Let $N(n,d)$ denote the minimum possible target dimension of a monomial sphere map of degree d. We show, in source dimension unequal to 2, that ${lim}_{d\to \mathrm{\infty }}\frac{m(n,d)}{N(n,d)}=n$. The limit is 4 when $n=2$. We discuss some complicated results obtained by coding when $n=2$.

Let $E_1$ and $E_2$ be elliptic curves in Legendre form with integer parameters. We show there exists a constant C such that for almost all primes, for all but at most C pairs of points on the reduction of $E_1\times E_2$ modulo p having equal x coordinate, at least one among $P_1$ and $P_2$ has a large group order. We also show similar abundance over finite fields of elements whose images under the reduction modulo p of a finite set of rational functions have large multiplicative orders.

The Dobiński set $\mathcal{D}$ is an exceptional set for a certain infinite product identity, whose points are characterized as having exceedingly good approximations by dyadic rationals. We study the Hausdorff dimension and logarithmic measure of $\mathcal{D}$ by means of the mass transference principle and by the construction of certain appropriate Cantor-like sets, termed willow sets, contained in $\mathcal{D}$.