We study the geometry of the component of the origin in the uniform spanning forest of $\mathbb{Z}^{d}$ and give bounds on the size of balls in the intrinsic metric.

We present the geometry behind counting twin prime polynomials in $\mathbb{F}_{q}[T]$ in general. We compute cohomology and explicitly count points by means of a twisted Lefschetz trace formula applied to these parametrizing varieties for cubic twin prime polynomials. The elliptic curve $X^{3}=Y(Y-1)$ occurs in the geometry, and thus counting cubic twin prime polynomials involves the associated modular form. In theory, this approach can be extended to higher degree twin primes, but the computations become harder.

The formula we get in degree 3 is compatible with the Hardy–Littlewood heuristic on average, agrees with the prediction for $q\equiv 2$ (mod 3), but shows anomalies for $q\equiv 1$ (mod 3).

A problem in representation theory of $p$-adic groups is the computation of the Casselman basis of Iwahori fixed vectors in the spherical principal series representations, which are dual to the intertwining integrals. We shall express the transition matrix $(m_{u,v})$ of the Casselman basis to another natural basis in terms of certain polynomials that are deformations of the Kazhdan–Lusztig R-polynomials. As an application we will obtain certain new functional equations for these transition matrices under the algebraic involution sending the residue cardinality $q$ to $q^{-1}$. We will also obtain a new proof of a surprising result of Nakasuji and Naruse that relates the matrix $(m_{u,v})$ to its inverse.

In this paper, we first derive the CR volume doubling property, CR Sobolev inequality, and the mean value inequality. We then apply them to prove the CR analogue of Yau’s conjecture on the space consisting of all pseudoharmonic functions of polynomial growth of degree at most $d$ in a complete noncompact pseudohermitian $(2n+1)$-manifold. As a by-product, we obtain the CR analogue of the volume growth estimate and the Gromov precompactness theorem.

The aim of this paper is to study the group of elliptic units of a cyclic extension $L$ of an imaginary quadratic field $K$ such that the degree $[L:K]$ is a power of an odd prime $p$. We construct an explicit root of the usual top generator of this group, and we use it to obtain an annihilation result of the $p$-Sylow subgroup of the ideal class group of $L$.

We study approximation of operators between Banach spaces $X$ and $Y$ that nearly attain their norms in a given point by operators that attain their norms at the same point. When such approximations exist, we say that the pair $(X,Y)$ has the pointwise Bishop–Phelps–Bollobás property (pointwise BPB property for short). In this paper we mostly concentrate on those $X$, called universal pointwise BPB domain spaces, such that $(X,Y)$ possesses pointwise BPB property for every $Y$, and on those $Y$, called universal pointwise BPB range spaces, such that $(X,Y)$ enjoys pointwise BPB property for every uniformly smooth $X$. We show that every universal pointwise BPB domain space is uniformly convex and that $L_{p}(\unicode[STIX]{x1D707})$ spaces fail to have this property when $p>2$. No universal pointwise BPB range space can be simultaneously uniformly convex and uniformly smooth unless its dimension is one. We also discuss a version of the pointwise BPB property for compact operators.

We study criteria for the uniform convergence of trigonometric series with general monotone coefficients. We also obtain necessary and sufficient conditions for a given rate of convergence of partial Fourier sums of such series.

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.

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.

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.