We study webs in quantum type C, focusing on the rank three case. We define a linear pivotal category $\textbf {Web}(\mathfrak {sp}_{6})$ diagrammatically by generators and relations, and conjecture that it is equivalent to the category $\textbf {FundRep}(U_q(\mathfrak {sp}_{6}))$ of quantum $\mathfrak {sp}_{6}$ representations generated by the fundamental representations, for generic values of the parameter q. We prove a number of results in support of this conjecture, most notably that there is a full, essentially surjective functor $\textbf {Web}(\mathfrak {sp}_{6}) \rightarrow \textbf {FundRep}(U_q(\mathfrak {sp}_{6}))$ , that all $\textrm {Hom}$ -spaces in $\textbf {Web}(\mathfrak {sp}_{6})$ are finite-dimensional, and that the endomorphism algebra of the monoidal unit in $\textbf {Web}(\mathfrak {sp}_{6})$ is one-dimensional. The latter corresponds to the statement that all closed webs can be evaluated to scalars using local relations; as such, we obtain a new approach to the quantum $\mathfrak {sp}_{6}$ link invariants, akin to the Kauffman bracket description of the Jones polynomial.

We construct a family of compactifications of the affine cone of the Grassmannian variety of $2$ -planes. We show that both the tropical variety of the Plücker ideal and familiar valuations associated to the construction of Newton–Okounkov bodies for the Grassmannian variety can be recovered from these compactifications. In this way, we unite various perspectives for constructing toric degenerations of flag varieties.

Let $a_1$ , $a_2$ , and $a_3$ be distinct reduced residues modulo q satisfying the congruences $a_1^2 \equiv a_2^2 \equiv a_3^2 \ (\mathrm{mod}\ q)$ . We conditionally derive an asymptotic formula, with an error term that has a power savings in q, for the logarithmic density of the set of real numbers x for which $\pi (x;q,a_1)> \pi (x;q,a_2) > \pi (x;q,a_3)$ . The relationship among the $a_i$ allows us to normalize the error terms for the $\pi (x;q,a_i)$ in an atypical way that creates mutual independence among their distributions, and also allows for a proof technique that uses only elementary tools from probability.

In [5], Chen and Yui conjectured that Gross–Zagier type formulas may also exist for Thompson series. In this work, we verify Chen and Yui’s conjecture for the cases for Thompson series $j_{p}(\tau )$ for $\Gamma _{0}(p)$ for p prime, and equivalently establish formulas for the prime decomposition of the resultants of two ring class polynomials associated to $j_{p}(\tau )$ and imaginary quadratic fields and the prime decomposition of the discriminant of a ring class polynomial associated to $j_{p}(\tau )$ and an imaginary quadratic field. Our method for tackling Chen and Yui’s conjecture on resultants can be used to give a different proof to a recent result of Yang and Yin. In addition, as an implication, we verify a conjecture recently raised by Yang, Yin, and Yu.

Wilson loop diagrams are an important tool in studying scattering amplitudes of SYM $N=4$ theory and are known by previous work to be associated to positroids. We characterize the conditions under which two Wilson loop diagrams give the same positroid, prove that an important subclass of subdiagrams (exact subdiagrams) corresponds to uniform matroids, and enumerate the number of different Wilson loop diagrams that correspond to each positroid cell. We also give a correspondence between those positroids which can arise from Wilson loop diagrams and directions in associahedra.

This paper studies a new Whitney type inequality on a compact domain $\Omega \subset {\mathbb R}^d$ that takes the form

$$ \begin{align*} \inf_{Q\in \Pi_{r-1}^d(\mathcal{E})} \|f-Q\|_p \leq C(p,r,\Omega) \omega_{\mathcal{E}}^r(f,\mathrm{diam}(\Omega))_p,\ \ r\in {\mathbb N},\ \ 0<p\leq \infty, \end{align*} $$

$\omega _{\mathcal {E}}^r(f, t)_p$

$f\in L^p(\Omega )$

$\mathcal {E}\subset \mathbb {S}^{d-1}$

$\mathrm {span}(\mathcal {E})={\mathbb R}^d$

$\Pi _{r-1}^d(\mathcal {E}):=\{g\in C(\Omega ):\ \omega ^r_{\mathcal {E}} (g, \mathrm {diam} (\Omega ))_p=0\}$

$\mathcal {E}$

$\Omega \subset {\mathbb R}^d$

$C^2$

$\Omega \subset {\mathbb R}^d$

$\mathcal {E}$

$\mathcal {N}_d(\Omega )\in {\mathbb N}$

$\mathcal {N}_d(\Omega )$

$\mathcal {E}$

$\mathrm {span}(\mathcal {E})={\mathbb R}^d$

$\Omega $

$r\in {\mathbb N}$

$p>0$

$\mathcal {N}_d(\Omega )=d$

$C^2$

$\Omega \subset {\mathbb R}^d$

$d=2$

$\Omega \subset {\mathbb R}^2$

$d\ge 3$

$\Omega \subset {\mathbb R}^d$

$d\ge 3$

$\Omega \subset {\mathbb R}^d$

$\mathcal {N}_d(\Omega )$

$\Omega $

$\Omega $

$\mathcal {E}\subset \mathbb {S}^{d-1}$

$\mathcal {E}$

$\Omega $

$r\in {\mathbb N}$

$0<p\leq \infty $

$\mathcal {N}_d(\Omega )$

$d\ge 3$

A slight modification of the proof of the usual Whitney inequality in literature also yields a directional Whitney inequality on each convex body $\Omega \subset {\mathbb R}^d$ , but with the set $\mathcal {E}$ containing more than $(c d)^{d-1}$ directions. In this paper, we develop a new and simpler method to prove the directional Whitney inequality on more general, possibly nonconvex domains requiring significantly fewer directions in the directional moduli.

Let G be a connected reductive group over a p-adic number field F. We propose and study the notions of G- $\varphi $ -modules and G- $(\varphi ,\nabla )$ -modules over the Robba ring, which are exact faithful F-linear tensor functors from the category of G-representations on finite-dimensional F-vector spaces to the categories of $\varphi $ -modules and $(\varphi ,\nabla )$ -modules over the Robba ring, respectively, commuting with the respective fiber functors. We study Kedlaya’s slope filtration theorem in this context, and show that G- $(\varphi ,\nabla )$ -modules over the Robba ring are “G-quasi-unipotent,” which is a generalization of the p-adic local monodromy theorem proved independently by Y. André, K. S. Kedlaya, and Z. Mebkhout.

We study harmonic analysis on the symmetric space $\text{GL}_n \times \text{GL}_n \backslash \text{GL}_{2n}$ . We prove several standard results, e.g. Shalika germ expansion of orbital integrals, representability of the Fourier transform of orbital integrals and representability of spherical characters. These properties are not expected to hold for symmetric spaces in general.

For a smooth strongly convex Minkowski norm $F:\mathbb {R}^n \to \mathbb {R}_{\geq 0}$ , we study isometries of the Hessian metric corresponding to the function $E=\tfrac 12F^2$ . Under the additional assumption that F is invariant with respect to the standard action of $SO(k)\times SO(n-k)$ , we prove a conjecture of Laugwitz stated in 1965. Furthermore, we describe all isometries between such Hessian metrics, and prove Landsberg Unicorn Conjecture for Finsler manifolds of dimension $n\ge 3$ such that at every point the corresponding Minkowski norm has a linear $SO(k)\times SO(n-k)$ -symmetry.

Gay and Kirby introduced trisections, which describe any closed, oriented, smooth 4-manifold X as a union of three 4-dimensional handlebodies. A trisection is encoded in a diagram, namely three collections of curves in a closed oriented surface $\Sigma $ , guiding the gluing of the handlebodies. Any morphism $\varphi $ from $\pi _1(X)$ to a finitely generated free abelian group induces a morphism on $\pi _1(\Sigma )$ . We express the twisted homology and Reidemeister torsion of $(X;\varphi )$ in terms of the first homology of $(\Sigma ;\varphi )$ and the three subspaces generated by the collections of curves. We also express the intersection form of $(X;\varphi )$ in terms of the intersection form of $(\Sigma ;\varphi )$ .

Let A be a two-dimensional abelian variety defined over a number field K. Fix a prime number $\ell $ and suppose $\#A({\mathbf {F}_{\mathfrak {p}}}) \equiv 0 \pmod {\ell ^2}$ for a set of primes ${\mathfrak {p}} \subset {\mathcal {O}_{K}}$ of density 1. When $\ell =2$ Serre has shown that there does not necessarily exist a K-isogenous $A'$ such that $\#A'(K)_{{tor}} \equiv 0 \pmod {4}$ . We extend those results to all odd $\ell $ and classify the abelian varieties that fail this divisibility principle for torsion in terms of the image of the mod- $\ell ^2$ representation.

Let M be a closed, connected, orientable topological $4$ -manifold, and G be a finite group acting topologically and locally linearly on M. In this paper, we investigate the spectral sequence for the Borel cohomology $H^*_G(M)$ and establish new bounds on the rank of G for homologically trivial actions with discrete singular set.

We prove $L^{p}$ -boundedness of oscillating multipliers on symmetric spaces of noncompact type of arbitrary rank, as well as on a wide class of locally symmetric spaces.