Polyharmonic maps of order k (briefly, k-harmonic maps) are a natural generalization of harmonic and biharmonic maps. These maps are defined as the critical points of suitable higher-order functionals which extend the classical energy functional for maps between Riemannian manifolds. The main aim of this paper is to investigate the so-called unique continuation principle. More precisely, assuming that the domain is connected, we shall prove the following extensions of results known in the harmonic and biharmonic cases: (i) if a k-harmonic map is harmonic on an open subset, then it is harmonic everywhere; (ii) if two k-harmonic maps agree on an open subset, then they agree everywhere; and (iii) if, for a k-harmonic map to the n-dimensional sphere, an open subset of the domain is mapped into the equator, then all the domain is mapped into the equator.

Consider an algorithm computing in a differential field with several commuting derivations such that the only operations it performs with the elements of the field are arithmetic operations, differentiation, and zero testing. We show that, if the algorithm is guaranteed to terminate on every input, then there is a computable upper bound for the size of the output of the algorithm in terms of the size of the input. We also generalize this to algorithms working with models of good enough theories (including, for example, difference fields).

We then apply this to differential algebraic geometry to show that there exists a computable uniform upper bound for the number of components of any variety defined by a system of polynomial PDEs. We then use this bound to show the existence of a computable uniform upper bound for the elimination problem in systems of polynomial PDEs with delays.

Immanants are functions on square matrices generalizing the determinant and permanent. Kazhdan–Lusztig immanants, which are indexed by permutations, involve $q=1$ specializations of Type A Kazhdan–Lusztig polynomials, and were defined by Rhoades and Skandera (2006, Journal of Algebra 304, 793–811). Using results of Haiman (1993, Journal of the American Mathematical Society 6, 569–595) and Stembridge (1991, Bulletin of the London Mathematical Society 23, 422–428), Rhoades and Skandera showed that Kazhdan–Lusztig immanants are nonnegative on matrices whose minors are nonnegative. We investigate which Kazhdan–Lusztig immanants are positive on k-positive matrices (matrices whose minors of size $k \times k$ and smaller are positive). The Kazhdan–Lusztig immanant indexed by v is positive on k-positive matrices if v avoids 1324 and 2143 and for all noninversions $i< j$ of v, either $j-i \leq k$ or $v_j-v_i \leq k$ . Our main tool is Lewis Carroll’s identity.

The main aim of this article is to show that normalised standard intertwining operator between induced representations of p-adic groups, at a very specific point of evaluation, has an arithmetic origin. This result has applications to Eisenstein cohomology and the special values of automorphic L-functions.

We compare the solutions of two Poisson problems in a spherical shell with Robin boundary conditions, one with given data, and one where the data have been cap symmetrized. When the Robin parameters are nonnegative, we show that the solution to the symmetrized problem has larger convex means. Sending one of the Robin parameters to $+\infty $ , we obtain mixed Robin/Dirichlet comparison results in shells. We prove similar results on balls and prove a comparison principle on generalized cylinders with mixed Robin/Neumann boundary conditions.

We compare crystal combinatorics of the level $2$ Fock space with the classification of unitary irreducible representations of type B rational Cherednik algebras to study how unitarity behaves under parabolic restriction. We show that the crystal operators that remove boxes preserve the combinatorial conditions for unitarity, and that the parabolic restriction functors categorifying the crystals send irreducible unitary representations to unitary representations. Furthermore, we find the supports of the unitary representations.

We generalize a classical extension result by Seeley in the context of Bastiani’s differential calculus to infinite dimensions. The construction follows Seeley’s original approach, but is significantly more involved as not only $C^k$ -maps (for ) on (subsets of) half spaces are extended, but also continuous extensions of their differentials to some given piece of boundary of the domains under consideration. A further feature of the generalization is that we construct families of extension operators (instead of only one single extension operator) that fulfill certain compatibility (and continuity) conditions. Various applications are discussed as well.

We give a formula for the cohomological invariants of a root stack, which we apply to compute the cohomological invariants and the Brauer group of the compactification of the stacks of hyperelliptic curves given by admissible double coverings.

Let $\mathcal {N}$ be the Nevanlinna class, and let B be a Blaschke product. It is shown that the natural invertibility criterion in the quotient algebra $\mathcal {N} / B \mathcal {N}$ , that is, $|f| \ge e^{-H} $ on the set $B^{-1}\{0\}$ for some positive harmonic function H, holds if and only if the function $- \log |B|$ has a harmonic majorant on the set $\{z\in \mathbb {D}:\rho (z,\Lambda )\geq e^{-H(z)}\}$ , at least for large enough functions H. We also study the corresponding class of positive harmonic functions H on the unit disc such that the latter condition holds. We also discuss the analogous invertibility problem in quotients of the Smirnov class.

We study the number of ways of factoring elements in the complex reflection groups $G(r,s,n)$ as products of reflections. We prove a result that compares factorization numbers in $G(r,s,n)$ to those in the symmetric group $S_n$ , and we use this comparison, along with the Ekedahl, Lando, Shapiro, and Vainshtein (ELSV) formula, to deduce a polynomial structure for factorizations in $G(r,s,n)$ .

We study the free metabelian group $M(2,n)$ of prime power exponent n on two generators by means of invariants $M(2,n)'\to \mathbb {Z}_n$ that we construct from colorings of the squares in the integer grid $\mathbb {R} \times \mathbb {Z} \cup \mathbb {Z} \times \mathbb {R}$ . In particular, we improve bounds found by Newman for the order of $M(2,2^k)$ . We study identities in $M(2,n)$ , which give information about identities in the Burnside group $B(2,n)$ and the restricted Burnside group $R(2,n)$ .

The notion of the truncated Euler characteristic for Iwasawa modules is a generalization of the the usual Euler characteristic to the case when the Selmer groups are not finite. Let p be an odd prime, $E_{1}$ and $E_{2}$ be elliptic curves over a number field F with semistable reduction at all primes $v|p$ such that the $\operatorname {Gal}(\overline {F}/F)$ -modules $E_{1}[p]$ and $E_{2}[p]$ are irreducible and isomorphic. We compare the Iwasawa invariants of certain imprimitive multisigned Selmer groups of $E_{1}$ and $E_{2}$ . Leveraging these results, congruence relations for the truncated Euler characteristics associated to these Selmer groups over certain $\mathbb {Z}_{p}^{m}$ -extensions of F are studied. Our results extend earlier congruence relations for elliptic curves over $\mathbb {Q}$ with good ordinary reduction at p.

We obtain a nontrivial upper bound for the multiplicative energy of any sufficiently large subset of a subvariety of a finite algebraic group. We also find some applications of our results to the growth of conjugates classes, estimates of exponential sums, and restriction phenomenon.