We explore some probabilistic applications arising in connections with $K$-theoretic symmetric functions. For instance, we determine certain corner distributions of random lozenge tilings and plane partitions. also introduce that are naturally related to the growth model. Our main tools dual Grothendieck polynomials normalized Schur

We show that the Grothendieck bialgebra of the semi-tower of partition lattice algebras is isomorphic to the graded dual of the bialgebra of symmetric functions in noncommutative variables. In particular this isomorphism singles out a canonical new basis of the symmetric functions in noncommutative variables which would be an analogue of the Schur function basis for this bialgebra.

Let D1 ⊂ D2 be domains in C. Under very mild conditions on D2 we show that there exist holomorphic functions f , defined on D1 with the property that f is nowhere extendible across ∂D1, while the graph of f over D1 is not complete pluripolar in D2 × C. This refutes a conjecture of Levenberg, Martin and Poletsky [6].

Let M be a closed connected spin manifold of dimension 2 or 3 with a fixed orientation and a fixed spin structure. We prove that for a generic Riemannian metric onM the non-harmonic eigenspinors of the Dirac operator are nowhere zero. The proof is based on a transversality theorem and the unique continuation property of the Dirac operator.

We show that if C is a class of graphs which is nowhere dense then rst-order model-checking is xed-parameter tractable on C. As all graph classes which exclude a xed minor, or are of bounded local tree-width or locally exclude a minor are nowhere dense, this generalises algorithmic meta-theorems obtained for these classes in the past (see [11, 13, 4]). Conversely, if C is not nowhere dense and ...

We use Baire categorical arguments to construct dramatically pathological locally Lipschitz functions. The origins of this approach can be traced back to Banach and Mazurkiewicz (1931) who independently used similar categorical arguments to show that \almost every continuous real-valued function deened on 0,1] is nowhere diierentiable". As with the results of Banach and Mazurkiewicz, it appears...

As a necessary step in construction of elliptic matrix models, which preserve the superintegrability property ?char, we suggest an deformation peculiar loci pk?n, play important role precise formulation this property. The suggestion is to define pk?n-loci as functions with right asymptotics at ??i?. If hypothesis correct, one can move substituting Schur and Macdonald characters by GNS stu...