We discuss a surprising relationship between the partially ordered set of Newton points associated with an affine Schubert cell and the quantum cohomology of the complex flag variety. The main theorem provides a combinatorial formula for the unique maximum element in this poset in terms of paths in the quantum Bruhat graph, whose vertices are indexed by elements in the finite Weyl group. Key to establishing this connection is the fact that paths in the quantum Bruhat graph encode saturated chains in the strong Bruhat order on the affine Weyl group. This correspondence is also fundamental in the work of Lam and Shimozono establishing Peterson’s isomorphism between the quantum cohomology of the finite flag variety and the homology of the affine Grassmannian. One important geometric application of the present work is an inequality which provides a necessary condition for nonemptiness of certain affine Deligne–Lusztig varieties in the affine flag variety.

We compare two different types of mapping class invariants: the Hochschild homology of an ${\mathit{A}}_{\infty }$ bimodule coming from bordered Heegaard Floer homology and fixed point Floer cohomology. We first compute the bimodule invariants and their Hochschild homology in the genus two case. We then compare the resulting computations to fixed point Floer cohomology and make a conjecture that the two invariants are isomorphic. We also discuss a construction of a map potentially giving the isomorphism. It comes as an open-closed map in the context of a surface viewed as a 0-dimensional Lefschetz fibration over the complex plane.

We estimate the p-harmonic measure of the part of the boundary of an infinite slab outside a cylinder with large radius in terms of the radius.

Given an affine surface X with rational singularities and minimal resolution ${\mathit{X}}^{\prime }$, the covering of the Artin component of the deformation space of X where simultaneous resolutions are achieved is Galois and the Galois group is the Weyl group W associated with the configuration of $(-2)$-curves on ${\mathit{X}}^{\prime }$. This gives the existence of actions of W on polynomial rings over $\mathbb{Z}$ where the ring of invariants is also polynomial. In turn, this leads to a description of the integral cohomology rings of flag varieties of type $\mathit{ADE}$ that extends the known description of the rational cohomology rings as rings of coinvariants for actions of W.

The isoperimetric inequalities for the expected lifetime of Brownian motion state that the ${\mathit{L}}^{\mathit{p}}$-norms of the expected lifetime in a bounded domain for $1\le \mathit{p}\le \infty$ are maximized when the region is a ball with the same volume. In this paper, we prove quantitative improvements of the inequalities. Since the isoperimetric properties hold for a wide class of Lévy processes, many questions arise from these improvements.

In [PR19], Peters and Regts confirmed a conjecture by Sokal [Sok01] by showing that for every $\mathrm{\Delta }\in {\mathbb{Z}}_{\ge 3}$, there exists a complex neighborhood of the interval $[0,{(\mathrm{\Delta }-1)}^{\mathrm{\Delta }-1}/{(\mathrm{\Delta }-2)}^{\mathrm{\Delta }})$ on which the independence polynomial is nonzero for all graphs of maximum degree Δ. Furthermore, they gave an explicit neighborhood ${\mathit{U}}_{\mathrm{\Delta }}$ containing this interval on which the independence polynomial is nonzero for all finite rooted Cayley trees with branching number Δ. The question remained whether ${\mathit{U}}_{\mathrm{\Delta }}$ would be zero-free for the independence polynomial of all graphs of maximum degree Δ. In this paper, we show that this is not the case.

We previously showed that the inverse limit of standard-graded polynomial rings with perfect (or semiperfect) coefficient field is a polynomial ring in an uncountable number of variables. In this paper, we show that the result holds with no hypothesis on the coefficient field. We also prove an analogous result for ultraproducts of polynomial rings.