Physical Sciences and Mathematics Commons^™

Frobenius Objects In The Category Of Spans, Ivan Contreras, Molly Keller, Rajan Amit Mehta

Mathematics Sciences: Faculty Publications

We consider Frobenius objects in the category Span, where the objects are sets and the morphisms are isomorphism classes of spans of sets. We show that such structures are in correspondence with data that can be characterized in terms of simplicial sets. An interesting class of examples comes from groupoids.

Our primary motivation is that Span can be viewed as a set-theoretic model for the symplectic category, and thus Frobenius objects in Span provide set-theoretic models for classical topological field theories. The paper includes an explanation of this relationship.

Given a finite commutative Frobenius object in Span, one can obtain …

Relaxed Wythoff Has All Beatty Solutions, Jon Kay, Geremías Polanco

Mathematics Sciences: Faculty Publications

We find conditions under which the P-positions of three subtraction games arise as pairs of complementary Beatty sequences. The first game is due to Fraenkel and the second is an extension of the first game to non-monotone settings. We show that the P-positions of the second game can be inferred from the recurrence of Fraenkel's paper if a certain inequality is satisfied. This inequality is shown to be necessary if the P-positions are known to be pairs of complementary Beatty sequences, and the family of irrationals for which this inequality holds is explicitly given. We highlight several games in the …

Neumann Problems For P-Harmonic Functions, And Induced Nonlocal Operators In Metric Measure Spaces, Luca Capogna, Josh Kline, Riikka Korte, Nageswari Shanmugalingam, Marie Snipes

Mathematics Sciences: Faculty Publications

Following ideas of Caffarelli and Silvestre in [20], and using recent progress in hyperbolic fillings, we define fractional p-Laplacians (−∆p) θ with 0 < θ < 1 on any compact, doubling metric measure space (Z, d, ν), and prove existence, regularity and stability for the non-homogenous non-local equation (−∆p) θu = f. These results, in turn, rest on the new existence, global Hölder regularity and stability theorems that we prove for the Neumann problem for p-Laplacians ∆p, 1 < p < ∞, in bounded domains of measure metric spaces endowed with a doubling measure that supports a Poincaré inequality. Our work also includes as special cases much of the previous results by other authors in the Euclidean, Riemannian and Carnot group settings. Unlike other recent contributions in the metric measure spaces context, our work does not rely on the assumption that (Z, d, ν) supports a Poincaré inequality.

The Transition Matrix Between The Specht And 𝔰𝔩3 Web Bases Is Unitriangular With Respect To Shadow Containment, Heather M. Russell, Julianna Tymoczko

Mathematics Sciences: Faculty Publications

Webs are planar graphs with boundary that describe morphisms in a diagrammatic representation category for 𝔰𝔩k ⁠. They are studied extensively by knot theorists because braiding maps provide a categorical way to express link diagrams in terms of webs, producing quantum invariants like the well-known Jones polynomial. One important question in representation theory is to identify the relationships between different bases; coefficients in the change-of-basis matrix often describe combinatorial, algebraic, or geometric quantities (e.g., Kazhdan–Lusztig polynomials). By ”flattening” the braiding maps, webs can also be viewed as the basis elements of a symmetric group representation. In this paper, we define …

Lipschitz Regularity For The Parabolic P-Laplacian In The Heisenberg Group, Luca Capogna, Giovanna Citti, Xiao Zhong

Mathematics Sciences: Faculty Publications

In this paper we establish the local Lipschitz regularity of weak solutions of a certain class of quasilinear, degenerate parabolic equations in the Heisenberg group Hn , or more in general in contact subRiemannian manifolds. In particular we extend to the non-stationary setting.

Toward Permutation Bases In The Equivariant Cohomology Rings Of Regular Semisimple Hessenberg Varieties, Megumi Harada, Martha Precup, Julianna Tymoczko

Mathematics Sciences: Faculty Publications

Recent work of Shareshian and Wachs, Brosnan and Chow, and Guay-Paquet connects the wellknown Stanley–Stembridge conjecture in combinatorics to the dot action of the symmetric group S_n on the cohomology rings H∗ (Hess(S, h)) of regular semisimple Hessenberg varieties. In particular, in order to prove the Stanley–Stembridge conjecture, it suffices to construct (for any Hessenberg function h) a permutation basis of H∗ (Hess(S, h)) whose elements have stabilizers isomorphic to Young subgroups. In this manuscript we give several results which contribute toward this goal. Specifically, in some special cases, we give a new, purely combinatorial construction of classes …