Tue, 31 May 2022

14:00 - 15:00
L6

Towards 3d mirror symmetry for characteristic classes

Richard Rimanyi
(UNC Chapel Hill)
Abstract

In the first half of the talk, we will explore the concept of a characteristic class of a subvariety in a smooth ambient space. We will focus on the so-called stable envelope class,  in cohomology, K theory, and elliptic cohomology (due to Okoukov-Maulik-Aganagic). Stable envelopes have rich algebraic combinatorics, they are at the heart of enumerative geometry calculations, they show up in the study of associated (quantum) differential equations, and they are the main building blocks of constructing quantum group actions on the cohomology of moduli spaces.

In the second half of the talk, we will study a generalization of Nakajima quiver varieties called Cherkis’ bow varieties. These smooth spaces are endowed with familiar structures: holomorphic symplectic form, tautological bundles, torus action. Their algebraic combinatorics features a new powerful operation, the Hanany-Witten transition. Bow varieties come in natural pairs called 3d mirror symmetric pairs. A conjecture motivated by superstring theory predicts that stable envelopes on 3d mirror pairs are equal (in a sophisticated sense that involves switching equivariant and Kahler parameters). I will report on a work in progress, with T. Botta, to prove this conjecture.

Tue, 23 Nov 2021
14:00
Virtual

PageRank on directed preferential attachment graph

Mariana Olvera-Cravioto
(UNC Chapel Hill)
Abstract

We study a family of evolving directed random graphs that includes the directed preferential model and the directed uniform attachment model. The directed preferential model is of particular interest since it is known to produce scale-free graphs with regularly varying in-degree distribution. We start by describing the local weak limits for our family of random graphs in terms of randomly stopped continuous-time branching processes, and then use these limits to establish the asymptotic behavior of the corresponding PageRank distribution. We show that the limiting PageRank distribution decays as a power-law in both models, which is surprising for the uniform attachment model where the in-degree distribution has exponential tails. And even for the preferential attachment model, where the power-law hypothesis suggests that PageRank should follow a power-law, our result shows that the two tail indexes are different, with the PageRank distribution having a heavier tail than the in-degree distribution.

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