arXiv subjects

Trishan Mondal

Publications and source records attributed to Trishan Mondal.

2 recordsLinked to original sources

arXiv2026

Braid Group Action on $D^b(\mathfrak{M}_η)$

We construct an action of the braid group on the bounded derived category of coherent sheaves on hypertoric varieties arising from hyperplane arrangements. Using wall-crossing equivalences associated to paths in the complexified complement of the hyperplane arrangement, we show that these equivalences under certain conditions yield a functor from the Deligne groupoid to the category of triangulated equivalences. This gives rise to a canonical representation of the fundamental group, which recovers the braid group, acting on $D^b(\mathfrak{M}_η)$. This is a summary of Brad Hannigan-Daley's PhD thesis.

math.AG↗

arXiv2025

Revisiting Pontryagin's Proof of Stable Stems 1 and 2

In this paper, we introduce fundamental notions of homotopy theory, including homotopy excision and the Freudenthal suspension theorem. We then explore framed cobordism and its connection to stable homotopy groups of spheres through the Pontryagin-Thom construction. Using this framework, we compute the stable stems in dimensions $0$, $1$, and $2$. This work is primarily expository, revisiting proofs from \cite{Pont1} with slight modifications incorporating modern notation. Furthermore, in the final section, we discuss 2-dimensional framed manifolds with Arf invariant one and examine why the result of \cite{Pont2} regarding $π_2^S$ is incorrect.

math.AT↗