Anish Chedalavada's Homepage

About Me:

I am a fifth year PhD student and Teaching Assistant in the Department of Mathematics at Johns Hopkins.

Research:

My current research focus is in spectral algebraic geometry and tensor-triangulated geometry. I am broadly interested in homotopy theory and the areas of geometry and algebra with which it interacts.

Preprints:

  1. Higher Zariski geometry, joint with Ko Aoki, Tobias Barthel, Tomer Schlank, and Greg Stevenson. (A more updated version is available here).
      We revisit the classical constructions of tensor-triangular geometry in the setting of stably symmetric monoidal idempotent-complete \(\infty\)-categories, henceforth referred to as 2-rings. In this setting, we produce a Zariski topology, a Zariski spectrum, a category of locally 2-ringed spaces (more generally \(\infty\)-topoi), and an affine spectrum-global sections adjunction, based on the framework of "\(\infty\)-topoi with geometric structure" as developed by Lurie in [DAGV]. Using work of Kock and Pitsch, we compute that the underlying space of the Zariski spectrum of a 2-ring recovers the Balmer spectrum of its homotopy category. These constructions mirror the analogous structures in the classical Zariski geometry of commutative rings (and commutative ring spectra), and we also demonstrate additional compatibility between classical Zariski and higher Zariski geometry. For rigid 2-rings, we show that the descent results of Balmer and Favi admit coherent enhancements. As a corollary, we obtain that the Zariski spectrum fully faithfully embeds rigid 2-rings into locally 2-ringed \(\infty\)-topoi. In an appendix, we prove a ``stalk-locality principle'' for the telescope conjecture in the rigid setting, extending earlier work of Hrbek.
  2. Affineness and reconstruction in higher Zariski geometry. (Draft, to be posted soon).
      We explain how the geometric framework introduced in [HZG] provides a universal property for the 2-rings of perfect complexes on qcqs spectral or Dirac spectral scheme. As an application, given a spectral or Dirac spectral scheme \(X\), this will give rise to a comparison morphism from the Zariski spectrum of \(\mathrm{Perf}_{X}\) to \(X\) itself; this supplies a new proof of the Balmer and Neeman-Thomason reconstruction of a qcqs scheme \(X\) from its space of thick subcategories, assuming the result for noetherian rings. As another application, we find spectral and Dirac spectral enhancements of support varieties arising for 2-rings in representation theory which “geometrize” the 2-rings that produce them. For example, given a finite group \(G\) over a field \(k\), this produces a “spectral support variety” \(\mathcal{V}_{G}\) such that \(\mathrm{Perf}_{\mathcal{V}_{G}}\) maps into the stable module category of \(kG\). We derive these results as a corollary of a general affineness criterion for 2-schemes, which states that 2-schemes are affine if and only if they are quasicompact and quasiseparated.

In preparation:

Research Talks:

  1. A derived refinement of a classical reconstruction theorem in tt-geometry, Oberwolfach Workshop 2338a: Tensor-Triangular Geometry and Interactions. [Notes], [Extended Abstract for Oberwolfach Report].

Teaching:

Past teaching:

JH

UIC

Notes:

I maintain a list of talks along with notes from various reading seminars that I have participated in here.

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