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Pier Pacchiarotti

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I am a PhD student at the University of Warwick under the supervision of Goncalo Tabuada and Martin Gallauer.

My main interests lie within Homotopy Theory and Algebraic Geometry, such as K-Theory, Motivic Homotopy Theory, Derived Algebraic Geometry, and (oo,n)-Categories more broadly. A copy of my CV is available upon request.

You can contact me at Pier-Federico.Pacchiarotti (at) warwick.ac.uk.

Besides Mathematics, I'm a scout, I'm into politics and highly appreciate figurative arts, literature, and hiking.

Research

Undergraduate Manuscripts
  • "On Levine's Coniveau Filtration", Dissertation for the Galilean School of Higher Education (Padua) under the supervision of Prof. Tom Bachmann (Mainz). [Full-Text]Link opens in a new window

Abstract: The goal of the dissertation is to formulate Levine's Coniveau Tower in the setting of Derived Algebraic Geometry and in a self-contained way. The first section includes prerequisites: after a brief introduction to Animated Algebraic Geometry to ease referencing, it follows a presentation of foundations of Motivic Homotopy Theory from an ``animated" standpoint. The latter is an elaboration on lectures by Hoyois and has the goal to motivate notions, terminology, and axioms that will appear in the main body. Moreover, a - yet unsatisfactory - digression on (co)dimension theory in animated algebraic geometry is included. In the second section, the main part is developed. Levine's machinery is reformulated in the language of oo-categories to achieve a formal construction for the Coniveau Tower provided a - yet conjectural - ``nice enough" notion of (co)dimension. Finally, the dissertation is complemented by an appendix on spectra, t-structures, and spectral sequences. We purposely omit Levine's versions of Chow's Moving Lemma, as their generalization to the derived setting is yet unclear; for instance, a suitable derived analogous of generic points has not been discovered yet.

  • "An introduction to Blow-Ups of Quasi-Smooth Derived Sub-Schemes", Algant Master's Thesis (Padua and Regensburg) under the supervision of Prof. Marc Hoyois (Regensburg).

    [Full-Text]Link opens in a new window

Abstract: After a quick review of classical blow-ups and regular closed immersions, the first part of the thesis aims at providing a self-contained introduction to both Animated Higher Algebra and Animated Algebraic Geometry by merging the approach of Lurie's HA, SAG with that illustrated in the introduction to Khan's PhD thesis. At every stage, a model-independent formulation is attempted. Thereafter, a review of the recent work by Khan-Rydh on derived blow-ups of quasi-smooth closed subschemes follows. Apart from familiarity with general oo-category theory and standard algebraic geometry, no other prerequisite is assumed. Thus, the thesis includes appendices reviewing free sifted completion of oo-categories, symmetric monoidal oo-categories, and the theory of oo-topoi.

  • "A proof of the Countable Telescope Conjecture for Module Categories", Summer Research Project (Charles University Prague) under the supervision of Prof. Jan Trlifaj (CU Prague). arXiv:2201.10347Link opens in a new window

Abstract: The goal is to review recent developments on applications of methods from Set-theoretical Homological Algebra to Representation Theory of modules. Starting from the study of (Flat) Mittag-Leffler modules, we investigated deconstructibility issues via left and right approximations of modules. This led us to review the recent contributions by Saroch and Angeleri-Hügel-Saroch-Trlifaj to the solution of the Countable Telescope Conjecture for Module Categories, as well as its applications to the Enochs’ Conjecture. Finally, we considered both Saroch's advancements in the introduction to his Habilitation Thesis and the parallel and very promising contramodule approach, applied by Positselski and Trlifaj to investigate the properties of small precovering classes of modules. The final outcome of the project is an attempt to present the aforementioned results in a self-contained unifying manner.

  • "A proof of the Flat Cover Conjecture", Bachelor's Thesis (Padua). [Full version upon request]

Abstract: Rewriting some notes of Jan Trlifaj, we investigated the theory of approximation of modules via envelopes and covers and by means of set-theoretical constructions in homological algebra. In this framework, the FCC states that each module admits an `optimal left approximation' via a flat cover, thus leading to the existence of minimal flat resolutions of modules.

Teaching

Travel

  • [Coming soon] PCMI GSS 2024 on Motivic Homotopy Theory, July 2024, Salt Lake City, Utah, USA.
  • [Coming soon] Conference "The Interplay of Geometric Group Theory and K-theory", June 2024, University of Southampton, UK.
  • Conference "Motives in Mainz", March 2024, University of Mainz, Germany.
  • Winter School on "Unstable Motivic Homotopy Theory", March 2024, University of Mainz, Germany.
  • Oberseminar "On Six-Operations Formalisms", January 2024, University of Regensburg, Germany.

Contributing with a talk on"Passage to Stacks".

  • Conference “Young Topologists’ Meeting 2023”, July 2023, EPFL Lausanne, Switzerland.
  • Conference “Homotopy theory, K-theory, and trace methods”, July 2023, Radboud University Nijmegen, The Netherlands.
  • Workshop “Motivic and non-commutative aspects of enumerative geometry”, July 2023, Radboud University Nijmegen, The Netherlands.
  • Summer School “Motives in Ratisbona”, September 2022, University of Regensburg, Germany.
  • Conference “Young Topologists’ Meeting 2022”, July 2022, University of Copenhagen, Denmark.

Reading Seminars

  • "Stable Homotopy Groups of Spheres and Motivic Homotopy Theory", January - July 2024, University of Warwick, UK.

Contributing with talks on "The Cofibre of tau, second take" and "The Special Fibre".