# Tanguy Vernet

### Biography

I did my PhD at the Chair of Arithmetic Geometry (ARG) of EPFL, under the supervision of Dimitri Wyss. I started in June 2024 as an IST-Bridge & Marie Skłodowska-Curie fellow (postdoc) in Tamás Hausel's group, at ISTA. My areas of interest are geometric representation theory and algebraic geometry. My research focuses on counts of quiver representations over rings of (truncated) power series and their links with p-adic integration, Donaldson-Thomas invariants of quivers and geometric realizations of quantum groups. See here for my resume and here for my new webpage.### Education

**PhD in Mathematics**

EPF Lausanne

2020-2024

**MSc in Mathematics (year 2)**

Université Paris-Sud, Orsay

2019-2020

**MSc in Mathematics (year 1)**

Ecole Polytechnique, Palaiseau

2018-2019

**BSc in Engineering**

Ecole Polytechnique, Palaiseau

2017-2018

**Civic service**

Ecole Polytechnique, Palaiseau

2016-2017

**Prep school in Mathematics and Physics**

Lycée Faidherbe, Lille

2014-2016

## Publications

### Selected publications

T. VernetIn this paper, I study counts of jets over the zero-fibre of quiver moment maps and their asymptotic behaviour. When the quiver is totally negative, I show that these counts converge to a p-adic integral, by proving that the corresponding fibre has rational singularities. I also generalise this result to a larger class of moduli spaces, which are locally modelled on quiver moment maps. |
Rational singularities for moment maps of totally negative quivers |

T. VernetThis paper focuses on counts of absolutely indecomposable quiver representations over rings of power series, generalising Kac polynomials. When the rank vector has entries at most one, I prove that the polynomial counting these representations has non-negative coefficients. Moreover, I show that jet spaces over zero-fibres of toric quiver moment maps have pure equivariant cohomology and that their Poincaré polynomials can be computed from the higher depth version of Kac polynomials. This is reminiscent of PBW-type isomorphisms on preprojective cohomological Hall algebras. |
Positivity for toric Kac polynomials in higher depth |

### Research

#### Talks

*Rational singularities for moment maps of totally negative quivers*,**Poster session in VBAC - Recent applications to the geometry of moduli spaces, Universität Duisburg-Essen**, Essen, Aug. 2023.*Rational singularities for moment maps of totally negative quivers*,**Poster session in workshop Derived categories, moduli spaces and counting invariants**, Imperial College, London, Jul. 2023.*The critical cohomological Hall algebra of a quiver with potential*,**Workshop on Vertex algebras and Hall algebras in enumerative geometry**, Skye, Apr. 2023.*Rational singularities for moment maps of totally negative quivers*,**Algebraic Geometry and Number Theory Seminar**, IST Austria, Klosterneuburg, Jan. 2023.*Rational singularities for moment maps of totally negative quivers*,**Number Theory and Algebraic Geometry Seminar**, KU Leuven, Leuven, Nov. 2022.*Rational singularities for moment maps of totally negative quivers*,**Séminaire Groupes, Représentations et Géométrie**, Université de Paris, Paris, Nov. 2022.*Rational singularities for moment maps of totally negative quivers*,**ARG Seminar**, EPFL, Lausanne, Nov. 2022.*Rational singularities for moduli of totally negative 2-Calabi-Yau categories*,**Poster session in summer school Recent perspectives in Hodge theory**, Centro De Giorgi, Pisa, Sept. 2022.*Counting quiver representations in higher depth and singularities of the moment map*,**Séminaire compréhensible (graduate seminar)**, Université Grenoble-Alpes, Grenoble, Mar. 2022.*Counting quiver representations in higher depth*,**Junior London Algebra Colloquium**, remote, Dec. 2021.*Counting quiver representations in higher depth*,**Young Algebraists’ Conference**, EPFL, Lausanne, Sept. 2021.*The geometry of indecomposable quiver representations - Counting over finite rings*,**Séminaire compréhensible (graduate seminar)**, Université Grenoble-Alpes, Grenoble, Dec. 2020.

### Teaching & PhD

**Spring 2023:**Bachelor thesis supervision,**Raphaël Parriaux: "Cayley-Bacharach theorems and conjectures"**, EPFL.**Spring 2023:**Main assistant for**Topics in Arithmetic Geometry**, by Dimitri Wyss, EPFL.**Fall 2022:**Main assistant for**Riemann Surfaces**, by Maryna Viazovska, EPFL.**Spring 2022:**Bachelor thesis supervision,**Matthew Dupraz: "The Weil conjectures for elliptic curves"**, EPFL.**Spring 2022:**Main assistant for**Algebraic curves**, by Dimitri Wyss, EPFL.**Fall 2021:**Main assistant for**Riemann Surfaces**, by Maryna Viazovska, EPFL.**Spring 2021:**Main assistant for**Algebraic curves**, by Dimitri Wyss, EPFL.**Fall 2020:**Main assistant for**Linear algebra**, by Christian Urech, EPFL.