Sabrina Pauli
I am currently a junior professor at the TU Darmstadt (6 year positiion). Before that, I was a substitute professor in Düsseldorf and a postdoc at the University of Duisburg-Essen, working with Marc Levine.
I received my Ph.D. in September 2020 from the University of Oslo, supervised by Paul Arne Østvær and Kirsten Wickelgren.
My research area is A1-homotopy theory, especially applications to enumerative geometry. For this I also use tropical geometry. I am also working on some computations in equivariant parametrized homotopy theory with my WIT group.
Address
TU Darmstadt
Schlossgartenstraße 7
64289 Darmstadt
Gebäude: S2 15
Raum: 442
E-Mail: pauli - at - mathematik.tu-darmstadt.de
Webpage
Education
September 2020: PhD at the University of Oslo
Fall 2019: Visiting Scholar at Duke University
June 2017: Masters in Mathematics at the University of Oslo
Mai 2015: Bachelor in Mathematics at the TU Darmstadt
Employment
Junior Professor at the TU Darmstadt since October 2023
Substitute Professor at the Heinrich-Heine-Universität April 2023 - September 2023
Postdoc at the Universität Duisburg-Essen October 2020 - March 2023
Research
A1-homotopy theory, also known as motivic homotopy theory developed by Morel-Voevodsky, is an emerging area of mathematics that unites the two worlds of algebraic geometry and algebraic topology by applying tools from algebraic topology, in particular homotopy theory, to algebraic varieties.
This provides new tools for attacking classical problems in algebraic geometry and related areas.
I started my research by studying examples of algebraic varieties that are A1-contractible, i.e., algebraic varieties that look like a point in this A1-homotopy category.
Currently, my research focuses on the application of A1-homotopy theory to enumerative geometry, leading to the new and rapidly growing field of A1-enumerative geometry, which allows to study questions in enumerative geometry over an arbitrary base field. I also use methods from tropical geometry for this.
Recently, with my WIT group, I have started to study and compute equivariant parametrized cohomology for G a finite group. This is cohomology with extended grading, which can be seen as an equivariant version of cohomology with local coefficients and was developed by Costenoble-Waner.
Publications
- A quadratically enriched correspondence theorem, with Andrés Jaramillo Puentes, accepted for publication in the Journal of the European Mathematical Society, arXiv
- Arithmetic counts of tropical plane curves and their properties, with Andrés Jaramillo Puentes, Hannah Markwig and Felix Röhrle, accepted for publication in Advances in Geometry, arXiv
- Quadratic Counts of Twisted Cubics, with Marc Levine, accepted for publication in Trends in Mathematics, book in memory of Alberto Collino, arXiv
- Bézoutians and the A^1-degree, with Thomas Brazelton and Stephen McKean, published in Algebra Number Theory 17, No. 11, 1985-2012 (2023). arXiv
- Quadratic types and the dynamic Euler number of lines on a quintic threefold, published in 2022 in Advances in Mathematics. https://doi.org/10.1016/j.aim.2022.108508 arXiv
- Computing A^1-Euler numbers with Macaulay2, published in Res. Math. Sci., 2023, arXiv
- Applications to A^1-enumerative geometry of the A^1-degree, with Kirsten Wickelgren,
Res Math Sci 8, 24 (2021). https://doi.org/10.1007/s40687-021-00255-6, arXiv
- A^1-contractibility of affine modifications, with Adrien Dubouloz and Paul Arne Østvær, published 2019 in International Journal of Mathematics. https://doi.org/10.1142/S0129167X19500691, arXiv
Preprints
Code
Sage code for the A^1-degree using the Bézoutian by Thomas Brazelton, Stephen McKean and me
Teaching
Here is an overview of the courses I was lecturing as a postdoc in Essen, substitute professor in Düsseldorf and junior professor in Darmstadt.
- Algebraic Topology II, summer 2024, TU Darmstadt
- Algebraic Topology I, winter 2023/24, TU Darmstadt
- Algebra, summer 2023, HHU Düsseldorf
- Topologie, winter 2021/22, Universität Duisburg-Essen
- Topologie, winter 2020/21, Universität Duisburg-Essen
Other writing and notes
I gave a mini-workshop at PCMI 2024 titled
Motivic explorations in enumerative geometry in which I explain how to do enumerative geometry over an arbitrary field and how to use tropical geometry to solve questions in this direction.
Here are my
lecture notes for the mini-workshop.
Here are some notes for the talks I gave at the Motives and Research Seminar in Essen.
- Notes for my talk in the Motives Seminar on Virtual fundamental classes in motivic homotopy theory
- Notes for my talk in the Motives Seminar on Caporaso-Harris recursion formula for place curves of positive genus
- Notes for my talk in the Motives Seminar on Caporaso-Harris formula for refined invariants
- Notes for my talk in the Motives Seminar on Milnor Witt sheaves and Chow Witt groups
- Notes for my talk in the Research Seminar titled Back to Irrationality Proofs
- Notes for my talk in the Motives Seminar titled Individual Power Operations: Formulas on the motivic Steenrod operations
Ph.D. Students
Talks
- Parametrized equivariant cohomology of the classifying space of C_2-line bundles, Seminar talk at the University of Bonn, 4.6.2024
- Quadratically enriched correspondence theorems, Seminar talk at ETH, 24.5.2024
- Quadratically enriched Gromov-Witten invariants and how to compute them, Workshop in Oberwolfach on Real and Logarithmic Enumerative Geometry, 06.05.2024
- A tropical Bézout theorem for curves over different base fields, Seminar talk in Stavanger, 12.04.2024
- The Segre type of a line on a degree 2n-1 hypersurface in P^{n+1}, Conference: Motives in Mainz, 20.03.2024
- Counting curves over different fields using tropical geometry, EDGE seminar, Edinburgh, 06.03.2024
- Kurvenzählen leicht gemacht: Wie man mit tropischer Geometrie algebraische Kurven zählt, Antrittsvorlesung, TU Darmstadt, 10.01.2024
- Counting curves over an arbitrary field using tropical geometry, Seminar at the University in Stavanger, 10.8.2023
- Counting curves over an arbitrary field using tropical geometry, Motivic and non-commutative aspects of enumerative geometry, 3.7.2023, video
- Tropical Methods in Refined Enumerative Geometry, Algebraic Geometry Seminar Düsseldorf, 19.06.2023
- Counting curves over an arbitrary field using tropical geometry, Conference talk at "HODGE THEORY, MIRROR SYMMETRY, AND PHYSICS OF CALABI-YAU MODULI" in Heidelberg, 15.6.2023, video
- A quadratically enriched correspondence theorem, Tropical methods in geometry in Oberwolfach, 16.5.2023, report
- A quadratically refined tropical Bézout theorem, Seminar Talk in Heidelberg, 12.05.2023
- Tropical methods in refined enumerative geometry, Seminar Talk in Wuppertal, 19.04.2023
- Counting curves over an arbitrary field using tropical geometry, Belgian-Dutch Algebraic Geometry Seminar in Nijmegen, 24.03.2023
- Counting curves in GW(k) using tropical geometry, Algebraic Geometry Seminar at Duke University, 03.03.2023
- A quadratically enriched tropical Bézout theorem, Algebra Seminar Tromsø, 23.02.2023
- A quadratically enriched correspondence theorem, Algebra Seminar Oslo, 23.02.2023
- Quadratically refined enumerative geometry, COW meets EAGLE in Birmingham, 02.02.2023, video
- Tropical methods in A1-enumerative geometry, Seminar Talk at the Universität Tübingen, 25.01.2023, notes
- A quadratically enriched Bézout theorem for tropical curves, Seminar Talk at the Universität Bielefeld, 18.01.2023, slides
- Quadratically enriched tropical intersections, Seminar talk at the Goethe Universität Frankfurt, 25.11.2022
- An introduction to A^1-enumerative geometry, Seminar talk at the Heinrich-Heine-Universität Düsseldorf, 28.10.2022
- Arithmetic enrichments in enumerative geometry, Conference on Arithmetic Algebraic Geometry, Darmstadt, 05.10.2022
- Arithmetic enrichments of classical results in enumerative geometry, Women in Arithmetic Geometry, 27.09.2022, notes
- A quadratically enriched tropical Bézout theorem, Motivic Geometry conference in Oslo, 10.08.2022, notes
- The Bézoutian and the A^1-degree, Seminar on AG and ramification at Peking university (online), 25.07.2022, notes
- Bott’s residue formula in A^1-enumerative geometry, K-theory, algebraic cycles and motivic homotopy theory at the Isaac Newton Institute in Cambridge, 04.07.2022, video, notes
- Tropical Methods in A^1-enumerative geometry, algebraic topology seminar at UCLA (online), 27.06.2022, notes
- Enriched tropical interesections, 5.4. and 12.4.2022, joint talk with Andrés Jaramillo Puentes at the Motives seminar in Essen, notes and video
- Quadratic counts of rational curves on a quintic threefold at the University of Pennsylvania Homotopy Theory Seminar (virtual), 25.02.2022, notes
- Quadratic enrichments of enumerative counts using Atiyah-Bott localization, AG Seminar at the Universität Regensburg, 25.11.2021
- Counting twisted cubics over an arbitrary field, Algebra Seminar at the University of Oslo, 30.09.2021, notes
- Atiyah-Bott localization in equivariant Witt cohomology and the count of twisted cubics on a quintic threefold in W(k), Summer School and Conference on The Six-Functor Formalism and Motivic Homotopy Theory, 20.09.2021
- An introduction to A^1-enumerative geometry, Branching from number theory: p-adics in the sciences, 30.08.2021, notes
- A^1-contractible varieties, Seminar on Motivic Homotopy Theory, 24.06.2021, notes
- Formulas for the A^1-degree, Seminar on A^1-topology, motives and K-theory, 25.03.2021, video, notes
- Enumerative geometry via the A^1-degree, 2020 Cascade Topology Seminar (Online), 26.09.2020, video
- Types of lines and dynamic Euler numbers enriched in GW(k), Motivic Geometry, 09.09.2020, video, notes
- Quadratic dynamic and excess intersection, Motives and what not (online conference), 29.07.2020, video, notes, example, example
- Types of lines and Euler Numbers enriched in GW(k), Geometry seminar at Duke University, 09.03.2020, notes
- Types of Lines and Euler Numbers enriched in GW(k), Workshop in Real Enumerative Geometry and Beyond, Vanderbilt University, 06.03.2020, notes
- A^1-enumerative geometry via the A^1-degree, University of Oslo, Motivic Hopf equations seminar, 12.02.2020, notes
- Lines on a Quintic Threefold, University of Southern California, 18.11.2019
- Lines on a Quintic Threefold, University of South Carolina, Algebra, Geometry, and Number Theory Seminar, 08.11.2019
- Counting Lines on Hypersurfaces in P^n over different fields, Young Topologists Meeting 2019, 22.07.2019
- A^1-homotopy theory and affine algebraic geometry, 1st Joint Meeting Brazil-France in Mathematics, 19.07.2019
- Types of Lines on Quintic Threefold and Beyond, PIMS Workshop on Arithmetic Topology, 12.06.2019
- A^1-contractible varieties, Affine Algebraic Geometry and Transformation Groups,
In honor of Lucy Moser-Jauslin's 60th Birthday, 28.05.2019
- A^1-contractible varieties, Women in Homotopy Theory and Algebraic Geometry II, 20.03.2019
- A^1-contractible varieties, NTNU Trondheim, 08.10.2018
- A^1-contractibility of Koras-Russell like varieties, Algebraic Geometry - Mariusz Koras in memoriam, 31.05.2018, video