Disputation: Sabrina Pauli

Doctoral candidate Sabrina Pauli at the Department of Mathematic, Faculty of Mathematics and Natural Sciences, is  defending the thesis Computations in A1-homotopy theory - Contractibility and enumerative geometry for the degree of Philosophiae Doctor.

Picture of the candidate.

Doctoral candidate Sabrina Pauli

The University of Oslo is partially closed. The PhD defence and trial lecture will therefore be fully digital and streamed directly using Zoom. The host of the session will moderate the technicalities while the chair of the defence will moderate the disputation.

Ex auditorio questions: the chair of the defence will invite the audience to ask ex auditorio questions either written or oral. This can be requested by clicking 'Participants -> Raise hand'. 

Trial lecture

16th September, 10:15, Zoom

"The Milnor Conjecture"

  • Join the trial lecture
    The meeting opens for participation just before 10:15 (a.m.) and closes for new participants approximately 15 minutes after the trial lecture has begun.


Main research findings

 

The zero set of a degree 3 polynomial in 3 variables is a surface. Already in 1849 Cayley and Salmon 
showed that on such a surface there are in general 27 lines. In algebraic geometry one studies those geometric spaces that are the zero sets of polynomials in several variables, called algebraic varieties.

Homotopy theory studies geometric spaces up to continuous deformations called homotopies. For example a disc can be deformed to a point. One calls such shapes that can be deformed to a point, contractible. An annulus is not contractible because it has a hole. To a homotopy theorist it looks like a circle. 

My thesis work is in the area of A1-homotopy theory which combines the two fields, algebraic geometry and homotopy theory. More precisely, in A1-homotopy theory we study algebraic varieties up to deformations. This yields new ways to study geometric questions as the following example shows.

The count of 27 lines is only true when one uses the complex numbers. There could for example only be 3 lines when one works with the real numbers, while to find the other 24, one needs the imaginary numbers. Techniques from A1-homotopy theory yield a new way to refine such geometric counts and get information over fields other than the complex numbers, for example the real or rational numbers.

In my thesis work, I provide several refinements of classical geometric counts, in particular the count of lines on a quintic threefold for which I introduce a refined version of dynamic intersection theory. Furthermore, I provide new methods to construct examples of A1-contractible varieties, that is, algebraic varieties that can be deformed to a single point.

Published Sep. 3, 2020 11:07 AM - Last modified Sep. 15, 2020 3:41 PM