Simon Felten

Simon Felten

I am a post-doc currently based at Columbia University, New York. Most of my work is in logarithmic deformation theory and applications to mirror symmetry.

The degeneration of the smooth quartic surface

The picture shows the union of four copies of the projective plane P² in projective space P³ arranged like a tetrahedron. Indeed, they are the four coordinate planes, so their union is defined by the homogeneous equation xyzw = 0 in P³. Let us denote this union by D. We obtain the four planes D as a degeneration of the smooth quartic surface E in P³ which is defined by the equation x⁴ + y⁴ + z⁴ + w⁴ = 0 in P³. This means we have a total space where the coordinates of P¹ x P³ are ([t:s],[x:y:z:w]), and we have a map f: X→P¹ given by the first projection. The degenerate surface E is the fiber f^-1(0) whereas the smooth surface E is the fiber f^-1(∞). It is deformed via the intermediate fibers f^-1(t) into D as t→0. The 24 marked points in the picture are the singularities of the total space X. Indeed, this variety is (in a neighborhood of D) smooth everywhere except in these points. When we consider f via f: (X,D)→(P¹,0) as a morphism of pairs (or more fancily of logarithmic schemes), these 24 points become the locus where f is not logarithmically smooth. Treating such singularities of logarithmic maps is the starting point for the theory of log toroidal families which I developed in my PhD thesis. This theory is a partial generalization of toric degenerations as studied thoroughly by Gross and Siebert and others. You can find more information about the example in the introduction of my Master Thesis.

Education

PhD studies

Mathematics, Johannes-Gutenberg-Universität Mainz
January 2017 - January 2021

Title: Log toroidal families
Advisor: Helge Ruddat

Master Studies

Mathematics with minor subject Theoretical Physics, Johannes-Gutenberg-Universität Mainz
October 2015 - July 2018

Degree: Master of Science
Thesis: Good differential forms for a family of schemes
Advisor: Helge Ruddat

Bachelor Studies

Mathematics with minor subject Theoretical Physics, Johannes-Gutenberg-Universität Mainz
April 2013 - January 2016

Degree: Bachelor of Science
Thesis: Normal structures, cohomology with supports, and logarithmic differential forms
Advisor: Helge Ruddat

Mathematics, Technische Universität Kaiserslautern
October 2011 - March 2012

Early Education

Secondary School, Gymnasium Saarburg
2005 - 2013

Primary School, Grundschule Irsch
2001 - 2005

Irsch is a quiet village at the Saar river near the German-Luxemburg border.

Awards and Scholarships

PhD Scholarship of the German National Academic Foundation
(Promotionsstipendium der Studienstiftung des deutschen Volkes)
April 2017 - March 2020

PhD Scholarship of the Carl-Zeiss Foundation
(Promotionsstipendium der Carl-Zeiss-Stiftung)
January 2017 - March 2020

Award for Excellent Master Theses of JGU Mainz
(Preis des Gutenberg-Lehrkollegs für herausragende Abschlussarbeiten)
February 2019

Leadership Team Award of JGU Mainz
January 2018

Scholarship of the German National Academic Foundation
(Stipendium der Studienstiftung des deutschen Volkes)
April 2013 - December 2016

Cowinner of the German National Mathematics Competition
(Bundessieger des Bundeswettbewerbs Mathematik)
January 2013

Thomas-Morus Award for excellent achievement in Latin and Greek
(Thomas-Morus-Preis für herausragende Leistungen in Latein und Griechisch)
2011

Publications / Preprints

You can find my articles also on the arXiv and on Google Scholar .

• Global logarithmic deformation theory

  Preprint

• Deformations of log Calabi-Yau pairs can be obstructed (with Andrea Petracci and Sharon Robins)

  Preprint

• Log toroidal families

  PhD thesis available at Gutenberg Open Science

• The logarithmic Bogomolov-Tian-Todorov theorem (with Andrea Petracci)

  Bull. Lond. Math. Soc. 54 (2022), no. 3, 1051 - 1066.

• Log smooth deformation theory via Gerstenhaber algebras

  manuscripta math. (2020)

• Smoothing toroidal crossing spaces (with Matej Filip, Helge Ruddat)

  Forum of Mathematics, Pi (2021) Vol. 9, e7

• Good differential forms for a family of schemes

  Master Thesis

• A Diophantine equation for sums of consecutive like powers (with Stefan Müller-Stach)

  Elem. Math. 70 (2015), no. 3, 117–124

Posters

• Good differential forms for a singular family of schemes

  This is a poster from 2018 explaining my master thesis.

• Almost log smooth families in the Gross-Siebert program and beyond

  This is a poster from 2017 explaining my PhD project in an early stage.

Others

• Aus dem Leben eines Mathematikers

  This is a Science Slam Talk in German. I describe the life of a PhD student in Mathematics.

• Wie man neue Begriffe einführt ...

  This article in German is directed at a general scientific audience and explains my PhD project. It has been written in 2017 when I just begun my PhD.

Miscellanea

• I did my PhD in Helge Ruddat's Emmy Noether Research Group, based at JGU Mainz.
• This is my profile on MathSciNet.