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
X = {sxyzw - t(x4 + y4 + z4 + w4) = 0} in P1 x P3
where the coordinates of P1 x P3 are ([t:s],[x:y:z:w]), and we have a map f: XP1 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 neighbourhood 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-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 .

Posters

Others

Miscellanea

Contact

Simon Felten
Department of Mathematics
Columbia University
2990 Broadway
New York, NY 10027
United States

felten.math@posteo.net