Personal Website - Anton Ullrich

Email
aullrich(at)andrew.cmu.edu
Location

Pittsburgh, Pennsylvania, USA

CMU, Wean Hall 7206
Arxiv
https://arxiv.org/a/ullrich_a_1.html
Orcid
0000-0003-4239-7061

About me

Since September 2025, I am a Walter Noll Postdoctoral Research Fellow at Carnegie Mellon University.

Previously, I was a doctoral student (PhD) at the Max Planck Institut for Mathematics in the Sciences in Leipzig, Germany, under the supervision of Prof. Felix Otto.

My research interests are focussed on parabolic differential equation, especially mean curvature flow and the heat flow on general spaces. Furthermore, I am also interested at the intersection of analysis with geometry, numerics, probability theory and graph theory.

Education

Curriculum Vitae

Link to the CV.

Education

Carnegie Mellon University, Pittsburgh, PA, United States
Sep 2025 — ongoing
Walter Noll Research Fellow, Postdoc mathematics
Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany
2022 — 2025
Ph.D. mathematics
PhD thesis: "Heat Flow, Mean Curvature Flow and Their Related Schemes",
advisor: Prof. Felix Otto
University of Bonn, Germany
2020 — 2022
Master-studies mathematics (grade 1.0, excellent)
Master's thesis: "Thresholding for generic mean curvature flow",
advisor: Prof. Tim Laux
University of Bonn, Germany
2017 — 2020
Bachelor-studies mathematics (grade 1.0, excellent)
Bachelor's thesis: "Starke isoperimetrische Ungleichung" (Sharp isoperimetric inequality),
advisor: Prof. Herbert Koch
awarded bachelor prize of the BMG (Bonn Mathematical Society)
Georg-Cantor-Gymnasium, Halle, Germany
2009 — 2017
Graduation: Abitur (grade 1.0, excellent)

Research interests

Geometric flows
Calculus of variations
Parabolic partial differential equations

Publications and paper

Peer-Reviewed

"Generic level sets in mean curvature flow are BV solutions",
2024
with Tim Laux, J. Geom. Anal. 34, 375 (2024). https://doi.org/10.1007/s12220-024-01819-y

Preprints

"Median filter method for mean curvature flow using a random Jacobi algorithm",
2024
with Tim Laux, arXiv preprint arXiv:2410.07776
"The heat flow on glued spaces with varying dimension",
2024
arXiv preprint arXiv:2406.09996

Research stays

University Heidelberg, Germany
Jul 2025
visiting Tim Laux
University Regensburg, Germany
Apr 2024
visiting Tim Laux
University Heidelberg, Germany
Apr 2023
visiting Tim Laux

Contributed Talks & Poster presentation

"Analysis AG seminar", Heidelberg, Germany
Jul 2025
Summer School "Geometric Analysis and PDEs", Poster, Chęciny, Poland
Jul 2025
"Dresden-Leipzig AG seminar day", Leipzig, Germany
Oct 2024
Conference "Free Boundary Problems: Theory and Application" - "Interfaces and Free Boundaries in Machine Learning", João Pessoa City, Brazil
Aug 2024
"Regiomontanus Seminar", Leipzig, Germany
Oct 2023
Summer School "Geometric Analysis and PDEs", Poster, Chęciny, Poland
Jul 2023
"Scientific retreat of the International Max Planck Research School (IMPRS)", Wittenberg, Germany
Jul 2023
"Scientific retreat of the IMPRS", Trebsen, Germany
Nov 2022

Organization

Organization FLINTAQ*-lunches
Oct 2024 - Aug 2025
Organization of Conference "IMPRS Combo III: Cosmology and Quantum Information Theory", Leipzig
Sep 2024
intended for all Ph.D. candidates in mathematics of the IMPRS
new concepts: ethics lectures combined with a discussion
Ph.D. representative for the Max Planck Institute for Mathematics in the Sciences
Oct 2023 - Apr 2025

Teaching experience

Supervision of a high-school student for the creation of a scientific work
Oct 2024 - Dec 2025
Teaching assistant for
2018 - 2021
"Linear algebra 1", "Linear algebra 2", "Analysis 3", "Introduction to robotics", "Introduction to partial differential equations" and "Functional analysis and PDE"

Special accomplishments

Studienstiftung des deutschen Volkes, Scholarship of the German Academic Scholarship Foundation
2020 - 2022
awarded to approximately 0.25% of the German university student population
Besondere Lernleistung (special academic accomplishment, a voluntary research project)
"2- und 3-dimensionale Gleichdicke" (curves and surfaces of constant width)"
2015 - 2017
Participation in olympiads in mathematics, physics, chemistry, computer science and latin, for instance:
2007 - 2017
International physics-olympiad, 3rd round
International chemistry-olympiad, 4th round
National computer science competition, 2nd round
Certamen Franckianum, 2nd round
National mathematics competition, 2nd round
Mathematics Olympiad, up to the selection exams for the International Mathematics Olympics
Scientific-practical-work
"Erstellung virtueller Welten mit Fraktalen" (Creation of virtual worlds using fractals)"
2014

Attended Conferences and Workshops

"Dresden-Leipzig AG seminar day", Leipzig, Germany
Oct 2024
Conference “Free Boundary Problems: Theory and Application”, João Pessoa City, Brazil
Aug 2024
"IMPRS Combo II: Numerics in Algebra & Analysis", Leipzig, Germany
May 2024
"n-Cities Seminar", Leipzig, Germany
May 2024
Workshop on “Calculus of Variations and Partial Differential Equations”, Regensburg, Germany
Apr 2024
"Scientific retreat of the IMPRS", Halle, Germany
Mar 2024
"Applications of Optimal Transportation", Oberwolfach, Germany
Feb 2024
"Workshop on Geometry and Machine Learning", Leipzig, Germany
Nov 2023
"IMPRS Combo I", Leipzig, Germany
Nov 2023
"Unfinished Business: Problems in Applied Geometric Measure Theory, Many-particle Systems, Epidemiology, and Phase Transitions", Bingen, Germany
Jul 2023
"Scientific retreat of the IMPRS", Wittenberg, Germany
Jul 2023
"Summer School on Geometric Analysis and PDEs", Chęciny, Poland
Jul 2023
"Calculus of Variations and Geometric Measure Theory", Pisa, Italy
Jun 2023
MPI MiS Summer School on "PDEs and Randomness", Leipzig, Germany
May 2023
Spring School "Recent trends in the mathematics of complex materials", Bonn, Germany
Mar 2023
"Théorie de la mesure géométrique et calcul des variations", Nancy, France
Dec 2022
"Scientific retreat of the IMPRS", Trebsen, Germany
Nov 2022
Hausdorff School: "Geometric Analysis", Bonn, Germany
Aug 2022
"Workshop on New Trends in Geometric PDEs", Münster, Germany
Nov 2021
Workshop on "Geometric and Applied Analysis", Hausdorff Center for Mathematics, Bonn, Germany
Aug 2021

Special knowledge

Languages

German: native
English: C1.1
Latin: advanced Latin certificate
French: A1.1

Programming languages

C: course work
C♯: private projects, game development
Python: private projects, course work, research projects
Matlab: course work, research projects
Rust: research projects
js, css, html: website design

Hobbies and interests

Literature
Programming & making games
Painting
Teaching for olympiad preparation
Board games

Association membership

eLeMeNTe e.V. (math association to organize olympiads and math camps)
QED e.V. (mathematics organization centered in Bavaria)
CDE e.V. (organizes camps and events for people who attended "Deutsche Schüler-Akademie")

Research

In this section, I present some of my research projects and directions together with their applications.
All the image and videos which are not explicitely specified are created by myself. I used matlab, python, rust and unity (C♯ and HLSL).

Diffusion of Heat - Heat Flow

A solution of the heat equation, the L2-gradient flow (GF) of the Dirichlet energy. It shows how heat distributes in an area.
Soap Bubbles - Volume preserving Mean Curvature Flow (VMCF)

Soap bubbles minimize their surface area under the constrained that they contain a fixed volume of air. This is modelled by mean curvature flow which is formally the gradient flow of the surface area / surface tension. In this video, the simulation is done by a median filter scheme together with a ray marcher and implementation of the eikonal equation and heat flow.
Fluid Phase Separation - Allen-Cahn Equation

Here, one can see the densities of two fluids (red and blue) which begin mixed but separate out over time. This is modelled by the Allen-Cahn equation which is also the L2-GF of the free Ginzburgh-Landau energy.
Hybrid Zones in Mice Genes - Allen-Cahn and SLFVPS

The picture is from doi:10.1038/hdy.1973.56 . It shows the hybrid zones in the alleles of mice in denmark. The formation of these can be modelled by the Allen-Cahn equation and simulated for instance Spatial Lambda Fleming Viot Processes with Selection (SLFVPS)
Metal Properties - Multiphase MCF

The local structure of metals dictates their material properties. For instance, if the metal has a lot of grains (grain boundaries are local dislocations in the orientation of the atoms), it will be very brittle. On the other hand, materials with few grains are very ductile. This is for instance used in cars, where the crumble zone gives in to pressure while the sides require more force and will break instead of bend. The experimental data is from link.
Water Droplets - VMCF Young Angles

Water droplets on different surfaces have varying contact angles depending on whether the surface is hydrophilic or hydrophobic. For instance, on a lotus paddle, the contact angle is very high while on glass it is shallow. This can be modelled with Young angles and volume preserving mean curvature flow.
Visualization MCF

The video shows a visualization of mean curvature flow in three dimensions with a two dimensional surface. Even though the initial data is smooth, singularities will develop in finite time.
Image Processing - Median filter

A median filter can be used to remove noise from an image without changing the relevant features and edges. The original cat image is under a CC0 license.
Soap Film Wire - MCF Dirichlet Data

If one lifts a metal wire out of a bath of soap, the remaining film will become a minimal surface. As these are the stationary solutions to MCF, one can model this behaviour with MCF with Dirichlet boundary conditions.
Unsupervised Learning MNIST Classification - Glued Manifolds, Multiphase MCF on Manifolds

In unsupervised learning, the objective is to learn the structure of data without labeling. Here, one has 60.000 hand-drawn digits as 28x28 greyscale images. The task is a classification task, i.e., one wants to have a program that gives each image the right label. For the modelling of the data one can use the union of manifold hypothesis and model it as a glued manifold. The algorithm can be created by a (potentially volume preserving) multiphase mean curvature algorithm to minimize the area between the sets.
Two-Balloon Experiment - VMCF

In this experiment, one has two connected balloons, one small, one large. Now the pressure inside these will average out and it turns out that the small balloon will shrink while the larger one grows. This can also be modelled and observed with volume preserving mean curvature flow.
Visualization of the MBO Scheme

The MBO or Thresholding scheme is a very efficient scheme to simulate mean curvature flow. It was developed by Merriman-Bence-Osher and uses the reaction diffusion character of mean curvature flow. On the euclidean space this scheme can be implemented using the discrete fast Fourier transform.