The Young PDEs Mini Symposium will take place on April 29 at Boltzmanngasse 9A, in the Schrödinger Lecture Hall of the Erwin Schrödinger International Institute for Mathematics and Physics. It aims to bring together PhD students and postdoctoral researchers in the field of partial differential equations. There will be six 20 minute presentations in English each followed by an equally long collegial discussion. Snacks will be provided.
The Young PDEs Mini Symposium is organized by Thomas Körber with assistance from Michael Eichmair. Among the Young PDE honoris causa, there will be Wilhelm Schlag from Yale University and Roland Donninger, David Fajman, and Michael Eichmair from the University of Vienna.
Program
8:30-9:10 Liam Urban
Nonlinear stability of Big Bang formation for FLRW solutions with hyperbolic spatial geometry within the Einstein scalar field system
The Strong Cosmic Censorship conjecture, in the context of cosmological spacetimes, posits that, for generic initial data, the Kretschmann curvature scalar blows up where causal geodesics become incomplete. In my talk, I will outline an approach (based on joint work with D. Fajman) to verify this conjecture in presence of Einstein scalar field matter for initial data close to that of Friedman-Lemaître-Robertson-Walker (FLRW) spacetimes with hyperbolic spatial geometry. This takes the form of showing the past nonlinear stability of the blow-up exhibited by such FLRW solutions (the Big Bang formation) within a system of hyperbolic and elliptic geometric PDEs. Crucially and in contrast to previous results, we actually do not rely on the specific spatial geometry to a very large degree, potentially making our approach viable for other, more involved Einstein systems.
9:10-9:50 Matthias Ostermann
Stable blowup in the whole space for geometric wave equations
A universal phenomenon among partial differential equations of wave-type is the spontaneous breakdown of their solutions. This non-linear effect may be encountered in evolution equations via blowup formation from smooth initial data in finite time. Such blowup profiles are known in closed form for, e.g., the co-rotational Wave Maps Equation and the equivariant Yang-Mills Equation in supercritical space dimensions. The central question for their role in the dynamics of the equations concerns their stability, i.e., if this blowup also arises for all “near” initial data. Until recently, stability results have only been available in the region of the backwards lightcone of the singularity. The first global stability result for co-rotational wave maps from (1 + 3)-dimensional Minkowski spacetime into the three-sphere was proved by P. Biernat, R. Donninger and B. Schörkhuber. One of the key insights is the construction of hyperboloidal similarity coordinates. In my joint work with R. Donninger, we have now expanded the methods to establish a new stability theory for blowup in the Yang-Mills equation in the whole space. This result holds in all odd space dimensions. A better understanding of the geometric and analytic interplay of hyperboloidal similarity coordinates with different equations of wave-type will be relevant for further progress and the problem whether a well-defined continuation of the Cauchy evolution after the singularity is possible.
9:50-10:30 David Wallauch
Strichartz estimates and blowup stability for energy critical nonlinear wave equations
Energy critical nonlinear wave equations exhibit many fascinating phenomena, among them finite time blowup of solutions which start from smooth and compactly supported initial data. Whenever one has a solution which exhibits finite time blowup, a natural question that arises is the stability of the singular behavior under perturbations of the initial data. In this talk, I will present the derivation of Strichartz estimates for radial wave equations with a potential in similiarity coordinates. These estimates are then used to establish the asymptotic blowup stability of the ODE blowup profile of the energy critical wave equation with a power nonlinearity under radial perturbations in dimensions 3≤ d≤ 6 at the lowest possible regularity.
10:30-11:10 Maciej Maliborski
Characteristic approach to the soliton resolution
I will present a toy model for studying the soliton resolution phenomenon. The soliton resolution conjecture states that global-in-time generic solutions of nonlinear dispersive wave equations resolve for late times into a superposition of decoupled nonlinear bound states (solitons) and radiation. Our main objective is to illustrate the advantages of employing outgoing null (or asymptotically null) foliations in analyzing the relaxation processes due to the dispersal of energy by radiation. Based on results from joint work with Piotr Bizoń and Bradley Cownden.
11:10-11:50 Thomas Körber
Uniqueness of large stable constant mean curvature spheres in asymptotically flat 3-manifolds
Stable constant mean curvature spheres encode important information on the asymptotic geometry of initial data sets for isolated gravitational systems. By the work of C. Nerz, the asymptotic region of such an initial data set is foliated by large stable constant mean curvature spheres. In this talk, I will present a short proof (joint with M. Eichmair) that, in the case where the initial data set has non-negative scalar curvature, the leaves of this foliation are the only large stable constant mean curvature spheres that enclose the center of the initial data set.
11:50 - 12:30 Irfan Glogić
Stable finite-time aggregation for the supercritical Keller-Segel model
We consider a parabolic-elliptic system which arises in modeling the bacterial growth or stellar dynamics. This model, which goes under the name of Keller-Segel, is known to admit a closed form self-similar blowup solution in all mass-supercritical dimensions. In this talk, I will report on my joint work in progress with Birgit Schörkhuber, in which we establish global nonlinear stability of the aforementioned blowup profile. This solves a more than two decades old open problem.
Young PDE speakers
Irfan Glogić (he/him)
- I am currently a postdoctoral researcher at the Faculty of Mathematics of the University of Vienna. As of March 2021, I have been PI for a stand-alone research project P34378 of the Austrian Science Fund (FWF). Before that, I was funded through the FWF project P30076 of Roland Donninger. Before coming to Vienna, I completed my doctoral studies at The Ohio State University in 2018, under the supervision of Ovidiu Costin. Before that, I completed my undergraduate studies at the University of Sarajevo.
- ORCID: 0000-0002-9382-9293
- Email: irfan.glogic@univie.ac.at
- Homepage: https://homepage.univie.ac.at/irfan.glogic/
Thomas Körber (he/him)
- I obtained my PhD at the University of Freiburg in 2020 under the supervision of Guofang Wang. During my studies, I spent time at the University of Melbourne and at University College London. Since June 2020, I have been a postdoctoral researcher at the University of Vienna in the START-Project Y963 of Michael Eichmair. Starting in May 2022, I will be PI of the Lise-Meitner project M3184 of the Austrian Science Fund (FWF) with Michael Eichmair as co-PI.
- ORCID: 0000-0003-1676-0824
- Email: thomas.koerber@univie.ac.at
- Homepage: https://mat.univie.ac.at/~koerber/
Maciej Maliborski (he/him)
- I am a postdoc at the Faculty of Mathematics of the University of Vienna. Previously I was a postdoc at the Physics Faculty of the University of Vienna. Before moving to Vienna, I was a Humboldt Research Fellow at the Albert Einstein Institute in Potsdam. I obtained my PhD in 2015 at the Jagiellonian University.
- ORCID: 0000-0002-8621-9761
- Email: maciej.maliborski@univie.ac.at
- Homepage: https://homepage.univie.ac.at/maciej.maliborski/
Matthias Ostermann (he/him)
- I am a PhD student at the Faculty of Mathematics of the University of Vienna working under the supervision of Roland Donninger. In my PhD project, I am exploring new methods for the study of global stability of blowup in non-linear wave equations such as wave maps and Yang-Mills models. Prior to this, I completed the Master of Science program in mathematics in 2020 and the Bachelor of Science programs in both mathematics and physics in 2019, all at the University of Vienna. I am also an alumni of the Vienna Master Class Mathematical Physics. Since January 2021, I have been appointed as a PraeDoc supported by the Vienna School of Mathematics.
- Email: matthias.ostermann@univie.ac.at
- Homepage: https://www.mat.univie.ac.at/~ostermann/
- Member of the Vienna School of Mathematics
Liam Urban (he/they)
- Since October 2021, I have been a doctoral student at the University of Vienna, supervised by David Fajman and Michael Eichmair. My doctoral research is funded by the START-Project Y963 and supported by the German Academic Scholarship Foundation. As an undergraduate, I studied and worked at the FAU Erlangen-Nürnberg before switching to the University of Vienna for my Master's.
- ORCID: 0000-0001-9185-9627
- Email: liam.urban@univie.ac.at
- Homepage: https://www.mat.univie.ac.at/~lurban/index.html
- Member of the Vienna School of Mathematics
David Wallauch (he/him)
- I am currently a PhD student at the University of Vienna under the supervision of Roland Donninger, supported by the project P30076 of the Austrian Science Fund (FWF). I obtained my bachelor degree (2018) and my master degree (2019) in mathematics at the University of Vienna. My research interests lie in the study of nonlinear wave equations and breakdown of solutions to such equations. More precisely, I am interested in the stability of finite time blowups under small perturbations.
- Email: david.wallauch@univie.ac.at
- Homepage: https://homepage.univie.ac.at/david.wallauch/
- Member of the Vienna School of Mathematics
Young PDE honoris causa
Roland Donninger (he/him)
- I am professor for analysis of partial differential equations at the Faculty of Mathematics of the University of Vienna. Previously, I held positions at the University of Bonn, at EPF Lausanne, and at the University of Chicago. I obtained my PhD in 2007 at the University of Vienna.
- ORCID: 0000-0002-4522-648X
- Email: roland.donninger@univie.ac.at
- Homepage: https://homepage.univie.ac.at/roland.donninger/
David Fajman (he/him)
- I am an assistant professor at the faculty of physics of the University of Vienna in the group of Gravitational Physics. I received my Diploma degrees in math and physics from Humboldt University of Berlin and did my PhD studies at the Max Planck Institute for Gravitational Physics - the Albert Einstein Institute - in Potsdam. After that I held a Postdoc position at the University of Vienna and was later the PI of an Austrian Science Fund project before taking up the assistant professorship.
- ORCID: 0000-0003-3034-6232
- Email: david.fajman@univie.ac.at
- Homepage: https://homepage.univie.ac.at/david.fajman/assets/player/KeynoteDHTMLPlayer.html#0
Michael Eichmair (he/him)
- As of March 2015, I have been Chair of Global Analysis and Differential Geometry at the Faculty of Mathematics of the University of Vienna. I obtained my PhD from Stanford University under the supervision of Richard Schoen in 2008. Then, I was a CLE Moore Instructor at the Massachusetts Institute of Technology and an Assistant Professor at the Swiss Federal Institute of Technology in Zurich. My research group in Vienna is supported by the START-Project Y963 of the Austrian Science Fund. I also lead the outreach project MmF at the University of Vienna. The current focus of MmF is on supporting Ukrainian students and teachers in Austria.
- ORCID: 0000-0001-7993-9536
- Email: michael.eichmair@univie.ac.at
- Homepage: https://mmf.univie.ac.at/team/michael-eichmair/
Wilhelm Schlag (he/him)
- I am a Phillips Professor of Mathematics at Yale University and PI of the NSF grant DMS-2054841. Previously, I held positions at The Institute for Advanced Study, at Princeton University, at the California Institute of Technology, and at the University of Chicago. I obtained my PhD in 1996 at the California Institute of Technology under the supervision of Thomas Wolff.
- ORCID: 0000-0002-6418-1715
- Email: wilhelm.schlag@yale.edu
- Homepage: https://gauss.math.yale.edu/~ws442/
Contact
Please contact Thomas Körber (he/him) with any questions related to the Young PDEs Mini Symposium.
