Stable hypersurfaces with prescribed mean curvature
Lead Research Organisation:
University College London
Department Name: Mathematics
Abstract
The area of a surface governs many physical phenomena. Nature tends to optimise shapes by finding equilibrium positions dictated by a minimality property- roughly speaking, it prefers to use as little area as possible. Well-known examples of this phenomenon are soap films. As early as the mid 19th century, the physicist Plateau conducted experiments in which he immersed a closed wire in and out of a soap solution. The resulting soap film is a minimal surface, i.e. it locally minimizes area among surfaces spanning the given wire (it avoids wasting soap). Of particular interest are configurations of ``stable'' equilibrium, i.e. under any slight perturbation the film will go back to its initial position. Similarly, in the case of soap bubbles, it is again a minimality property of area that dictates their shape (e.g. spherical bubbles), with the difference that this time the minimality is achieved under the constraint of a fixed enclosed volume (how much air the bubble contains): the surface obtained is characterized by having constant mean curvature (CMC). The mean curvature of a soap film or bubble is a geometric quantity that is proportional to the pressure difference on the sides of the film.
The optimising behaviour observed in these examples is ubiquitous in nature (for example, bees use hexagonal cells because this requires the minimal amount of wax for tiling a planar portion); the following is a further example, taken from capillarity theory, and it is very relevant to the present project.
Consider a stable equilibrium configuration for a liquid that is surrounded by air, subject to surface tension and to the action of external body forces, such as gravitational energy. By a principle of energy optimization, the equilibrium configuration is once again dictated by a partial differential equation whose geometric content is to prescribe the mean curvature of the interface (the surface that separates liquid and air). More precisely, in the absence of gravity or other external forces, the condition is that the mean curvature is constant (CMC surfaces); in the presence of a non-zero potential, for example, a gravitational one, the mean curvature is prescribed up to an additive constant by the value of the potential.
Modern geometry is not limited to surfaces in three-dimensional space and this has allowed, and will for time to come, far-reaching applications, from relativity theory and black holes to engineering. It is therefore natural to introduce hypersurfaces (a generalization to arbitrary dimensions of a surface in three-dimensional space) of dimension n that sit in an ambient space of dimension n+1. In mathematics this ambient space is a Riemannian manifold, i.e. a space with compatible notions of length and angle that permit the computation of area, volume, etc.
In this project I study stable hypersurfaces whose mean curvature is prescribed by a given function on the ambient Riemannian manifold (special cases of which include minimal and constant-mean-curvature hypersurfaces). The project aims to address the fundamental geometric question of existence of closed hypersurfaces of this type in arbitrary closed Riemannian manifolds, employing an analytic framework (regularity and compactness results) that I recently developed. The successful completion of this project will be a pathway towards a more complete understanding of interfaces between liquids and air (as in the capillarity model above).
The optimising behaviour observed in these examples is ubiquitous in nature (for example, bees use hexagonal cells because this requires the minimal amount of wax for tiling a planar portion); the following is a further example, taken from capillarity theory, and it is very relevant to the present project.
Consider a stable equilibrium configuration for a liquid that is surrounded by air, subject to surface tension and to the action of external body forces, such as gravitational energy. By a principle of energy optimization, the equilibrium configuration is once again dictated by a partial differential equation whose geometric content is to prescribe the mean curvature of the interface (the surface that separates liquid and air). More precisely, in the absence of gravity or other external forces, the condition is that the mean curvature is constant (CMC surfaces); in the presence of a non-zero potential, for example, a gravitational one, the mean curvature is prescribed up to an additive constant by the value of the potential.
Modern geometry is not limited to surfaces in three-dimensional space and this has allowed, and will for time to come, far-reaching applications, from relativity theory and black holes to engineering. It is therefore natural to introduce hypersurfaces (a generalization to arbitrary dimensions of a surface in three-dimensional space) of dimension n that sit in an ambient space of dimension n+1. In mathematics this ambient space is a Riemannian manifold, i.e. a space with compatible notions of length and angle that permit the computation of area, volume, etc.
In this project I study stable hypersurfaces whose mean curvature is prescribed by a given function on the ambient Riemannian manifold (special cases of which include minimal and constant-mean-curvature hypersurfaces). The project aims to address the fundamental geometric question of existence of closed hypersurfaces of this type in arbitrary closed Riemannian manifolds, employing an analytic framework (regularity and compactness results) that I recently developed. The successful completion of this project will be a pathway towards a more complete understanding of interfaces between liquids and air (as in the capillarity model above).
Planned Impact
The project is completely self-contained, it addresses a fundamental geometric problem with an underlying and fundamental analytic component and an expected impact on applied problems. In addition to the PI, it will involve a PDRA that will be chosen and trained, thereby adding expertise in the UK in the area of geometric analysis. Moreover, part of the proposed research will be conducted in collaboration with other UK-based mathematicians. In view of these considerations, the successful completion of the proposal will foster a UK-based ``school'' with expertise in hot topics at the interface between partial differential equations and geometry, in line with the aims outlined in the 2010 International Review of Mathematical Sciences and the 2016 EPSRC Mathematical Sciences Community Overview Documents. The importance of this area of research can be stressed by recalling a few major breakthroughs and awards over the past dozen years, starting with Perelman's proof of the geometrization and Poincare conjectures (Fields medal 2006), until the more recent proofs by Marques-Neves of the 50-year-old Willmore conjecture, by Brendle--Schoen of the 1/4-pinching conjecture, and the very recent proof of the positive mass theorem in all dimensions by Schoen-Yau; the award to Nirenberg and Nash of the Abel Prize in 2016 for ``seminal contributions to the theory of nonlinear partial differential equations and its applications to geometric analysis'' testifies the importance placed on progress in the area.
From an economic and societal perspective, the creation of a "school", the knowledge produced and the techniques developed will be key attractive features for future MSci, MPhil and PhD students interested in geometric analysis to choose the UK over other world-leading institutions based abroad. At the same time, the completion of the present project will provide an occasion to train students in geometric analysis, by introducing ad hoc 4th year and graduate courses motivated by the achievements obtained; this will enhance the competitiveness of home-grown talent in view of future hirings in the UK, a direction encouraged by the 2010 International Review of Mathematical Sciences. Implicitly, by reinforcing the leading role of the UK in geometric analysis, the success of the project will make academic institutions in the UK less prone to losing their leaders to abroad institutions, in agreement with the objectives of the 2016 EPSRC Mathematical Sciences Community Overview Documents.
The exchanges between the analytic and geometric aspects in this project will provide an occasion for the team to interact with leading researchers (based in the UK and abroad) from the analysis and applied communities. The 3-day workshop proposed will provide a platform for such interactions. In addition to the dissemination component, with the presence of internationally affirmed mathematicians, the workshop will feature an outlook component: the activities will provide an occasion to identify future directions of research that are of interest to several mathematical communities and will strengthen cross-disciplinary links. At the same time, the workshop will be beneficial for the London EPSRC Centre for Doctoral Training, both as a learning opportunity for students and to enhance the identification of new and interesting research projects, including those for potential PhD theses.
From an economic and societal perspective, the creation of a "school", the knowledge produced and the techniques developed will be key attractive features for future MSci, MPhil and PhD students interested in geometric analysis to choose the UK over other world-leading institutions based abroad. At the same time, the completion of the present project will provide an occasion to train students in geometric analysis, by introducing ad hoc 4th year and graduate courses motivated by the achievements obtained; this will enhance the competitiveness of home-grown talent in view of future hirings in the UK, a direction encouraged by the 2010 International Review of Mathematical Sciences. Implicitly, by reinforcing the leading role of the UK in geometric analysis, the success of the project will make academic institutions in the UK less prone to losing their leaders to abroad institutions, in agreement with the objectives of the 2016 EPSRC Mathematical Sciences Community Overview Documents.
The exchanges between the analytic and geometric aspects in this project will provide an occasion for the team to interact with leading researchers (based in the UK and abroad) from the analysis and applied communities. The 3-day workshop proposed will provide a platform for such interactions. In addition to the dissemination component, with the presence of internationally affirmed mathematicians, the workshop will feature an outlook component: the activities will provide an occasion to identify future directions of research that are of interest to several mathematical communities and will strengthen cross-disciplinary links. At the same time, the workshop will be beneficial for the London EPSRC Centre for Doctoral Training, both as a learning opportunity for students and to enhance the identification of new and interesting research projects, including those for potential PhD theses.
People |
ORCID iD |
Costante Bellettini (Principal Investigator) |
Publications
Hiesmayr F
(2020)
A Bernstein theorem for two-valued minimal graphs in dimension four
C. Bellettini
(2021)
Existence of hypersurfaces with prescribed mean curvature
Bellettini C
(2023)
Hypersurfaces with mean curvature prescribed by an ambient function: Compactness results
in Journal of Functional Analysis
Bellettini C
(2024)
Embeddedness of min-max CMC hypersurfaces in manifolds with positive Ricci curvature
in Nonlinear Differential Equations and Applications NoDEA
Bellettini C
(2023)
Embeddedness of liquid-vapour interfaces in stable equilibrium
in Interfaces and Free Boundaries
Bellettini C
(2019)
Curvature estimates and sheeting theorems for weakly stable CMC hypersurfaces
in Advances in Mathematics
Bellettini C
(2023)
Multiplicity-1 minmax minimal hypersurfaces in manifolds with positive Ricci curvature
in Communications in Pure and Applied Mathematics
Bellettini C
(2023)
Multiplicity-1 minmax minimal hypersurfaces in manifolds with positive Ricci curvature
in Communications on Pure and Applied Mathematics
Bellettini C
(2021)
Sets with constant normal in Carnot groups: properties and examples
in Commentarii Mathematici Helvetici
Bellettini C
(2022)
Generic existence of multiplicity-1 minmax minimal hypersurfaces via Allen-Cahn
in Calculus of Variations and Partial Differential Equations
Description | The award is ongoing and research progress has been very successful. The essential preliminary steps were completed in 2019. In 2020/21 the key goal of the award, i.e. proving the existence of closed hypersurfaces with prescribed mean curvature, has been achieved in 2021. The second and key step was the more geometric one, where the object of interest is produced. It relies on the preliminary part, which is more analytic. The preliminary part has the scope of producing a set of conditions that are checkable in the applications and that guarantee that a certain object is a closed hypersurface with prescribed mean curvature. The second step involves a so-called minmax-construction that leads to an object on which the aforementioned conditions can be successfully tested. The preliminary part is also interesting for its own sake (regardless of its use in the next step of this project), in particular it has applications in the calculus of variations for models of liquid in stable equilibrium, subject to a gravitational (or other) field; this was also investigated further in 2021. Some aspects of the award that have been delayed due to the Covid pandemic, specifically those that involved travelling. In particular, the organisation of a series of lectures and of a workshop have been delayed, due to protracted travel restrictions and to the difficulty in coordinating mathematicians from different countries (with restrictions that were usually different between Country and Country). |
Exploitation Route | It is too early to give a complete answer. Surely the findings so far will contribute to the increase of academic knowledge in the fields of analysis and geometry. It is likely that they will spark new collaborative research work, in fact three projects (in which I am involved with some PhD students at my institution and with collaborators in the US) are progressing in a promising fashion. |
Sectors | Education,Other |
Description | Membership of the Institute for Advanced Study |
Amount | $20,000 (USD) |
Funding ID | National Science Foundation Grant No. DMS-1638352 |
Organisation | National Science Foundation (NSF) |
Sector | Public |
Country | United States |
Start | 01/2019 |
End | 04/2019 |
Description | Curvature estimates and sheeting theorems under weak stability |
Organisation | Princeton University |
Country | United States |
Sector | Academic/University |
PI Contribution | I collaborated with O. Chodosh and N. Wickramasekera on the theme of curvature estimates and sheeting theorems under weak stability. Contribution were based on expertise in geometry and geometric measure theory. |
Collaborator Contribution | Contributions of all parties were of mathematical nature, based on complementing expertise in geometry and geometric measure theory. |
Impact | Publication in Advances in Mathematics. |
Start Year | 2018 |
Description | Curvature estimates and sheeting theorems under weak stability |
Organisation | University of Cambridge |
Country | United Kingdom |
Sector | Academic/University |
PI Contribution | I collaborated with O. Chodosh and N. Wickramasekera on the theme of curvature estimates and sheeting theorems under weak stability. Contribution were based on expertise in geometry and geometric measure theory. |
Collaborator Contribution | Contributions of all parties were of mathematical nature, based on complementing expertise in geometry and geometric measure theory. |
Impact | Publication in Advances in Mathematics. |
Start Year | 2018 |
Description | Existence of curves of prescribed geodesic curvature in compact surfaces |
Organisation | Brown University |
Country | United States |
Sector | Academic/University |
PI Contribution | Intellectual input, expertise in the fields of semilinear PDEs, minmax, differential geometry. |
Collaborator Contribution | Intellectual input, expertise in the fields of semilinear PDEs, minmax, differential geometry. |
Impact | None yet. |
Start Year | 2020 |
Description | Existence of curves of prescribed geodesic curvature in compact surfaces |
Organisation | Stanford University |
Country | United States |
Sector | Academic/University |
PI Contribution | Intellectual input, expertise in the fields of semilinear PDEs, minmax, differential geometry. |
Collaborator Contribution | Intellectual input, expertise in the fields of semilinear PDEs, minmax, differential geometry. |
Impact | None yet. |
Start Year | 2020 |
Description | Existence of prescribed-mean-curvature hypersurfaces |
Organisation | University of Cambridge |
Country | United Kingdom |
Sector | Academic/University |
PI Contribution | Expertise in the fields of semilinear PDEs, minmax methods and geometric-measure-theory. Intellectual input. Write-up of the output. |
Collaborator Contribution | Expertise in the fields of semilinear PDEs, minmax methods and geometric-measure-theory. Intellectual input. Write-up of the output. |
Impact | The collaboration led to the preprint "The inhomogeneous Allen--Cahn equation and the existence of prescribed-mean-curvature hypersurfaces", currently submitted for publication. |
Start Year | 2018 |
Description | Prescribed-mean curvature hypersurfaces: regularity and compactness |
Organisation | University of Cambridge |
Country | United Kingdom |
Sector | Academic/University |
PI Contribution | Mathematical contributions based on the expertise in the relevant fields of geometry and topology. |
Collaborator Contribution | Mathematical contributions based on the expertise in the relevant field of geometric measure theory. |
Impact | C. Bellettini, N. Wickramasekera Stable constant-mean-curvature integral varifolds of codimension 1: regularity and compactness C. Bellettini, N. Wickramasekera Stable prescribed-mean-curvature integral varifolds of codimension 1: regularity and compactness These are preprints (2019), submitted for publication. |
Start Year | 2018 |
Description | American Mathematical Society meeting |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Postgraduate students |
Results and Impact | Two presentations at the American Mathematical Society sectional meeting in March 2019 on the topic "Stable prescribed-mean-curvature integral varifolds of codimension 1: regularity and compactness", which is the object of a preprint produced in 2019 within the scope of the award "Stable hypersurfaces with prescribed mean curvature". The presentations sparked interesting research questions. |
Year(s) Of Engagement Activity | 2019 |
Description | Article on Mathematics today |
Form Of Engagement Activity | A magazine, newsletter or online publication |
Part Of Official Scheme? | No |
Geographic Reach | National |
Primary Audience | Professional Practitioners |
Results and Impact | In a co-authored article on Mathematics today, a non-specialist magazine, I described the mathematical contributions of K. Uhlenbeck that led to the award of the Abel Prize 2019. The Prize was awarded for Uhlenbeck's groundbreaking contributions to mathematics, that created the field of geometric analysis as we know it today and motivated a plethora of new research directions, with impact felt until today. In particular, the research themes in the award "Stable hypersurfaces with prescribed mean curvature" present parallels and connections with the questions addressed by Uhlenbeck. |
Year(s) Of Engagement Activity | 2019 |
Description | BOWL |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Other audiences |
Results and Impact | I gave a talk at the BOWL Geometry Seminar, which includes attendees from Brussels, Oxford, London, Warwick, where I discussed recent work that I had completed on "existence of hypersurfaces with prescribed mean curvature". |
Year(s) Of Engagement Activity | 2021 |
Description | Bath Analysis seminar |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | National |
Primary Audience | Other audiences |
Results and Impact | I gave a talk at the Bath Analysis Seminar, where I discussed recent work that I had completed on "existence of hypersurfaces with prescribed mean curvature". |
Year(s) Of Engagement Activity | 2021 |
Description | ETH Geometric analysis and calibrated geometries |
Form Of Engagement Activity | Participation in an activity, workshop or similar |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Postgraduate students |
Results and Impact | Geometric analysis and calibrated geometries was both a summer school and a workshop. I gave a mini course of 4 hours on Regularity and compactness questions in calibrated geometric analysis, mainly aimed at PGRAs and graduate students. There was a lot of mathematical exchange and discussions, and some possibilities for collaborations were explored. |
Year(s) Of Engagement Activity | 2022 |
URL | https://math.ethz.ch/fim/activities/conferences/past-conferences/2022/geometric-analysis-and-calibra... |
Description | GMTA Workshop at UCL |
Form Of Engagement Activity | Participation in an activity, workshop or similar |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Postgraduate students |
Results and Impact | I organised the workshop Geometric Measure Theory and Applications at UCL. The primary purpose was to ease PhD students and PDRAs into a set of very useful techniques in geometric measure theory that have been used for a number of years but that are rarely part of graduate programs, and their (relations with and) applications to analysis and geometry. The workshop was organised with two mini-courses and two long talks, all quite related. There was ample space for discussion. There were about 20 attendees, largely PhD students, from Oxford, Cambridge, Warwick, Bath, London, Pisa, Princeton. Feedback was very positive, with PhD students very satisfied with the format and with the learning outcomes. There were initial conversations for future collaborations (including with myself). |
Year(s) Of Engagement Activity | 2022 |
URL | http://www.homepages.ucl.ac.uk/~ucahcbe/GMT_Workshop_22.html |
Description | Hebrew University Geometry Seminar |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Postgraduate students |
Results and Impact | I gave a talk at the Geometry Seminar of the Hebrew University in Jerusalem, where I discussed recent work that I had completed on "existence of hypersurfaces with prescribed mean curvature". |
Year(s) Of Engagement Activity | 2021 |
Description | Institute for Advanced Study Seminar |
Form Of Engagement Activity | A formal working group, expert panel or dialogue |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Postgraduate students |
Results and Impact | I gave a presentation on the topic "Stable prescribed-mean-curvature integral varifolds of codimension 1: regularity and compactness", which is the object of a preprint produced in 2019 within the scope of the award "Stable hypersurfaces with prescribed mean curvature". About forty attendees (international experts, scholars and postgraduates) engaged with my presentation, which sparked questions that suggested future directions of research. |
Year(s) Of Engagement Activity | 2019 |
Description | NCTS international GMT seminar |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Postgraduate students |
Results and Impact | NCTS International Geometric Measure Theory Seminar focuses on the regularity and singularity theories for submanifolds of Riemannian manifolds and some of their applications. It is a virtual event built around presentations on progress in geometric measure theory, attended by international experts, postgraduates, and graduate students. I discussed recent work that I had completed on "existence of hypersurfaces with prescribed mean curvature", followed by Q&A. |
Year(s) Of Engagement Activity | 2022 |
URL | https://sites.google.com/ncts.ntu.edu.tw/international-gmt-seminar |
Description | Oberwolfach Workshop Calculus of Variations 2020 |
Form Of Engagement Activity | Participation in an activity, workshop or similar |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Postgraduate students |
Results and Impact | The workshop "Calculus of Variations" is hosted every second year by the prestigious MFO (Oberwolfach, Germany). It brings together international experts of the field for presentations on the latest advances in the field, with the aim of fostering interactions and further projects. I gave a presentation and subsequently had excellent conversations with other participants. |
Year(s) Of Engagement Activity | 2020 |
Description | Oberwolfach Workshop on Partial Differential Equations 2021 |
Form Of Engagement Activity | Participation in an activity, workshop or similar |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Other audiences |
Results and Impact | The workshop gathers international experts, postgraduates and graduate students, whose field of expertise is Partial differential equations. The workshop permits the presentation of recent advances in the field though a series of talks, and further incentivises collaborations and the flow of new ideas through discussions among participants. |
Year(s) Of Engagement Activity | 2021 |
URL | https://publications.mfo.de/handle/mfo/3879 |
Description | Oxbridge PDE conference |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | National |
Primary Audience | Postgraduate students |
Results and Impact | This was the eleventh annual two and a half day conference held alternately in Oxford and Cambridge, focusing on analysis and PDE. Academics and PhD students from the two universities, together with external speakers, were invited to give talks. I presented my recent advances in the existence of prescribed-mean-curvature hypersurfaces, followed by Q&A. |
Year(s) Of Engagement Activity | 2022 |
URL | https://www.maths.ox.ac.uk/node/40583 |
Description | Oxford Geometry and Analysis Seminar |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | Regional |
Primary Audience | Postgraduate students |
Results and Impact | I discussed Hypersurfaces with prescribed-mean-curvature, specifically their existence and properties, and engaged in discussion with colleagues on aspects of the results. |
Year(s) Of Engagement Activity | 2022 |
URL | https://www.maths.ox.ac.uk/events/past/641 |
Description | Princeton University Seminar |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Postgraduate students |
Results and Impact | I gave a talk at the Princeton Differential Geometry and Geometric Analysis Seminar, where I discussed recent work that I had completed on "Multiplicity-1 minmax minimal hypersurfaces". |
Year(s) Of Engagement Activity | 2020 |
Description | Rome Tor Vergata |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Other audiences |
Results and Impact | I gave two talks at the Rome Tor Vergata, one at the Analysis Seminar and one at the Colloquium, where I discussed recent work that I had completed on "existence of hypersurfaces with prescribed mean curvature". The seminar was aimed at experts in the field, while the colloquium was aimed at a general mathematical audience. |
Year(s) Of Engagement Activity | 2021 |
Description | Rutgers Geometric Analysis seminar |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Other audiences |
Results and Impact | I gave a talk at the Geometric analysis seminar of Rutgers University, where I discussed recent work that I had completed on "existence of hypersurfaces with prescribed mean curvature". |
Year(s) Of Engagement Activity | 2021 |
Description | Stanford University/UCSD Geometric Analysis Colloquium |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Postgraduate students |
Results and Impact | I gave a talk at the joint Stanford/San Diego Geometric Analysis Colloquium, where I discussed recent work that I completed on "Multiplicity-1 minmax minimal hypersurfaces". |
Year(s) Of Engagement Activity | 2020 |
Description | University of Chicago Seminar |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Postgraduate students |
Results and Impact | I gave a talk at the geometric analysis seminar at the University of Chicago, where I discussed recent work that I had completed on "multiplicity-1 hypersurfaces in compact manifolds with positive Ricci curvature". |
Year(s) Of Engagement Activity | 2021 |
Description | University of Warwick seminar |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | Regional |
Primary Audience | Postgraduate students |
Results and Impact | The Mathematics Institute of the University of Warwick hosts a weekly analysis seminar, inviting speakers of the field to discuss recent advances in their research. I discussed recent work that I had completed on "existence of hypersurfaces with prescribed mean curvature", followed by interesting questions and discussions with attendees. |
Year(s) Of Engagement Activity | 2022 |
URL | https://warwick.ac.uk/fac/sci/maths/research/events/seminars/areas/analysis/2021-2022/ |
Description | Workshop on Geometric PDEs at Warwick |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Postgraduate students |
Results and Impact | This workshop is part of a UK network covering all aspects of the rigorous study of Partial Differential Equations. This event will focus on PDEs with a geometric flavour, including topics such as minimal surfaces, harmonic maps, mean curvature flow, free boundary problems and variational problems for eigenvalues. Speakers and attendees were international. There were numerous opportunities for exchange of ideas and exploration of research collaborations. |
Year(s) Of Engagement Activity | 2022 |
URL | https://warwick.ac.uk/fac/sci/maths/research/events/2022-2023/geometricpde/ |