Rigidity and Small Divisors in Holomorphic Dynamics
Lead Research Organisation:
Imperial College London
Department Name: Mathematics
Abstract
The simplest non-linear systems are driven by quadratic polynomials. That is, "time n" of a state is determined by a quadratic polynomial of "time n-1" of that state. However, despite over a century of intense study, the dynamical features of even quadratic formulae remain far from well understood. For example, complex quadratic polynomials with "small divisors", which may be used to model resonance phenomena, still exhibit mysterious behaviour in many cases.
There has been extensive research on the dynamics of quadratic polynomials over the last three decades. Often, sophisticated tools from different disciplines of mathematics are needed to describe the fine dynamical features of these maps. Usually, a set of such tools is introduced to study the dynamics of a type of quadratic maps, but leads to the successful study of non-linear systems of that type. Thus, an effective set of tools for the study of quadratic polynomials provide the basis of extensive research in the wider area of non-linear systems.
In this project, I develop a new set of tools from different disciplines of mathematics to provide a comprehensive description of the dynamics of certain types of quadratic polynomials. This develops effective techniques from analysis, geometry, and more sophisticated mathematical machinery such as renormalisation and Teichmuller theory.
I will achieve the following major goals.
(1) Small divisors:
A main goal of this research is to introduce a systematic approach to obtain a comprehensive understanding of the dynamics of quadratic polynomials with small divisors. This provides the first examples of such systems with unstable behavior at the center of resonance, whose dynamical behaviour is completely understood.
The Julia set of a quadratic polynomial is the unstable locus of its dynamics. A recent remarkable result of X. Buff and A. Cheritat states that there are quadratic polynomials with small divisors which have observable (positive area) Julia sets. A central problem in the presence of small divisors is to determine arithmetic conditions on the rotation number that leads to observable Julia sets. The proposed research makes major advances on this problem.
(2) Rigidity and density of Hyperbolicity:
The quadratic polynomials that exhibit a certain well understood dynamical behaviour are called hyperbolic. There is a remarkable property, anticipated by P. Fatou in 1920's, stating that any quadratic polynomial may be perturbed to a nearby one with hyperbolic behaviour (by small changes in coefficients in an appropriate normalisation).
The project studies some deep analytic properties of a renormalisation technique to confirm this conjecture for certain types of quadratic polynomials (a Cantor set of parameters). This programme suggests a refined quantitative (in spirit of continued fractions) version of this conjecture to hold.
(3) Generalized Feigenbaum maps:
Period doubling bifurcation is a remarkable phenomenon that appears in the family of quadratic polynomials with real coefficients. There is a wide range of analogous, but more complicated, phenomena that occur when one considers quadratic polynomials with complex coefficients. This reflects the complicated structure of the Mandelbrot set. The dynamical features of such maps with real coefficients have been deeply studied in a period of intense research in 1980's and 90's, while the ones with complex coefficients are largely unexplored. The research proposal uses renormalisation techniques and develops innovative analytical methods to present a detailed description of the dynamics of such a map near degenerate bifurcations.
I will carry out some parts of this major project in collaboration with the leading experts of holomorphic dynamics: A. Avila (Rio, Brazil and Paris, France), X. Buff (Toulouse, France), A. Cheritat (Bordeaux, France), and M. Shishikura (Kyoto, Japan).
There has been extensive research on the dynamics of quadratic polynomials over the last three decades. Often, sophisticated tools from different disciplines of mathematics are needed to describe the fine dynamical features of these maps. Usually, a set of such tools is introduced to study the dynamics of a type of quadratic maps, but leads to the successful study of non-linear systems of that type. Thus, an effective set of tools for the study of quadratic polynomials provide the basis of extensive research in the wider area of non-linear systems.
In this project, I develop a new set of tools from different disciplines of mathematics to provide a comprehensive description of the dynamics of certain types of quadratic polynomials. This develops effective techniques from analysis, geometry, and more sophisticated mathematical machinery such as renormalisation and Teichmuller theory.
I will achieve the following major goals.
(1) Small divisors:
A main goal of this research is to introduce a systematic approach to obtain a comprehensive understanding of the dynamics of quadratic polynomials with small divisors. This provides the first examples of such systems with unstable behavior at the center of resonance, whose dynamical behaviour is completely understood.
The Julia set of a quadratic polynomial is the unstable locus of its dynamics. A recent remarkable result of X. Buff and A. Cheritat states that there are quadratic polynomials with small divisors which have observable (positive area) Julia sets. A central problem in the presence of small divisors is to determine arithmetic conditions on the rotation number that leads to observable Julia sets. The proposed research makes major advances on this problem.
(2) Rigidity and density of Hyperbolicity:
The quadratic polynomials that exhibit a certain well understood dynamical behaviour are called hyperbolic. There is a remarkable property, anticipated by P. Fatou in 1920's, stating that any quadratic polynomial may be perturbed to a nearby one with hyperbolic behaviour (by small changes in coefficients in an appropriate normalisation).
The project studies some deep analytic properties of a renormalisation technique to confirm this conjecture for certain types of quadratic polynomials (a Cantor set of parameters). This programme suggests a refined quantitative (in spirit of continued fractions) version of this conjecture to hold.
(3) Generalized Feigenbaum maps:
Period doubling bifurcation is a remarkable phenomenon that appears in the family of quadratic polynomials with real coefficients. There is a wide range of analogous, but more complicated, phenomena that occur when one considers quadratic polynomials with complex coefficients. This reflects the complicated structure of the Mandelbrot set. The dynamical features of such maps with real coefficients have been deeply studied in a period of intense research in 1980's and 90's, while the ones with complex coefficients are largely unexplored. The research proposal uses renormalisation techniques and develops innovative analytical methods to present a detailed description of the dynamics of such a map near degenerate bifurcations.
I will carry out some parts of this major project in collaboration with the leading experts of holomorphic dynamics: A. Avila (Rio, Brazil and Paris, France), X. Buff (Toulouse, France), A. Cheritat (Bordeaux, France), and M. Shishikura (Kyoto, Japan).
Planned Impact
The field of Dynamical Systems plays a central role in the development of Mathematics and Physics. It is widely applied in many disciplines to study long term behaviour of environmental, economic, and social systems, such as predicting average values of observables over long periods of time. Thus, the impacts of advances in dynamical systems on everyday life is very important.
The proposed project concerns foundational work to make breakthroughs on central conjectures of Dynamical Systems. It introduces and develops techniques from different disciplines of mathematics such as analysis, geometry, and Diophantine approximation, to provide a comprehensive understanding of highly complicated dynamical behaviours. The flexibility of the methods developed in this programme are very likely to be utilised in a wide range of applications. In particular, the proposed research impacts the areas listed below.
(1) The techniques developed in this research can be used to describe the dynamics of systems with resonances. Resonances are prevalent phenomena in almost periodic events, from the rising of the sun each day to more complicated electromagnetic waves. They often lead to mysterious behaviours. Many such systems, even when given by simple formulae like quadratic polynomials, have remain far from understood to date. One of the main aims of this project is to introduce a systematic approach to successfully study such systems modeled by quadratic formulae on the complex plane. The project answers questions such as whether the set of unstable states are observable (have non-zero probability of occurring).
(2)The project introduces cost effective algorithms for simulating highly complicated non-linear systems. These are dynamical systems arising from complicated bifurcation patterns. Through the developments of this project, I plan to develop software for simulating such systems and making them widely accessible through the internet. I plan to deliver lectures addressing the general public to share the excitements of these ideas and the challenges involved.
(3)The project introduces effective methods that immediately impact many areas such as shape analysis and medical imaging (in healthcare industries), electrical impedance tomography, analysis of water waves. One of the main building blocks of the proposed project is to develop effective analytic methods to describe fine geometric features of the solutions of non-linear partial differential equations, and to obtain optimal estimates on the dependence of the solution of such equations on the data. These methods can be used to establish estimates on conformal mappings and in conformal geometry which have found wide ranges of applications listed above.
The proposed project concerns foundational work to make breakthroughs on central conjectures of Dynamical Systems. It introduces and develops techniques from different disciplines of mathematics such as analysis, geometry, and Diophantine approximation, to provide a comprehensive understanding of highly complicated dynamical behaviours. The flexibility of the methods developed in this programme are very likely to be utilised in a wide range of applications. In particular, the proposed research impacts the areas listed below.
(1) The techniques developed in this research can be used to describe the dynamics of systems with resonances. Resonances are prevalent phenomena in almost periodic events, from the rising of the sun each day to more complicated electromagnetic waves. They often lead to mysterious behaviours. Many such systems, even when given by simple formulae like quadratic polynomials, have remain far from understood to date. One of the main aims of this project is to introduce a systematic approach to successfully study such systems modeled by quadratic formulae on the complex plane. The project answers questions such as whether the set of unstable states are observable (have non-zero probability of occurring).
(2)The project introduces cost effective algorithms for simulating highly complicated non-linear systems. These are dynamical systems arising from complicated bifurcation patterns. Through the developments of this project, I plan to develop software for simulating such systems and making them widely accessible through the internet. I plan to deliver lectures addressing the general public to share the excitements of these ideas and the challenges involved.
(3)The project introduces effective methods that immediately impact many areas such as shape analysis and medical imaging (in healthcare industries), electrical impedance tomography, analysis of water waves. One of the main building blocks of the proposed project is to develop effective analytic methods to describe fine geometric features of the solutions of non-linear partial differential equations, and to obtain optimal estimates on the dependence of the solution of such equations on the data. These methods can be used to establish estimates on conformal mappings and in conformal geometry which have found wide ranges of applications listed above.
People |
ORCID iD |
Davoud Cheraghi (Principal Investigator / Fellow) |
Publications
Cheraghi Davoud
(2015)
Satellite renormalization of quadratic polynomials
in arXiv e-prints
Cheraghi D
(2015)
A proof of the Marmi-Moussa-Yoccoz conjecture for rotation numbers of high type
in Inventiones mathematicae
Cheraghi Davoud
(2016)
Geometric complex analysis
Cheraghi Davoud
(2017)
Topology of irrationally indifferent attractors
in arXiv e-prints
Broecker
(2017)
Mathematics Of Planet Earth: A Primer
Mycek P
(2017)
Iterative solver approach for turbine interactions: application to wind or marine current turbine farms
in Applied Mathematical Modelling
Avila A
(2018)
Statistical properties of quadratic polynomials with a neutral fixed point
in Journal of the European Mathematical Society
Cheraghi Davoud
(2019)
Hairy Cantor sets
in arXiv e-prints
Davoud CHERAGHI
(2019)
Typical orbits of quadratic polynomials with a neutral fixed point: Non-Brjuno type
in Annales scientifiques de l'École normale supérieure
Cheraghi Davoud
(2020)
Lacunary series, resonances, and automorphisms of $\mathbb{C}^2$ with a round Siegel domain
in arXiv e-prints
Title | Siegel disks |
Description | Maximal linearisation domains of non-linear systems have been produced. It involves writing intelligent software codes that simulated some time -consuming tasks in shorter periods of times. The current programme requires 5 full-working days for a normal computer to obtain a single image. |
Type Of Art | Image |
Year Produced | 2015 |
Impact | Helped with developing mathematical methods to study the dynamics of non-linear systems. |
URL | http://wwwf.imperial.ac.uk/~dcheragh/Siegel.html |
Description | We have developed powerful mathematical methods based on renrmalisation ideas to tackle a central problem in mathematics that was left open since 1970's. At this point we have obtained a complete topological description of long term behaviour of a fundamental system with resonant behaviour. This is collectively known as the problem of small divisors. We have also discovered deep connections between the dynamics of holomorphic maps with infinitely many renormalsation structures of satellite type (generalisation of period doubling Feigenbaum phenomena) and the problem of small divisors. We have discovered and explained the appearance of optimal arithmetic conditions in these seemingly unrelated settings. |
Exploitation Route | This originates a method to study the long term behaviour of some non-linear systems that were out of reach until recently. |
Sectors | Aerospace, Defence and Marine,Energy,Financial Services, and Management Consultancy,Healthcare,Manufacturing, including Industrial Biotechology |
URL | https://arxiv.org/abs/1706.02678 |
Description | Helped to established a new Journal in Mathematics, |
Geographic Reach | Multiple continents/international |
Policy Influence Type | Participation in a guidance/advisory committee |
Impact | I have helped to establish the new journal in mathamatics, Journal of the Iranian Math Society. I also serve as an editor of this jounral since 2018. The establishment of this journal helps to support under represented groups of people in Mathematics. |
URL | http://jims.ims.ir/ |
Description | Memeber of staff at UMI Unite Mixte Internationale (CNRS/Imperial joint laboratory) |
Geographic Reach | Europe |
Policy Influence Type | Participation in a guidance/advisory committee |
Impact | I am a member of the advosory board of a new institute which facilitates movement of mathematicians netween the French and British universities. We have a regular stream of mathematicians from France to the UK, and from the UK to France, with stays of three to six months. This has led to many collaborations across the Chanel. |
URL | http://www.imperial.ac.uk/abraham-de-moivre/people/academic-staff/ |
Description | Departmental Platform Grant |
Amount | £8,000 (GBP) |
Organisation | Imperial College London |
Sector | Academic/University |
Country | United Kingdom |
Start | 05/2016 |
End | 09/2016 |
Description | European Partners Fund |
Amount | £5,000 (GBP) |
Organisation | Imperial College London |
Sector | Academic/University |
Country | United Kingdom |
Start | 08/2017 |
End | 10/2019 |
Description | H2020 |
Amount | € 183,454 (EUR) |
Organisation | European Research Council (ERC) |
Sector | Public |
Country | Belgium |
Start | 09/2018 |
End | 09/2020 |
Description | H2020- Marie-Curie Individual fellowships |
Amount | € 224,933 (EUR) |
Funding ID | LYP-RIG - GAP-837602 |
Organisation | European Research Council (ERC) |
Sector | Public |
Country | Belgium |
Start | 06/2019 |
End | 07/2021 |
Description | London Math Society Scheme 1 grants |
Amount | £1,670 (GBP) |
Funding ID | 11446 |
Organisation | London Mathematical Society |
Sector | Academic/University |
Country | United Kingdom |
Start | 08/2015 |
End | 12/2015 |
Description | London Mathematical Society Scheme 1 grants |
Amount | £4,000 (GBP) |
Organisation | London Mathematical Society |
Sector | Academic/University |
Country | United Kingdom |
Start | 01/2018 |
End | 05/2018 |
Description | Centralisers of polynomials with a parabolic fixed point |
Organisation | Imperial College London |
Country | United Kingdom |
Sector | Academic/University |
PI Contribution | This was a PhD thesis carried out under my supervision. The grant to support a PhD student was promised if the EPSRC application became successful. |
Collaborator Contribution | The partner was a PhD student who worked on the topic under my supervision. |
Impact | A PhD thesis (230 pages) written during this supervision. We introduces a new line of research on local symmetries of analytic systems, and make a number of fundamental conjectures on the topic. |
Start Year | 2018 |
Description | Complex Feigenbaum phenomena of high type |
Organisation | Imperial College London |
Country | United Kingdom |
Sector | Academic/University |
PI Contribution | Davoud Cheraghi is the PhD supervisor of the candidate (M. Pedramfar). |
Collaborator Contribution | The PhD student is collaborating on some aspects of the research proposal submitted to EPSRC. |
Impact | The collaboration has led to a clear programme to describe the behaviour of one of the most complicated phenomena in non-linear dynamics; the presence of renormalisation structures of satellite type. This has lead to a manuscript about 120 pages, and will be divided into two articles to be submitted for publication. |
Start Year | 2016 |
Description | Computational complexity of Lorenz attractors |
Organisation | Meteorological Office UK |
Country | United Kingdom |
Sector | Academic/University |
PI Contribution | In this collaboration with Dr Gabriel Rooney we investigate the use of renormalisation methods in simulations related to weather forecasting and environmental changes in oceans. I provide expertise from dynamical systems. |
Collaborator Contribution | The collaborator provides expertise on applications. |
Impact | We have outlined a PhD project on this collaboration, which is funded by a CDT at Imperial College London. |
Start Year | 2017 |
Description | Convergence of conformally balanced trees to dendrites. |
Organisation | University of Zurich |
Country | Switzerland |
Sector | Academic/University |
PI Contribution | My team has introduced the main topic of this collaboration, and contributes to the conceptual development of the project. |
Collaborator Contribution | the Collaborator contributes to the conceptual and technical aspect of the project. |
Impact | A joint paper is emerging from this collaboration. |
Start Year | 2020 |
Description | Dimension paradox of irrationally indifferent attractors |
Organisation | Imperial College London |
Department | Department of Mathematics |
Country | United Kingdom |
Sector | Academic/University |
PI Contribution | In collaboration with Dr Yang Fei and Alexandre de Zotti we have successfully studied the metric properties of the attractors of analytic maps with resonant behavior. A journal article on this study is prepared and will be submitted for publication soon. I have played a central role in drawing the overall strategy, and providing the foundational ingredients for the work. |
Collaborator Contribution | My collaborators have been mostly contributing to thechnical aspect of the project. |
Impact | This will lead to a journal article of about 40 pages. |
Start Year | 2016 |
Description | Dimension paradox of irrationally indifferent attractors |
Organisation | Nanjing University (NJU) |
Department | Department of Mathematics |
Country | China |
Sector | Academic/University |
PI Contribution | In collaboration with Dr Yang Fei and Alexandre de Zotti we have successfully studied the metric properties of the attractors of analytic maps with resonant behavior. A journal article on this study is prepared and will be submitted for publication soon. I have played a central role in drawing the overall strategy, and providing the foundational ingredients for the work. |
Collaborator Contribution | My collaborators have been mostly contributing to thechnical aspect of the project. |
Impact | This will lead to a journal article of about 40 pages. |
Start Year | 2016 |
Description | Endomorphisms of C2 with a wandering domain tending to a Cantor set |
Organisation | Imperial College London |
Department | Department of Mathematics |
Country | United Kingdom |
Sector | Academic/University |
PI Contribution | This is a collaboration with Professor Sebastian van Strien, DR Trevor Clark, and Fabrizio Bianchi. I have sketched the main strategy of the project where we investigate the existence of wandering domains in higher dimensional analytic spaces. |
Collaborator Contribution | Provide technical details from real analysis and higher dimensional complex analysis. |
Impact | If successful, this will lead to a journal article. |
Start Year | 2017 |
Description | Hairy Cantor sets in the plane |
Organisation | Imperial College London |
Department | Department of Mathematics |
Country | United Kingdom |
Sector | Academic/University |
PI Contribution | We discovered a new topological object in the plane with universal features similar to the Cantor set, that is, the set can be described by a number of axioms, and any two such objects in the plane are ambiently homeomorphic. Surprisingly, such objects which have not been identified to date, are prevalent in analytic dynamics. A dense set of rational maps on the bifurcation locus, preserves such an object. |
Collaborator Contribution | Technical aspects of the project have been carried out by my PhD student Mohammad Pedramfar. |
Impact | A journal paper about 30 pages is written and submitted for publication. |
Start Year | 2018 |
Description | Siegel disks with boundaries of Hausdorff dimension two |
Organisation | Imperial College London |
Department | Department of Life Sciences |
Country | United Kingdom |
Sector | Academic/University |
PI Contribution | In this project we investigate the existence of maximal linearisation domains with large boundaries, that is, of Heusdorff dimension two. This is a joint project with my research associate Dr Alexandre De Zotti. |
Collaborator Contribution | Provides technical support. |
Impact | The collaboration will lead to a journal paper. |
Start Year | 2016 |
Description | Topology of isentropes in a two parameter family of unimodal maps |
Organisation | Imperial College London |
Department | Department of Mathematics |
Country | United Kingdom |
Sector | Academic/University |
PI Contribution | This is project with my colleague at Imperial college London where we study the global deformation structures in the parameter space of polynomials. I provide techniques from complex analysis. |
Collaborator Contribution | The partner provides techniques from real dynamics. |
Impact | Expect to finish a paper about 20 pages, of which 10 pages written down to date. |
Start Year | 2016 |
Description | parabolic and eliptic fixed points with trivial centralisers |
Organisation | University of Zurich |
Country | Switzerland |
Sector | Academic/University |
PI Contribution | This is a collaborative research with Professor Artur Avila from the University of Zurich. In this collaboration, we prove the existence of holomorphic maps with a fixed point of parabolic type, whose local holomorphic centraliser is trivial. |
Collaborator Contribution | The result from the parabolic case is employed to prove the existence of holomorphic maps with an irationally indifferent fixed points whose local centraliser near the fixed point is trivial. This answers a long standing open problem in the field of complex dynamics. |
Impact | this has lead to a Journal paper. |
Start Year | 2019 |
Description | simultanious linearisation of commuting germs tangent to rotations |
Organisation | Pierre and Marie Curie University - Paris 6 |
Country | France |
Sector | Academic/University |
PI Contribution | expertise in iterations of complex analytic maps of the complex plane |
Collaborator Contribution | expertise in iterations of real analytic maps of the circle |
Impact | expect a journal paper resulting from this collaboration |
Start Year | 2017 |
Description | Five lectures on dynamical systems |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | National |
Primary Audience | Postgraduate students |
Results and Impact | 15 postgraduate students attended five lectures of two hours each. This was part of a Centre for doctoral training in mathematics of planet earth. |
Year(s) Of Engagement Activity | 2015 |
Description | Five lectures on dynamical systems for Mathematics of Planet Earth |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | National |
Primary Audience | Postgraduate students |
Results and Impact | Delivered five lectures (of two hours each) to nonspecialists in Mathematics of Planet Earth |
Year(s) Of Engagement Activity | 2016 |
Description | Junior Analysis seminars |
Form Of Engagement Activity | Participation in an activity, workshop or similar |
Part Of Official Scheme? | No |
Geographic Reach | National |
Primary Audience | Postgraduate students |
Results and Impact | I initiated the Junior Analysis seminars at Imperial College, in order to bring the PHD students working in broader analysis area together in order to broaden their scope, practice communicating their work, and identify phd students at other institutions within the UK. The meeting occurs every Friday for two hours, 2-3, and 3:30 to 4:30. |
Year(s) Of Engagement Activity | 2016,2019 |
URL | https://www.imperial.ac.uk/pure-analysis-and-pdes/seminars/jas/ |
Description | London Analysis and Probability seminar |
Form Of Engagement Activity | Participation in an activity, workshop or similar |
Part Of Official Scheme? | No |
Geographic Reach | National |
Primary Audience | Professional Practitioners |
Results and Impact | I was one of the organisers of the regular biweekly seminars on broader analysis are in the London region. This is a joint seminar between Imperial College London, University College London, Kings College London, and Queen Mary University London. I am the contact point for Imperial college London. |
Year(s) Of Engagement Activity | 2017,2018 |
Description | MPE CDT Sandpit meeting |
Form Of Engagement Activity | Participation in an activity, workshop or similar |
Part Of Official Scheme? | No |
Geographic Reach | National |
Primary Audience | Industry/Business |
Results and Impact | In this meeting experts from different areas of natural sciences came together to discuss techniques that could be useful for the study of environmental problems. Most of the participants came from industry and national organisations like MET Office Thames Water. My discussion with some of the participants from MET Office has lead to a proposal for a PhD thesis in CDT on Mathematics of Planet Earth. In the meeting I presented a brief description of how renormalisation methods could be used in more efficient programming methods in whether forecasting. |
Year(s) Of Engagement Activity | 2016 |
Description | Organisation of a conference |
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 | This was an international conference to bring Scandinavian experts in analysis in contact with dynamics experts in the UK. The meeting was highly successful in exchanging key challenges and the required tools, and due to high level of interest from both sides, it is planned that the meeting will take place every other year, alternating between UK and Finland. |
Year(s) Of Engagement Activity | 2018 |
URL | https://personalpages.manchester.ac.uk/staff/tuomas.sahlsten/analysisdynamics/ |
Description | Parameter problems in analytic dynamics |
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 | This was a major international conference on analytic dynamics which brought together the leading experts in the field. The event has helped us with attracting the best in the field and some of the best in the research area wish to visit our group at later stages. For instance, we have already made applications for Professor Francois Berteloot (University of Toulouse, France) to spend six months at Imperial College, and Professor Genadi Levin (University of Jerusalem, Israel) has requested to spend a six month sabbatical at Imperial College. We have also had a number of very strong applicants for junior level positions at Imperial College. |
Year(s) Of Engagement Activity | 2016 |
URL | http://wwwf.imperial.ac.uk/~dcheragh/PPAD/Conference.html |
Description | Pure Analysis and PDE |
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 was the organiser of the regular weekly research seminars on Pure Analysis and PDE in the department of mathematics. |
Year(s) Of Engagement Activity | 2017 |