 This project represents a five-year slice of research in connection with the Langlands program and its implications for number theories specifically. The Langlands program is a vast area of mathematics that involves hundreds of people around the world and it has many different facets. The Langlands program is the insight that two branches of mathematics, two areas of mathematics, one having to do with symmetry and geometry and the other one having to do with numbers, reflect each other in a very precise way and with minor adjustments his insights have proved to be right 50 years later. Then there's another part and this is an attempt to apply the insights of the Geometric Langlands program. It's particularly strong here at the IHES to the MODP Local Langlands program that's an ongoing project within number theory. One of the hopes was that this would help to give it a new start or get to the start in another direction. So those are the aspects of the project. It started in January of 2016 here. This particular occasion is in honor of the French School of Automorphic Forms. Automorphic forms is really one of the very central topics in number theory and it has been one of the most successful topics in recent years. The starting point of IHES was modeled on the Institute for Advanced Study in Princeton. Paris is the most concentrated center of specialists that has been and most of these speakers are people who were trained in the in the Paris School. Freedom of research is really the DNA of an institute. When someone is coming here he has complete freedom. It's one of the places where people go to think without distractions. It's important for people to be in the same place to get work done. People came here to collaborate several times and I also traveled to report on this work. There's nothing that replaces face-to-face interaction. This ERC project right now is supporting three postdocs to do her original research. Mathematics is a subject that needs time and needs concentration. I'm very happy because of the grant. I like some other young researchers and never worried about the financial support. So I have a chance during the five years to attend a lot of conferences to see the most recent progress in mathematics and to find interesting things. So I'm really grateful to Michael and to the other grant. The opportunity to pursue your own ideas is just fantastic. Lately I have been focusing on using some tools that come really from differential geometry to tackle some problems in number theory and I needed some time to essentially learn an area that is not exactly mine and this was a very good environment to do that. At the typical conference people talk about the problems they've solved. These conferences the aim was to approach existing problems from new perspectives. The idea is that people with different points of view and different backgrounds by sharing their insights would be able to find new perspectives and then formulate the problems in new ways. The first one was in PISA and that was connected specifically with this geometrization project, the one relating the Geometric Languages Program to the arithmetic. Here it was the first time that we had people coming from derived algebraic geometry coming into the game and this brilliant idea that Michael was to gather people from these two really separate domains together to try to think together and maybe try to see whether this connection could kind of go far enough to maybe give us to figure out. This was a very different kind of conferences. In fact I've never been to such conferences. We are trying to work a Languages Program but there is no conjecture. You don't know what it should be so maybe you can find a good formulation using new techniques, techniques that the people in the Languages Program do not know. To me at that time I was really a young mathematician so it was really exciting and well you know meeting new people is always very interesting mathematically because well they come with new ideas and other backgrounds. Knowing this I mean you really want to discuss with them and try to bridge and maybe try to prove things together and so coming up back to my home institution I was discussing more with colleagues coming from topology for instance which I wouldn't say wouldn't have kind of come up in my mind at some point but definitely not that fast. The second one was held in Greece last year on a different aspect more along the lines of automorphic motives. Again it was people from different perspectives to bear people who were more I would say automorphic and people who were more arithmetic that was another I think rather successful conference and the third one this is more traditional in the sense that the concerns that are being discussed are for the most part the sorts of things that were present to the mind of the very first people in the field the Languages in particular. The history of the field in mathematics is always important and doing this conference it's sort of seeing how the subject has broadened. In the middle of the planning then we learned that the Roger Godement had died and so that seemed to be inappropriate focus for the conference because he was one of the very first people to recognize the importance of Languages work. I saw in the room lots of young people so probably people in a doctorate or postdoctorate and I think for them this conference will be a really important step in their career and in their research. Well actually even at this conference I anticipate that new directions will be developed. I'm particularly excited by Jim Arthur's talk I think everybody found that very stimulating because he ended in a surprising way with sketching or hinting at a completely new direction in the Languages program. The first project that was completed was one jointly with Richard Taylor, Jack Thorn and Kaiwen Lan. That suggested some new directions and then a few years after that I started working with two colleagues, Kebhard Bechler and Shekhar Kare. We started talking to Jack Thorn who had been on this previous project and very quickly the project accelerated and grew into a paper. It's going to continue beyond the duration of this particular ERC project and other people are getting involved as well. I wrote a paper with Harold Gropner on this material. Some of the postdocs have been working directly in this project. Lynn Jones my student and she's become my collaborator and she has taken over one big chunk of the original proposal and she solved part of it in her thesis and the rest of it is being finished in our current collaboration with Harold Gropner. So a lot of grants and actually our project even the part two of Michael's project has been spending afterwards so we still have many things can be done I think and there are many applications also of our work. It mostly refers to a different paper that's in process right now with another group of collaborators and that's a paper on pediatric health functions essentially. That was actually closely related to the the topic of the conference in Greece which has led to another new project. So that again was not at all anticipated in the original project but it's an outgrowth of it. And finally the geometrization projects well that is still largely speculative. We've clarified some of the issues and then the relations between the issues that was mostly the topic of the seminars at the Math Institute in Berkeley. Also the conference in PISA was mostly devoted to that and I also think that these listed projects have more to do with one another. Insights from one part of the project have been reflected in another one. The hope is that by bringing people together the obstacles to further understanding will be identified and then in the course of identifying them some sense of what is needed to overcome the obstacles will become clearer. We have already done more than what has written in the project and after going to details we actually find something more can be done. I find that my work has become much more varied. Makes you not scared of trying things even if they're a bit out of your comfort zone. Undoubtedly the ability to work with so many different collaborators and to organize activities with so many different people has led me to think about things in a way I wouldn't have otherwise. The airsogress makes a difference.