## Final Report Summary - MALOA (From Mathematical Logic To Applications)

The project MALOA ran from 1 October 2009 to 30 September 2013. It was coordinated by the University of Leeds, had 8 Full Partners from universities in the UK, France, Germany, and the Czech Republic, and three Associated Partners (University of East Anglia, and the industrial partners Onera and British Telecommunications plc). The purpose of the project was to provide a broad doctoral-level training in logic, combining the complementary strengths of the partners to ensure that each Fellow received training in a range of areas of mathematical logic, whilst becoming aware of the potential for industrial applications. The network provided training for 18 Early Stage Researchers who held fellowships, mostly for a full 36 months, to do a PhD with one of the partners, and many of these PhDs have already been completed, with more to be finished soon. MALOA also provided 24 shorter Early Stage Researcher fellowships, of between 3 and 10 months, to enable Fellows doing a PhD anywhere to visit the network and receive training for an extended period, or to start a PhD with the network. Many of these short-term appointees participated in MALOA training events well beyond the period of their appointment, built links to other MALOA Fellows and to MALOA senior researchers; for some, their MALOA appointment, though short, has been a key period from which their PhD developed.

The branches of logic covered by the network, and corresponding Workpackages, were: Model Theory; Logical Aspects of Complexity Theory; Proof Theory; Computability Theory; Set Theory; and Real-valued Logics. The network recruited Fellows in all of these areas, and made substantial progress on most of the Tasks in each Workpackage, with significant unforeseen research successes in these areas which were beyond the planned Tasks. Particular highlights include the following:

(a) Major advances by A. Chernikov (at the CNRS partner based in Lyon) in model theory. These are some of the most exciting recent developments in the subject, showing that many ideas from model-theoretic stability theory apply under much weaker hypotheses, and thereby have much greater scope for application. Chernikov has also investigated mathematically interesting examples, such as ultraproducts of p-adically closed fields. He has written 7 papers through support by MALOA (two already published).

(b) A remarkable model-theoretic proof by Z. Ghadernezhad in Muenster (with Tent) that there is a simple group with a BN pair which is not an algebraic group. The fundamental concept due to Tits in the 1960s of a BN pair was intended to unify features of simple matrix groups over varying fields, and it is striking to have an example of completely different flavour.

(c) Major progress due to Vlitas (Paris) in set-theoretic Ramsey theory. Among many fine results, he has proved a self-dual theorem which implies at the same time the Carlson-Simpson and the classical Ramsey theorem.

(d) Mueller (Prague) obtained many results in (proof) complexity theory concerning the lengths of proofs in propositional proof systems. This is a fundamental area intimately connected to the P vs NP problem (one of the seven Clay Foundation Millennium Problems) and has potential industrial applications, for example through Satsolvers.

(e) Important results in proof theory. Among other successes, this includes: progress by Ziegler (Leeds) on large cardinal axioms in constructive set theory (a version of set theory suited to intuitionistic logic); and results of Rinaldi (Munich) on connections between commutative algebra and formal topology.

The network-wide training of MALOA took place through joint workshops, and through secondments of Fellows to other partners, with all the main joint events (workshops, final conference, special semesters) taking place essentially as planned, apart from some changes in the dates and venues. Near the end of each of the first three years there was a one-week Training Workshop, in which there were 3-4 short courses in key current topics in logic, accessible to all the ESRs, along with a small number of individual invited talks, and presentations by the ESRs. In the final year there was a 5-day MALOA `Final Conference', with some 61 participants, including 15 invited talks (7 from outside the network), and talks by ESRs. This acted simultaneously as a training event for the Fellows, and as an opportunity for them to meet a wider range of leading researchers and present their research, sometimes in conjunction with lectures by senior MALOA researchers. In addition, there were three short `Research Workshops' (Logic and Complexity, (Prague, 2011), Set Theory (Muenster, 2012), and Model Theory (Manchester, 2013)), in areas where there was a concentration of MALOA activity.

Two unusual but highly successful events were the 3-month Special Semester in Model Theory (Lyon, Spring 2011) and the Special Semester in Logic and Complexity Theory (Prague, Autumn 2011). For these events, all the MALOA Fellows working in the relevant field spent time at the host centre, along with some additional short-term MALOA appointees, local PhD students, and some other PhD students (not MALOA-funded) who found other funding to visit the host. In each case, there was a sequence of short courses in the area, some by invited speakers, and extended joint activity of the ESRs -- for example in Lyon an ESR seminar in model theory. These events gave an exceptional opportunity for the ESRs (and others) to build their research background in their chosen area and work together in a stimulating environment, and it has led to the development of an international network of highly-trained and well-interconnected early career researchers.

With one exception due to illness, all of the ESRs appointed for at least a year held secondments at one of the other partners, mostly for at least 3 months, some also having short research visits to logic centres beyond the MALOA network. In some cases, these secondments were linked to the Special Semesters, but in other cases an ESR would visit another centre to work with a senior researcher, disseminate their work, and also use the secondment to make short further visits elsewhere for dissemination. As examples, Rinaldi has been based in Munich, but through secondments worked extensively with Schuster, a senior researcher in Leeds, whilst Ghadernezhad (Muenster) both participated in the Lyon special semester in model theory, and visited Leeds for extended discussions with Macpherson and others.

The core research training (PhD supervision, taught courses, complementary training) was host-based, with a range of courses and seminars in logic available at each centre. In addition to some complementary training at the centres (e.g. courses in languages, IT, careers skills, teaching) some complementary training was provided at the Training Workshops and Final Conference -- for example on giving poster presentations, on aspects of giving talks (computer-based versus traditional approaches) and on postdoctoral opportunities.

The Associated Partners contributed to the project both through advice (participation in the Supervisory Board) and through training, with talks and/or courses by representives of all the Associated Partners at Workshops and the Final Conference. As a result of MALOA, the industrial partner Onera is a partner in a number of recent and current project proposals, leading to extended future interaction between Onera and academic logicians, and BT was also a named partner in a major recent logic-related proposal from Leeds. The Associated Partners also contributed to individual ESR research projects: for example Evans (UEA) has collaborated with Ghadernezhad (Muenster), and Dzamonja (UEA) interacted with ESRs in set theory in Paris on many visits to Paris.

Many of the ESRs are still completing their PhDs, but others have taken promising further career steps. For example, Chernikov and Vlitas have prestigious postdoctoral fellowships respectively in Paris and University of Illinois at Urbana-Champaign, Mueller will hold a 2-year postdoctoral fellowship in Tokyo following one in Toronto, Kestner has a permanent academic position at UCLan, and Elsner has a position in industry related to the development of advanced automated teaching materials. With many more MALOA ESRs likely to follow successful careers both in academic life and in other sectors, we are confident that MALOA has achieved its core goals: to produce high-quality and influential research; to build links between industry and researchers in logic; to provide additional stimulus to the partners, to their other PhD students, and to PhD students based elsewhere; and, above all, to give a broad training to a talented international cohort of researchers, who will pursue successful careers, in Europe or linked to Europe, in varied sectors, thereby strengthening the European research base.