future of topology in Zagreb

These days (January 2011), the *Seminar in topology* in Zagreb celebrates its 50th birthday! We should congratulate this strong tradition! The mathematics department of the university (link) is divided into 6 divisions or cathedras, and currently Topology is the smallest one; some topologists are in other faculties in Zagreb. Traditionally, the strongest field of topological research in Zagreb is shape theory where the school lead by academician Prof. Sibe Mardešić (which extends beyond Croatia) has been the leading in the world, especially in its golden period in 1970s and 1980s, when the group was numerous with the seminar attended by over 20 people in its best sessions. Historically developed under Zagreb influence, there is now a very strong and diverse topology school in Ljubljana, Slovenia (2 hours from Zagreb by train) and a more specialist group in Split, Slovenia, focused on shape theory. In past there were monthly traditional seminars Zagreb-Ljubljana as well, now less frequent, and the students and faculty from Split attended the research seminars on Monday morning in Zagreb, with travel support from Split University and from the national projects. The travel support from the university, e.g. for attending graduate courses for students has been cut recently. Another difficulty is that several principal researchers in Zagreb retired (one more retirement in a year or so) while very few young people chose to work in topology in last decade or two.

One of the great things about the seminar in topology in Zagreb has been its welcoming, non-exclusive character. People of various background, level, age and career were welcomed to attend and present their research but also learning topics. The attention to detail has always been welcome. The seminar sessions were long, up to 90 minutes each. This informal and inclusive atmosphere, without pressure and imposing, is a feature which should be praised and cherished. However, after shrinking of the topology community in last 2 decades, oen can not expect that the extent of the activities can resume spontaneously without some strong initiative and some awareness and consensus from the Croatian topology community. A couple of years ago or so, the math department accepted the strengthening of the topology division as a priority; however this seems not to be very radical: for example there has been no initiative to employ a strong mid-career topologist in open international search. No strategy about the interaction and interdisciplinary aspects between topology and other disciplines was discussed in the department or in the Croatian Mathematical Society. Such a wider crystalization of vision, goals and means is needed.

The following categories overlap to some extent.

Students get interested in the topics they hear often about and which they understand or can imagine. Thus the presence of well taught and refreshing university courses is our central responsibility. Besides, young researchers care weather they see the connections to other field, problems and applications. While every individual should have a freedom of choice, fostering mainstream topics of relevance for surrounding disciplines, with modern treatment and emphasis makes students feel secure about their career, the meaning of their work and support from their peers. With easier access to the free literature via internet and visits from the researchers from other centers, it is now easier to expose students to broader picture than before; good and enthusiastic mentorship to lead through the sea of information is however crucial. Fortunately, there are several good students or starting researchers at the moment (quote); however one needs to enforce the vigorous image of topology perpetually both for them and for the future students.

**Teaching**: It is easier to lecture one lecture twice or to a wider audience, than two different lectures. We may contemplate collaborative and guest teaching between Zagreb and nearby centers, especially Ljubljana and Split. To accomplish this one needs to plan coherently similar coursework in two centers and to ensure the additional support for travel expenses of the guest lecturers.

**Research**: This is now easy and obvious. Good researchers typically have good connection to the researchers abroad.

It is not sufficient to have some critical mass of topologists working each alone on esoteric topics; it is more beneficial to formulate problematics in which several researchers can collaborate within research, seminars, journal clubs and advanced teaching. Formulation of such topics should be founded on

- the tendencies in modern mathematics
- existing background of researchers
- potential of the research environment (including international collaborations)

All the three aspects are not trivial to observe and analyse, and it needs a common effort. Also, one should try to develop topology by fostering neighboring, largely missing, fields in Zagreb, like differential and noncommutative geometry, algebraic geometry and category theory.

The concentration of the expertise in the shape theory in Croatia is its main comparative advantage. It should not be an imperative for new researchers, but some effort into finding fresh and relevant topics which do use or advance the shape theory will be beneficial. The shape research should not isolate itself in traditional forms only, but also interplay with other mathematics. For example, studying formulations of strong shape using higher categories by Batanin and by Lurie, could lead to interesting results. There are also new categories where shape theory is under development, like operator algebras and topological stacks.

It is not sufficient to have a number of isolated topology researchers and interested students. It is important to have leaders with vision. The mathematics department at the moment lacks more strong topology researchers in mid career. A concentrated effort to ensure an employement of one or a couple of researchers from outside, possibly internationally could radically improve the situation. A new employee could be simultaneously a researcher in a nearby field, say symplectic or contact geometry.

With the extension of homotopic methods into new fields (motivic homotopy theory, derived algebraic geometry, higher topos theory and so on) and the ongoing spectacular merger of homotopy theory and higher category theory, it is crucial to employ or grow at least one leader in modern homotopy theory (simplicial homotopy theory, Quillen categories, coherent homotopy, infinity-categories). Currently Croatia has no specialists even in mainstream homotopy theory, including stable and rational homotopy nor in differential or symplectic topology; of course, some expertise was fostered for the purposes of research in geometric topology.

Ljubljana has more weekly seminars in modern pure mathematics than Zagreb currently. The attendance of people from other fields is smaller than in earlier periods; seminars in some areas have split into subseminars which have almost disjoint attendance lists. Croatian Mathematical Society has a specialized colloqium, featuring guests, but no journal club, big tendencies or interdisciplinary seminars. Researchers from diverse fieldsto conqueer new and modern fields. For example, survey lectures on works by Fields medalists can open eyes of many in the community.

We need to understand the topology in broader sense and include people who cover more than one specialty, e.g. thanks to the modern overlap of homotopy theory with derived algebraic topology, higher category theory etc. If the anachronic division of the department into 6 cathedras prevents such understanding, it should be actively opposed by the research community.

Recently, topological ideas systematically entered other fields, especially via abstraction through category theory; understanding this is beneficial for researchers as a basis for better choice of research problems and strategies; awareness of the wider scope of modern topology in mathematics is crucial for getting more support from colleagues in other areas. For instance, many categories in algebra and geometry embed into bigger categories of presheaves of spaces, which allow application of methods of abstract homotopy theory, thus leading to the homotopy theory of schemes in algebraic geometry, model category for Oka principle in complex analytic geometry, homotopy theory of operator algebras, topological stacks and so on. These methods are in principle simple and universal, and are easier to contemplate in their modern form than in partial elementary treatments; they are relevant to previously distant fields like anomalies and topological quantum field theories in physics, intensive type theory in foundations and so on.

Information is now easier to access, and to communicate; it is easier to collaborate internationally, not only via email but more recently also via wikis, as is this nLab. People can ask questions to open communities like recently established MathOverflow and nForum. See math resources.

Last revised on April 17, 2012 at 07:54:21. See the history of this page for a list of all contributions to it.