Lecture on the occasion of the 50th anniversary of the Department of Mathematics

**Looking back on 50 years of mathematics at the Paris Lodron University of Salzburg**

Maximilian Thaler

Dear guests of our 50th anniversary celebration, dear colleagues!

I have been entrusted with the honourable task of taking a brief look at the history of mathematics at our university, for which I would like to thank the organising team.

As the title of my presentation says, it will be a retrospective, not a strict historical treatise. The mere fact that I myself was an active member of the Institute of Mathematics / Department of Mathematics for forty of the fifty years to be celebrated today means that my presentation is not free of personal choice and weighting.

Despite the resulting misgivings, I was happy to take on the presentation. Its primary concern will be to emphasise that the Institute of Mathematics/the Department of Mathematics has very successfully mastered its diverse tasks in the fifty years of its existence. The range of tasks of such an institution, like that of the university as a whole, is extremely broad. Meeting all requirements in a balanced way is the great challenge of spanning the spectrum from internationally competitive research in theory and application to the sensitive area of university teaching and the cultivation of interdisciplinarity through to administrative and service tasks. The successful endeavour to broadly integrate the various fields of activity is one of the strengths on which mathematics in Salzburg can be congratulated today. I will report on the associated success stories below.

The structure of my review is designed in such a way that I will first look at the old university, the establishment of the new one, the initial phase of the Institute of Mathematics and the further personnel development. The rest will mainly consist of addressing individual development strands and initiatives that together make up the success.

Due to the time constraints, the individual points can only be addressed briefly. I apologise for the fact that some points worth mentioning are not addressed and that the achievements mentioned are often not mentioned by name. As the current situation and role of the Department of Mathematics has been discussed by the two previous speakers and the current, very invitingly designed homepage provides information about it, my review is limited almost exclusively to the period up to 2012, the year that saw a new beginning for the department.

From its beginnings until 2004, our organisational unit was referred to as the ‘Institute’; in the course of the implementation of the University Act 2002, it was renamed the ‘Department’. This did not result in a substantial change in the composition of the organisation.

**A LOOK BACK AT THE OLD UNIVERSITY (1622-1810)**. A review of mathematics at Paris Lodron University should not remain entirely without reference to the older history of the university. As is well known, the University of Salzburg is already in its second edition. The old university was ceremonially opened on 8 October 1622 by Archbishop Paris Lodron. It was therefore founded during the lifetime of icons such as Galileo Galilei, Johannes Kepler, René Descartes and Pierre Fermat. The university was abolished in 1810 after Salzburg came under Bavarian rule. The abolition was widely regretted in Salzburg, and the sad news was officially announced to those affected on Christmas Eve of all days.

Throughout the entire existence of the old university, mathematics was part of the two-year teaching programme of the Faculty of Philosophy, which all students had to complete. In his essay ‘Die Pflege der Naturwissenschaften an der alten Universität’ in the commemorative publication ‘Universität Salzburg 1622-1962-1972’, Walter Del-Negro cites a continuous list of 29 professors of mathematics, all of them fathers from Benedictine monasteries, who made up the teaching staff of the Faculty of Philosophy and Theology in confederation. The three professors Bernard Stuart (1706-1755), Dominikus Beck (1732-1791) and Ulrich Schiegg (1752-1810) are briefly described here.Further sources are mainly the enjoyable book ‘Aus Salzburgs Hoher Schule geplaudert’ by Christoph Brandhuber, the director of the Salzburg University Archive, and the ‘Verzeichniß aller akademischen Professoren zu Salzburg vom Jahre 1728 bis zur Aufhebung der Universität. Mit Nachrichten von ihrem Leben und ihren Schriften, herausgegeben von einem Mitgenossen derselben,’ Salzburg 1813. The comrade is Judas Thaddäus Zauner, professor of the Faculty of Law at the old university, historian and namesake of Zaunergasse in the city of Salzburg.

**Bernard Stuart,** came from the Scottish monastery of St. Jakob in Regensburg and worked in Salzburg from 1733 to 1741 as a professor of mathematics and court building director. Leopoldskron Palace was built according to his plans. In general, many of the mathematics professors mentioned were very successful in various technical fields of application. Zauner writes about Bernard Stuart: ‘In November 1733, he took over the teaching of mathematics at the high school in Salzburg, and the more he endeavoured to make the teachings of his science applicable to civic life, especially to construction, the greater his success.’ But he is also responsible for other technical achievements. For example, a magnificent clock designed by him can be admired in the DomQuartier. He enjoyed a scientific reputation far beyond the borders of Salzburg, which earned him a call to St. Petersburg. He eventually became abbot of his home monastery.

**Dominikus Beck,** came from the Swabian monastery of Ochsenhausen. According to Judas Thaddäus Zauner, in 1766 ‘he was given the teaching position of mathematical sciences and experimental physics at the university, which he administered until the last day of his life with as much zeal as benefit and favour.’ Technically very adept, he built and improved water buildings and machines on behalf of Archbishop Hieronymus Colloredo, who held him in high esteem, erected the first lightning conductor in Salzburg at Mirabell Palace and held public lectures on experimental physics, which were open to interested people of ‘both sexes’ and were also popular with members of the Mozart family. A wax bust of Dominikus Beck is kept in St Peter’s Abbey, but its exhibition in the former Museum Mathematicum was vetoed by his fellow professors.

**Ulrich Schiegg,** Dominikus Beck’s successor, came from the Ottobeuren monastery. He was Professor of Mathematics, Physics, Astronomy and Agriculture at the University of Salzburg from 1791 to 1800. The following selection of his writings, the first of which reports on the ascent of a hot-air balloon, also provides information about his wide-ranging interests: (1) Nachricht über einen aerostatischen Versuch, welcher in dem Reichsstifte Ottobeuren vorgenommen worden den 22. Jenner 1784 (2) Brief instructions for the thorough learning of arithmetic; dedicated to young students (3) Saving wood and time in brewing pans, brandy ovens and wash kettles (4) On friction and rigidity of ropes as obstacles to movement in machines. In addition to theorems from applied mathematics, physics, practical philosophy, morality and natural law (5) The journey up the Glockner. This last essay resulted from Ulrich Schiegg’s participation in the first ascent of the Großglockner in 1800, which he accompanied scientifically. In the same year, he was recalled to his home monastery despite intensive efforts to keep him in Salzburg. He later went on to have an impressive career as a surveyor of the Bavarian provinces in Franconia.

Let us briefly look back to the 17th century. In the book cited above, Brandhuber reports that a concept of a mathematics lecture was created at the University of Salzburg in 1653, which is one of the few documents on the teaching of arithmetic in the 17th century, and he points out that the teaching of arithmetic included the use of calculating machines. This addresses two fields that have also occupied mathematics in Salzburg extensively over the last 50 years: the design of courses and the promotion of the use of computers. Both will be discussed here.

**RE-ESTABLISHMENT OF THE UNIVERSITY (1962)** Today’s University of Salzburg was established by a resolution of the Austrian National Council on 5 July 1962. From a purely legal point of view, this was an amendment to the Higher Education Organisation Act of 1955. The minor change with a major impact was that the entry ‘the Faculty of Catholic Theology in Salzburg’ under the heading ‘Existing universities’ was replaced by the entry ‘the University of Salzburg’. The Faculty of Theology, together with the Faculty of Philosophy, had been converted into a lyceum following the abolition of the old university, but had already been elevated to an independent university institution again in 1850.

The new university was conceived as a classic comprehensive university with four faculties: theology, philosophy, law and medicine. In addition to the Faculty of Theology, the Faculty of Philosophy was already established with the founding law and began its first lectures on 6 April 1964. On 21 April 1964, the Academic Senate decided to name the new university after Archbishop Paris Lodron and on 14 November 1964, the university and its first elected rector (Egon Lendl, Geography) were formally inaugurated.

The Faculty of Law was established in 1965; despite intensive efforts, the Faculty of Medicine was not realised. The Faculty of Arts was split into the Faculty of Humanities and the Faculty of Natural Sciences by the University Organisation Act 1975, the former was renamed the Faculty of Cultural and Social Sciences in 2004.

In the years following the founding of the university, new institutes were successively established at the Faculty of Philosophy. In the second half of the 1960s, the establishment of an Institute of Mathematics was on the agenda. One may wonder what the motives were for deciding in favour of mathematics. Based on the arguments in favour of establishing the new university, the main reason was probably to safeguard teacher training in mathematics. This assumption is supported by the fact that the first draft of study regulations for mathematics, based on the study laws of 1966 and 1971, initially only provided for a teacher training programme at diploma level for Salzburg. It is to the credit of the Mathematics Study Commission of the 1970s that it was possible to secure the mathematics major, commonly referred to as the mathematics diploma, for Salzburg.

**BEGINNINGS OF THE INSTITUTE OF MATHEMATICS AND LOCATIONS** After mathematics courses had already been offered in the 1966/67 academic year on the basis of a visiting professorship, regular study programmes in mathematics began in the winter semester of 1967/68 with August Florian as full professor and Wilhelm Fleischer as assistant. The first major annual lecture was algebra, followed by analysis the following year.

Fritz Schweiger was appointed as a further full professor in 1969, Peter Zinterhof in 1972 and Peter Gerl in 1974. The mathematical research areas represented were geometry, metric number theory and ergodic theory, function theory, stochastics and numerics as well as uniform distribution and harmonic analysis. The appointment of further assistants was just as important for the development of the Institute of Mathematics: Johann Linhart in 1969, Walter Bauer and Roland Fischer in 1970, Johann Stegbuchner and Karl-Josef Parisot in 1973, Ferdinand Österreicher and Maximilian Thaler in 1976, Johannes Czermak and Gerhard Racher in 1977, Franz Kinzl and Peter Hellekalek in 1978. Apart from R. Fischer, who was appointed full professor at the University of Klagenfurt in 1974, and K.-J. Parisot, who moved to the Institute for Didactics of Natural Sciences, the above-mentioned persons formed the academic staff when the Institute of Mathematics was constituted on 15 December 1978 in accordance with the University Organisation Act 1975. This marked the first completion of the staffing structure in mathematics. The number of permanent positions was later de facto based on the number of staff at that time. A look at the current homepage of the Department of Mathematics shows that this has essentially remained the case to this day. The range of research areas has become increasingly colourful as a result of personnel development and the expansion of interests, which will be discussed below.

In the early years, the Institute of Mathematics was housed at Porschestraße 8 and 1 (near the main railway station). In 1975, it moved to Petersbrunnstraße 19 (Nonntal), which was rented from St Peter’s Abbey and has since changed its appearance considerably. The atmosphere there was very familiar and appreciated by teachers and students alike. The move to the new building in Freisaal in 1986 was therefore accompanied by a certain amount of melancholy. Shortly after the mathematics department moved out, Siemens Programme and System Development moved into the institute building in Petersbrunnstraße, so that some of our graduates at the time took their first professional steps in the premises where they had taken their mathematics exams shortly before.

**PERSONNEL DEVELOPMENT** This review must also leave it at brief details with regard to further personnel development. The number of professorships was increased in 1982 with the appointment of J. Czermak as Associate Professor of Logic and Foundations of Mathematics. On the other hand, the positions of F. Schweiger and P. Zinterhof were initially partially transferred to the Didactics of Natural Sciences and Computer Science respectively, and were fully transferred with the implementation of the University Act 2002. However, both colleagues have remained closely and actively involved in mathematics. A. Florian retired in 1996 and was succeeded in 2001 by Christian Buchta, whose second specialism alongside stochastic geometry is actuarial mathematics. – Since the mid-1980s, the establishment of a professorship for statistics/stochastics has been included in the various job plans and forecasts. A related concern was to align mathematics even more closely with the needs of the Faculty of Natural Sciences. This goal was achieved in 2012 with the appointment of Arne Bathke. Good things take time.

Our colleague W. Fleischer, the longest-serving member of the College since 1978, sadly passed away in 1982. All those who knew him will agree with the honourable words in the obituary by F. Schweiger in the journal uni-aktuell, 1982/83 no.1. The position was filled by Ingeborg Bittner and, following her universally regretted departure, by Gerhard Larcher, who was appointed to the University of Linz in 2000, where he holds the Chair of Financial Mathematics and Applied Number Theory. At the beginning of the 1990s, two new assistant positions were created, to which Michael Revers and Reinhard Wolf were appointed. The death of J. Stegbuchner, who was torn from a busy life in 1998, was a bitter loss. His versatile and successful work is summarised in the obituary by F. Kinzl, printed in the last issue of the Arbeitsberichte des Mathematischen Instituts (1999, issue 1). The position of J. Stegbuchner was filled by Wolfgang Schmid.

The wave of retirements that began in 2007 led to a staff shortage, to which the Rectorate initially responded by allocating temporary positions: a postdoc position held by Maria Alice Bertolim and temporary professorships that were filled by Johannes Schoißengeier, Christoph Haberl, Horst Heck, Clemens Fuchs and Andreas Schröder. With the 2013-2015 target agreement, the department was then fundamentally reorganised and personnel consolidation was sustainably advanced by filling the professorships of Technical Mathematics, Discrete Mathematics and Analysis with A. Schröder (2013), C. Fuchs and Verena Bögelein (2014) as well as the appointment of the current assistant professors Wolfgang Trutschnig (2013), Lothar Banz (2014), Volker Ziegler (2014) and Simon Blatt (2015).

A broader retrospective would definitely have to mention the other members of staff and the many lecturers at our institute/department and recognise their substantial contributions. The students, who have played a decisive role in the success story of mathematics in Salzburg, should also be given their own chapter in the history book of the institute/department.

**ADMINISTRATION AND ACADEMIC FUNCTIONS** The following have served as institute directors and department heads: A. Florian (1978-81 and 1988-92), P. Gerl (1981-84 and 1992-94), J. Czermak (1984-88), P. Zinterhof (1994-99), G. Larcher (1999-2000), W. Bauer (2000-04), P. Hellekalek (2004-06), F. Schweiger (2006-09), C. Buchta (2009-12), M. Thaler (2012-14), A. Bathke (2014-15). A. Schröder has been Head of the Department of Mathematics since 2015.

The chairpersons of the Study and Curriculum Commission for Mathematics were: F. Schweiger (1972-74), P. Gerl (1974-78), J. Linhart (1978-94 and 2005- 09), M. Thaler (1994-99), F. Österreicher (1999-2001 and 2004-05), W. Schmid (2001-04), R. Wolf (2009-2013). C. Fuchs has headed the Curriculum Commission since 2013.

This is an appropriate place to express our thanks and appreciation to the many secretarial teams at the Institute of Mathematics/Department of Mathematics. Their reliability, commitment and advisory impulses have made a decisive contribution to the success of the work and the creation of a good working atmosphere. The external image of the Institute/Department has also benefited greatly from the competence and friendliness of the secretarial teams.

It is not without pride that several senior academic posts have been filled by mathematicians. F. Schweiger was Dean of the Faculty of Natural Sciences for many years (1977-79, 1985-87 and 1999-2004), P. Zinterhof in the 1979-81 term of office, and Arne Bathke has held this office since 2015. F. Schweiger was also Rector of our University from 1987 to 1989. His speech in the Grosses Festspielhaus at Pope John Paul II’s meeting with the scientific community in June 1988 became known far beyond the borders of the academic world. F. Schweiger reported on its content in uni-aktuell, 1987/88 no. 9.

An entry in the roll of honour would also have been deserved by the many personalities from the wider university administration who have accompanied and continue to accompany the work of the institute/department with helpful and competent advice and action.

Dealing with the historical development of an institute/department necessarily includes an examination of the university organisation laws, of which mathematics in Salzburg has experienced four so far, enacted in 1955, 1975, 1993 and 2002. Of central importance for the period under review was the University Organisation Act 1975, which, in addition to the complete restructuring of the university, provided for the inclusion of all university groups in the decision-making processes at the various levels and, in particular, granted members of the academic mid-level faculty a high degree of autonomy in shaping the university. As can also be seen from this review, this opportunity was utilised decisively at our institute. The chronicle must also note that a great deal of working time went into the implementation and realisation of the organisational laws, which is little documented. There will be an opportunity to refer to the numerous reforms of the study laws and their equally labour-intensive implementation.

**RESEARCH** It should be noted that this review is mainly limited to the period up to 2012. The appointment of four new professorships since then has significantly changed the research landscape. A quick overview of the composition, specialisation and activities of the current working groups can be found on the Department of Mathematics homepage, as mentioned at the beginning of this report. Informative snapshots of the current research and teaching situation at the Department of Mathematics can be found in the brochure ‘Excursions in Mathematics’ from 1988 and the article ‘Mathematics in Salzburg’ by C. Fuchs, P. Hellekalek and M. Thaler in the Internationale Mathematische Nachrichten No. 222 (April 2013). The 15 volumes of the Yearbook of the University of Salzburg provide a good insight into the research activities of the Institute from the academic year 1968/69 to the academic year 1996/97.

**Fields of work **There is not enough space here to recognise and statistically evaluate the very productive research work that has been carried out at the institute/department over the many years. It remains only to mention a few important aspects and activities. The following fields of work were represented: Approximation theory, image processing, CAD systems, didactics and methodology of mathematics, dynamic systems and ergodic theory, functional analysis, function theory, geometry (discrete geometry, convex geometry, distance geometry, stochastic geometry, computer geometry, computational geometry), uniform distribution and metric number theory, harmonic analysis, information theory and cryptography, combinatorics and graph theory, Logic and foundations of mathematics, Monte Carlo and quasi-Monte Carlo methods, neural networks, numerical mathematics (number-theoretical methods, parallel algorithms, numerics of differential equations), econometrics, operations research, pseudo-random numbers, statistics (test theory, information-theoretical methods, robust and parameter-free statistics), actuarial mathematics, probability theory and stochastic processes, entertainment mathematics.

In the documents for the evaluation of the department in 2004/05, which will be discussed later, it was rightly emphasised that a large part of the research areas represented were characterised by proximity to stochastics, which provided a broad scientific communication basis. The research results have been recorded in countless publications and communicated locally and internationally in equally countless lectures. From 1979 to 1999, the Mathematical Institute regularly published working reports, which mainly served to disseminate and exchange preprints. These were continued by the Mathematical Reports, which were published on a case-by-case basis.

Contributions to applied research were made particularly in the areas of statistics and physical and scientific mathematical modelling, as well as in the predominantly computer-related areas that will be reported on later. The statistical issues dealt with resulted in many cases from the extensive consultancy work carried out continuously by W. Bauer and F. Österreicher in particular.

**Habilitations** A good yardstick for the academic performance of an institute/department is undoubtedly the number of habilitations achieved. Mathematics in Salzburg has an impressive record in this respect, with Wilhelm Fleischer (1973), Roland Fischer (1974), Johannes Czermak (1976), Johann Linhart (1977), Ferdinand Österreicher (1978), Walter Bauer (1979), Johann Stegbuchner (1980), Gerhard Racher (1982), Maximilian Thaler (1984), Wolfgang Woess (1986), Peter Hellekalek (1986), Franz Kinzl (1989), Gerhard Larcher (1989), Reinhard Wolf (1997), Michael Revers (2001), Wolfgang Schmid (2001), Erika Hausenblas (2003), Stefan Wegenkittl (2003). The latter two habilitations were the result of pure project work, as was that of Karl Entacher (2001), who was also able to achieve the research results for his habilitation within the framework of projects at the Department of Mathematics. The topics of the habilitation theses, several of which were honoured with the Christian Doppler Prize, can all be assigned to the research areas listed above.

**Projects** In the years from 1990 onwards, there was very lively project activity at the institute/department, starting with FWF projects in logic and geometry. Project work was particularly dense in the area of applied number theory/computational mathematics (uniform distribution, digital networks, number-theoretical numerics, parallel algorithms, pseudo-random numbers, cryptography, Monte Carlo and quasi-Monte Carlo methods). Further projects were dedicated to topics from the fields of metric number theory, ergodic theory, numerics of stochastic partial differential equations and automated analysis of paintings. In addition to the aforementioned habilitations, numerous dissertations, three of them sub auspiciis praesidentis, emerged from these projects, which were the building blocks for impressive careers.

**Scientific exchange** There is much to report about workshops, conferences and congresses that have taken place at the Department of Mathematics. The colloquia on discrete geometry and convex geometry organised by A. Florian, J. Linhart and C. Buchta were very popular. There were conferences and workshops on distance geometry, algebra, logic, ergodic theory, metric number theory and applied number theory/computational mathematics. Every four years, the Austrian Mathematical Society (ÖMG) organises a mathematics congress together with the German Mathematical Society (DMV). In 1977 and 1997, the Mathematical Institute in Salzburg organised the congress, the former with no less than 800 participants. Detailed reports on this, as well as on an Austrian Mathematics Meeting in Salzburg in 1983, can be found in the International Mathematical News No. 117 (November 1977), No. 177 (April 1998) and No. 136 (June 1984) published by the ÖMG. The ÖMG-DMV Congress will also be held in Salzburg this year.

The many encounters in the context of guest visits and guest lectures were also very enriching and fruitful. The yearbook reports repeatedly state that around twenty domestic and foreign mathematicians were invited to give guest lectures each year. In the 1970s, a ‘lecture book’ was compiled, which contains handwritten summaries of lectures and today represents a small treasure. A particularly prominent entry is the one from 10 November 1978: ‘Combinatorische Probleme der Geometrie’ (Combinatorial Problems of Geometry) by Paul Erdös, who is known to be one of the most important mathematicians of the 20th century.

In the first two decades of the Institute’s existence, great care was taken to foster scientific communication between the Institute’s members across the boundaries of their specialisms. This purpose was served, among other things, by a weekly Privatissimum, which was usually attended by all scientific members of the Institute. This was later replaced on the one hand by the ‘Mathematical Forum’ set up by P. Gerl, which aimed at faculty-wide scientific exchange, and on the other hand by private seminars on specialised areas of project and working groups.

**Honorary doctorate** As a worthy conclusion to this brief research review, ‘our’ honorary doctorates are listed below. The year in which the honorary doctorate was awarded by the University of Salzburg is added: Edmund Hlawka (1916-2009), University of Vienna (1981), László Fejes Tóth (1915-2005), Hungarian Academy of Sciences (1991), Jonas Kubilius (1921-2011), Vilnius University (1992), Rolf Schneider (born 1940), University of Freiburg (2004), Peter Gruber (1941-2017), Vienna University of Technology (2010).

**SPECIFIC EMPHASES IN TEACHING PRACTICE** Teaching has always had a high priority at our institute/department, fuelled by the endeavour to do justice to the principles and objectives of university studies, such as the unity of research and teaching and academic professional training in the strictest sense of the word. This section reports on specific emphases in teaching practice resulting from these principles. The development of curricula as a formal basis will be the subject of the next section.

Questions of teaching design were consistently a central topic of discussion among the teachers, both within and outside the institutionalised forums. The discussions about teaching practice were sometimes quite passionate and refreshingly controversial. The always very lively exchange with students and their numerous substantial impulses and suggestions contributed significantly to achieving a high degree of effectiveness. A correspondingly high level of energy and creativity was channelled into the didactic and methodological design of courses, the creation of teaching materials and the conception of special lectures.

**Specializations** The areas of specialization established over the years have been a hallmark of our educational offering. They are listed in the next section and arranged chronologically. In general, the specializations enabled a stronger application orientation on the one hand and a deeper penetration of a sub-area of mathematics on the other. Students with particular academic ambitions could thus be optimally prepared for international competition. It is very pleasing that the program was well received and that the goals set, such as an academic career, were achieved in an impressive manner on several occasions.

**Seminar weekends and Excursions** The open exchange between lecturers and students was particularly encouraged by courses on non-university premises, such as the stochastics weekends at the Veitenhof in Kössen/Tyrol, organized by F. Österreicher and M. Thaler, the logic and teacher training weekends in Mittersill, organized by J. Czermak and co-supervised by Karl-Josef Fuchs, as well as the geometry seminars at Lambach Abbey and Puchheim Monastery, which continue to be organized by C. Buchta and his team. One of the triggers for the early activities of this kind was the regret repeatedly expressed by students that no excursions are included in the mathematics degree course. To a certain extent, the desire for excursions was also met by visiting various facilities as part of courses, such as statistics lectures, where students could apply what they had learned in practice.

**Teacher training and further education seminars** A great deal of thought and organizational work has gone into the design and continuous improvement of teacher training courses. An important building block in this regard was the establishment of specialized courses specifically tailored to the teaching profession, a practice with which we played a pioneering role compared to similar institutes in Austria (and beyond). – Even at an early stage, activities were initiated at our institute that are now institutionalized by the School of Education, such as maintaining contact with schools. At the invitation of school institutions, numerous training seminars were held for teachers, some lasting several days, with the subject of stochastics/statistics being particularly popular.

**Close links to didactics and philosophy** Inseparably linked to the efforts just described is the very close and fruitful collaboration with the Department of Mathematics Education, which continues to be formally anchored through the partial assignment of K.-J. Fuchs to the Department of Mathematics. This success story began with the extension of F. Schweiger’s professorship in May 1973. The 1971-73 yearbook reports: “In the course of negotiations with Prof. Dr. Fritz Schweiger to ward off a call to the University of Vienna, a fourth chair for Mathematics was established and Prof. Schweiger’s chair was renamed Mathematics with special consideration of the methodology and didactics of teaching.” The incorporation of the mathematics didactics working group into the Institute for Didactics of Natural Sciences in 1978 did not stop the lively exchange. – Another success story grew out of the close contact in research and teaching that J. Czermak maintained with the Institute of Philosophy / Department of Philosophy of the GW/KGW Faculty. The combination of mathematics and philosophy has captivated numerous students and has had an extremely fruitful effect in both directions.

**CURRICULUM DEVELOPMENT** The creation and updating of curricula forms a significant part of the work balance of the 50 years of mathematics at the PLUS. This brief report provides an opportunity to draw attention to other specific concerns of the institute/department.

**Starting point and first major reform** At the time the institute was founded, the teacher training program and the doctoral program in mathematics were established, regulated by ordinances from 1937 and 1945 respectively. While the doctoral program today represents the third level of the Bologna system, it was designed as a one-level (!) program at the time. The path to the two-tier system with diploma and doctorate was paved by the General University Studies Act 1966 together with the Federal Act on Humanities and Natural Sciences Studies 1971. On May 18, 1972, the Study Commission (now the Curriculum Commission) for Mathematics met for the first time. The first curriculum in accordance with the new laws, which included diploma and teaching degrees, came into force in the summer semester of 1978. The long development period was caused by the parallel development of the study regulations, which required a high degree of consensus from all locations of the respective study programs in Austria. In Salzburg, mathematics was one of the first subjects with a new curriculum for diploma and teacher training. The curriculum for the new doctoral program came into force in the winter semester of 1980.

**Revisions and spealisations **The 1978 curriculum underwent several substantial revisions. Taking into account developments in the field of computers, IT training was made compulsory in the teacher training program as early as 1981, which can be seen as a pioneering achievement. The 1985 version of the curriculum brought an important and lasting change with the introduction of the area of specialization “Systems Analysis and Mathematical Modelling” on the initiative of P. Zinterhof and the area of specialisation “Statistics and Probability Theory” on the initiative of F. Österreicher and M. Thaler. In the first half of the 1990s, the curriculum was again fundamentally revised with the primary aim of bringing mathematics studies closer to the requirements of computer science. The range of specializations also changed: “Computational Mathematics” and “Discrete Mathematics” took the place of the first-mentioned specialization.

In 1997, the previous study laws were replaced by the University Studies Act (UniStG), which required all curricula to be redrafted. The new law provided for a consultation procedure that had to involve numerous bodies from politics, business and society in the creation of the curricula. A survey of a large group of our graduates in various fields of employment, conducted under the leadership of F. Österreicher, proved to be particularly fruitful and was concluded with a very stimulating plenary discussion. The new curriculum came into force in the winter semester 2000.

The amendment to the UniStG, which enabled the introduction of the Bologna structure with Bachelor’s and Master’s degrees (later renamed Bachelor’s and Master’s degrees), came in the middle of the work on the new curriculum. The Mathematics Study Commission quickly decided to take this step and the corresponding curriculum came into force in 2001. It was the first according to the new structure at a mathematics location in Austria and one of the first at the University of Salzburg.

A major innovation in terms of content was the focus on “financial and actuarial mathematics”, which C. Buchta developed into a full actuarial training course in the following years. The practical relevance and topicality of the course content soon attracted participants from numerous countries, making the project another success story.

The 2001 version of the curriculum provided for two master’s degrees, which were combined into one degree in 2006. A further redesign took place in 2008 after it became clear that the curriculum could no longer be implemented in its existing form due to the reduction in staff resulting from the wave of retirements. Without going into more detail about further developments, it should be mentioned that the plans for the Bachelor’s and Master’s degree programs in Mathematics were redesigned in 2013 in line with the new staffing situation and adapted to the framework curricula that had been developed in the meantime in 2016.

**Teaching degree and doctorate** With the UniStG 1997, a separate study commission was set up at faculty level for the teacher training program. The first curriculum approved by this commission came into force in 1999 and has undergone numerous revisions in the following years. Particularly pleasing for mathematics were the 2000 version, which introduced computer science and computer science management as a new subject, and the 2009 version, which introduced physics as a new subject. As early as the 1970s, mathematics had campaigned vigorously for the establishment of physics as a teacher training course, but was narrowly defeated. The teacher training curriculum has also undergone significant changes since 2013.

The doctoral study program, supervised by the Doctoral Study Commission at faculty level, has also seen numerous curriculum versions over the years, always guided by the concern to keep the scientific level of dissertations high and to make students competitive in the academic world.

The main burden of curriculum development and, as a rule, the planning and organization of teaching operations lay with the respective chairs of the study and curriculum committees. Special thanks are due to J. Linhart for his many years of extremely prudent and competent performance of this function.

**COMPUTER-RELATED SUCCESS STORIES** The history of mathematics at the PLUS coincides with the period of rapid development in the computer sector. As already mentioned, the Institute of Mathematics/Department of Mathematics was very open to this development and took it into account in a variety of ways. Here, too, only a brief and incomplete report can be given.

**Computer Science** The founding of the Department of Computer Science at our university on the initiative of P. Zinterhof is a success story that goes beyond mathematics. As reported, P. Zinterhof was appointed to the University of Salzburg in 1972. In the 1968/69 yearbook, the Dean of the Faculty of Philosophy had already stated in connection with the intended acquisition of an electronic data processing system for the University of Salzburg: “Above all, it has also become clear that the supervision of a larger computer must be in the hands of a representative of applied mathematics and that therefore the acquisition of such a system can only be considered in connection with the establishment of such a chair.” After his appointment, P. Zinterhof took over the management of the Interfaculty Institute Computer Center, which was founded in January 1972, and was then head of the successor institution, the Center for Electronic Data Processing, for many years.

At the instigation of P. Zinterhof, an Institute for Systems Analysis and a Research Institute for Software Technology were founded in September 1986. This was accompanied by the plan to establish a computer science program at the University of Salzburg. Once the groundwork had been laid, the curriculum for a computer science course was developed in an astonishingly short time. It was formally approved by an extended mathematics study commission, and the trial course was able to begin in the winter semester of 1988. The Institute for Systems Analysis was the nucleus of today’s Department of Computer Science, and the study experiment was the nucleus of today’s computer science studies. Mathematics was heavily involved in teaching in the early years, a contribution that paid off in many ways. – Politics and business were extremely interested in the establishment of computer science in Salzburg. Please refer to the interview with P. Zinterhof in uni-aktuell, 1986/87 no. 9.

**Computer-aided design (CAD)** The Memo Plot project for the development of CAD systems for the PC was also a success story linked to progress in the computer sector, especially that of small computers. It was started by J. Stegbuchner in 1983 as part of a research contract from the private sector and continued and expanded with enormous commitment until his death in 1998. CAD systems were developed for the construction of tools, machines and plants, for architecture and construction, advertising graphics, repro technology as well as geodesy and cartography.

Long-term members of the Memo Plot team were W. Bauer, J. Linhart and F. Kinzl. Memo Plot became a showcase project for applied research and knowledge transfer far beyond its practical objectives. Numerous trade fair visits, CAD seminars and specialist conferences have accompanied the team’s work. The public’s interest in this company can be measured by the fact that the project was presented at a press conference, as reported by uni-aktuell in December 1984. F. Kinzl gives a detailed account of the history of Memo Plot in his obituary of J. Stegbuchner (Arbeitsberichte 1999, issue 1). Please also refer to the article “CAD Einsatz in Geodäsie – Kartographie” by J. Stegbuchner in Ausflügen in die Mathematik, 1988.

Mathematics has also benefited from Memo Plot in many ways, for example in terms of equipment and training in the computer field. Today, the awarding of the Hans Stegbuchner Prize is an annual occasion to remember Memo Plot. In 1994, J. Stegbuchner donated a prize for outstanding academic achievements from his company’s funds, which was renamed the Hans Stegbuchner Prize after his death. After the bequeathed funds were exhausted, the University of Salzburg made the award possible for a time. For some years now, the prize has thankfully been financially supported by the Raiffeisenverband Salzburg.

**Applied number theory** Another success story associated with the development of the computer is the strong promotion of the research area “Applied Number Theory” since the early 1990s, namely by P. Hellekalek, G. Larcher, W. Schmid and P. Zinterhof, the later initiators of the area of specialization Computational Mathematics. In addition to the combination of theoretical research with application and knowledge transfer and the repeated organization of major scientific conferences, this area is characterized, as already mentioned, by very intensive project work, which has made an important contribution to the training and promotion of young scientists. The Institute of Mathematics/the Department of Mathematics has also benefited from the Applied Number Theory in terms of equipment and training in the field of computers.

**EVALUATION AND NEW START** Although the last two success stories mentioned here are separated by several years, they are closely linked in terms of subject matter. In 2004/05, at the suggestion of the Austrian Mathematical Society (ÖMG), the then Ministry of Education, Science and Culture carried out an evaluation of the research and teaching programs in mathematics at Austrian universities. Participation was optional for the institutions concerned. Although an extensive evaluation of the Faculty of Natural Sciences had taken place shortly beforehand, our department decided to take part. The local organizational work was carried out by P. Hellekalek in his dual function as head of department and head of the Salzburg branch of the ÖMG. After several months of intensive work, the department produced a comprehensive report on research, teaching and external impact. The evaluation commission visited the department on January 30, 2005 (it was actually a Sunday!) and the final report was presented to the Ministry on July 6, 2005. The evaluation objective formulated by the ÖMG, namely the creation of an “objective basis for future resource decisions”, was thus achieved.

The documents compiled for the evaluation are an impressive snapshot of the achievements of the Department of Mathematics in research and teaching, especially as the period in question, the years 2001-2003, was a very active project phase, C. Buchta had taken up his post in 2001 and the curricula had been redesigned and future-oriented at the turn of the millennium. The most important aspect of the evaluation in the longer term, however, was that in the course of the process, broad and comprehensive consideration was given to the possible future development of the department and a far-reaching consensus was reached on what its cornerstones should be. As a result of the age structure of the department members, it was clear that the department would have to be “restructured” in the following years.

The results and implications were summarized in a development plan for the Department of Mathematics. At the heart of this plan was the proposal to create professorships specializing in stochastics, discrete mathematics and analysis in addition to the existing professorship specializing in geometry. As already mentioned and as a glance at the current mathematics homepage shows, the vision from back then has now been realized. The number four probably still resonates with the founding years of the Institute of Mathematics. The idea of a professorship in Technical Mathematics, which was also successfully implemented, only came into focus with the establishment of the Bachelor’s degree course in Engineering Sciences in the 2006/07 academic year.

The new start at the Department of Mathematics mentioned in the section “Personnel development” was made possible by the Rectorate as part of the 2013-2015 target agreement. At the time, the author of this review was responsible for the creation of concepts and their coordination with the Rectorate’s plans as head of department. Even though the evaluation initiated by the ÖMG had already taken place some time ago, it had a strong influence on this phase, if only through the awareness that the far-reaching decisions that had to be made quickly would be met with broad approval in the department.

**ACKNOWLEDGEMENTS **Without wishing to shift responsibility for the content of this review and any incorrect information in the slightest, I would like to thank W. Bauer, C. Buchta, J. Czermak, A. Florian, H. Hagenauer, P. Hellekalek, F. Kinzl, J. Linhart, F. Österreicher, M. Revers, W. Schmid, F. Schweiger, R. Wolf and P. Zinterhof for their willingness to answer questions and for the informative exchange about the past of mathematics in Salzburg. The documents provided were also extremely helpful. Thanks are also due to Christoph Brandhuber, the head of our university archives, for his expert advice.

**SOURCES**

[1] C. Brandhuber: Aus Salzburgs Hoher Schule geplaudert. Hundert Mini-Traktate unter einen Hut gebracht. Reihe uni:bibliothek (Hg. U. Schachl-Raber). Müry Salzmann, Salzburg-Wien 2012.

[2] Verzeichniß aller akademischen Professoren zu Salzburg vom Jahre 1728 bis zur Aufhebung der Universität mit kurzen Nachrichten von ihrem Leben und ihren Schriften. Herausgegeben von einem Mitgenossen derselben. In der Mayrischen Buchhandlung, Salzburg 1813.

[3] Universität Salzburg 1622–1962–1972. Festschrift. Herausgegeben vom Akademischen Senat der Universität Salzburg, Redaktion Hans Wagner und Barbara Wicha. Universitätsverlag Anton Pustet, Salzburg 1972.

[4] Die Paris Lodron Universität Salzburg. Geschichte Gegenwart Zukunft (Hg. R. Reith). Müry Salzmann, Salzburg-Wien 2012.

[5] Ausflüge in die Mathematik. 21 Jahre Institut für Mathematik der Universität Salzburg (Hg. J. Czermak). Abakus Verlag, Salzburg 1988.

[6] C. Fuchs, P. Hellekalek und M. Thaler: Mathematik in Salzburg. Internat. Math. Nachrichten **222** (April 2013), 33–42.

[7] Jahrbuch der Universität Salzburg, Studienjahre 1968/69–1996/97 (15 Bände).

[8] Vorlesungsverzeichnis/Handbuch der Universität Salzburg, ab Sommersemester 1963.

[9] uni-aktuell/PLUS: Die Zeitschrift der Universität Salzburg/Paris Lodron Universität Salzburg, 1974– 1992/1992–2004.

[10] Mitteilungsblatt der Universität Salzburg, ab Oktober 1975.

[11] Arbeitsberichte des Instituts für Mathematik der Universität Salzburg, 1979–1999; Mathematische Berichte, ab 1999.

[12] Unterlagen der Studien- und Curricularkommission Mathematik an der Universität Salzburg, 1972– 2010 (J. Linhart, F. Österreicher, W. Schmid, M. Thaler).

[13] Private Unterlagensammlung von J. Czermak. Institut/Fachbereich Mathematik der Universität Salzburg.

[14] Internationale Mathematische Nachrichten (IMN). Herausgegeben von der Österreichischen Mathematischen Gesellschaft. Hefte Nr. 117, 136, 177, 222.