# Obituary: Professor Erik Kjær Pedersen has passed away

It was with great sadness that the members of the department learned that Erik Kjær Pedersen passed away on May 24, 2020, at a hospital in Florida after a long period of illness. Erik was 74 years old.

From 2007 until his retirement in 2016, Erik was a highly revered head of department, who played a transformative role in the development of the department and of Danish mathematics more generally. During a topology conference in Münster in 2006, he was encouraged by Jesper Grodal and Ib Madsen to apply for the newly opened position of Head of Department at the Department of Mathematical Sciences in Copenhagen. Erik recognized the great opportunities that the position offered, so he immediately phoned his wife Inger and they agreed to uproot their life in the US and return to Denmark. An expected first grandchild likely weighed in the decision, too, but the episode is characteristic of the enormous commitment with which he met the job as head of department. Later, in 2014, when sudden efforts were made to merge the department with the Department of Computer Science, it affected him so deeply that he was hospitalized for several days.

Under Erik's knowledgeable and committed leadership, the Department of Mathematical Sciences underwent an exceptional development on all fronts. The department's international standing was improved significantly, so that it now competes favorably with the best European mathematics departments. On a number of occasions, it has also competed successfully with top US institutions. The strengthened research profile brought along an explosion in the number of external grants, many of which were highly prestigious.

During Erik's tenure as head of department, members of the department thus obtained three ERC Advanced Grants, two ERC Consolidator Grants, two ERC Starting Grants, a Center of Excellence and a Niels Bohr Professorship from the Danish National Research Foundation, three Excellence Grants from the University of Copenhagen, three Sapere Aude grants from the Danish Research Council, a grant from the Danish National Advanced Technology Foundation, a Center Grant from the Villum Foundation, and a number of other grants.

A total of twenty-three faculty members in the department were PI on projects with an end date in the period 2013-2017. While, in 2006, external funding amounted to DKK 7.3 million, this income had grown in 2016 to DKK~35 million, a fivefold increase during Erik's tenure.

During this period, the number of PhD students and postdocs also saw a dramatic increase, so that the department today boasts a flourishing and very lively environment of young researchers from all over the world. This has made the department an attractive and inspiring place to work for mathematicians of all ages, and over time it will give the department a very large network of international contacts.

The permanent faculty also saw significant change during Erik's tenure. Upon his arrival at the department in 2007, he presented a list of prominent Danish mathematicians, including a statistician, whom he wanted to hire. Most on the list were employed abroad, in many cases at top universities. It was an ambitious recruitment plan, but when Erik retired in 2016, everyone on the list was employed at the University of Copenhagen with one exception. In parallel to this plan, Erik pursued what he called an opportunity-driven employment policy in which he used his large international network to identify excellent researchers who might be recruited. The hirings were made with the clear objective of creating a balance between research areas comparable to that of leading mathematical institutions, but most importantly, Erik pursued a very responsible employment policy and only made appointments that strengthened the research profile of the department.

Erik came to Copenhagen after 18 years as Professor at Binghamton University in Upstate New York, where he also served as head of department for two periods. One of his most important visions was to bring the way that the department operates as close to that of an American university department as was possible under Danish law. In his view, this was a prerequisite for the department to be able to be a player at a high level internationally. Erik found the existing German-inspired system with only a handful of full professors to be destructive for the department's interests. He also considered it a prime reason that several prominent Danish mathematicians had left the country for careers abroad. Therefore, he instituted the American system, where all members of the permanent faculty can expect, over time, a promotion to full professor. Another noticeable consequence of this new modus operandi is the annual postdoc call that covers all research areas represented in the department. The call attracts a large number of highly qualified applicants and competes favorably with top institutions worldwide.

The department's educational responsibilities were also of great importance to Erik. In 2009, the study program of mathematics was revised significantly with a focus on offering more freedom of choice. The revision led to a significant increase in the number of MSc graduates in mathematics, and as a by-product, the number of MSc graduates in statistics also grew dramatically over the same period. In addition, the number of annual PhD graduates grew from two to fifteen, while Erik was at the helm.

Erik had a very positive impact on the culture in the department. He believed that all areas of mathematics are important for a well-functioning department, as long as all groups constantly strive to improve. Erik's respect for high quality research regardless of the field caused the mutual understanding and the respect between the various groups at the department to grow substantially. Out of a department with limited contact and considerable suspicion between research groups, not least between the pure and applied branches, grew an environment with an unusual team spirit and a common ambition to become better, not least by good recruitments. Erik had gift for making people feel appreciated and for bringing the best out of them.

Finally, as a consequence of Erik's responsible management of the finances, the department has a sound economy, including substantial savings.

## Erik's research career

By his own reckoning, Erik's tenure as head of the Department of Mathematical Sciences at the University of Copenhagen was the most significant achievement of his career. However, he was also an internationally highly regarded research mathematician. His research career began at the department of mathematics at Aarhus University, which Sven Bundgaard built in the 1960's, and which Erik would later use as a model for his own chairmanship. There, he was a part of the large group of Danish mathematicians, who were ecudated by Leif Kristensen, and who came to play an influential role in mathematics on an international level within the field of topology.

Erik’s first mathematical work was a dissertation, written in collaboration with Leif Kristensen, on two-stage Postnikov systems and secondary cohomology operations. This was an important area of research in the 1960's with Frank Adams as the leading figure.

After Aarhus, Erik moved to Chicago, where he enrolled in the graduate program at the University of Chicago 1971–1974 with Richard Lashof as his advisor. During his studies at Chicago, he changed focus from the more algebraic aspects of topology to the geometric study of manifolds. He became an expert in topological manifolds, which, as an area, saw explosive growth following the investigation by Rob Kirby and Larry Siebenmann of the relationship between piecewise linear manifolds and topological manifolds.

In 1956, John Milnor had made the astonishing and unexpected discovery that, on the 7-sphere, there exists non-standard smooth structures. More precisely, there are 24 different subsheaves of the sheaf of continuous functions that, locally, are isomorphic to the subsheaf of standard smooth functions. Milnor’s discovery launched the area of surgery theory, whose methods give a phenomenally effective way of describing all manifolds, up to isomorphism, within a given homotopy type. Surgery theory and the concurrent evolution of topological $K$-theory via Bott periodicity and the Atiyah-Singer index theorem became a dominant field of research from the 1960's onwards. In its original formulation, surgery theory only dealt with compact manifolds, but later on, a larger group of mathematicians, including Erik, expanded the theory to the non-compact case under the slogan of Controlled Topology.

In 1974, Erik returned from Chicago to a position at Odense, where he along with Hans Jørgen Munkholm became a driving force in the development of the new mathematics department. During his time in Odense, Erik's research interests turned to algebraic $K$-theory. In a landmark paper from 1984, Erik introduced a new definition of negative $K$-groups, which, in retrospect, may be his most influential mathematical contribution. The negative $K$-groups had been introduced by Hyman Bass in the 1960's, and in the early 1970's, Doug Anderson and Wu-Chung Hsiang, Frank Quinn, and others uncovered their geometric relevance as obstruction groups for various geometric questions, all related to the relationship between piecewise linear and topological manifolds. Kirby’s torus trick, which is a compactification process, was a central tool. The central player in Erik’s new definition of the negative $K$-groups was the category of $\mathbb{Z}^n$-graded modules with bounded homomorphisms as morphisms. More precisely, it is Whitehead’s group $K_1$ of this category that defines the negative $K$-group $K_{1-n}$.

After close to fifteen years in Odense, Erik left Denmark for the professorship at Binghamton University, where his closest colleagues were Steve Ferry and Tom Farrell. There, Erik developed, in collaboration with first Chuck Weibel and Ferry and later Ian Hambleton and others, the theory of Controlled Surgery. The $\mathbb{Z}^n$-graded modules from the 1984 paper were replaced by non-compact manifolds equipped with a reference map to Euclidean $n$-space or, more generally, to a fixed metric space. The allowed morphisms are the continuous or smooth maps that, measured in the metric space, are bounded or controlled. With this construction, the necessity of Kirby’s torus trick was eliminated. Controlled Surgery proved to be a very fruitful point of view and became, in the hands of Erik, Hambleton, Quinn, Andrew Ranicki and others, a dominant research area from the 1990's and onwards. In addition to his contributions in the development of Controlled Topology, Erik, in collaboration with other mathematicians, has used the theory to solve a number of concrete problems of a geometric nature. Notably, Erik’s long collaboration with Hambleton has resulted in a pair of spectacular results on symmetries of manifolds.

The first result gave a complete answer to a question raised by Terry Wall. In the mid-1950's, Milnor had discovered that the dihedral group $D_{2q}$ cannot act freely on a sphere of any dimension. This was a delicate question, since it is possible for the dihedral group to act freely on a topological space of the same homotopy type as a sphere. In the same vein, Wall’s question concerned countable groups that admit a free action on a topological space of the same homotopy type as a sphere, namely, he asked if such a group can act freely on the product of an sphere and a Euclidean space. Erik and Hambleton’s complete resolution of this question is published in the first volume of J. Amer. Math. Soc.

The second result concerns Georges de Rham’s Similarity Problem. In 1935, de Rham asked whether two real linear representations of a finite group that are conjugate via a homeomorphism are also linearly equivalent. De Rham seemed to think that this was the case, but in 1981, Sylvan Cappel and Julius Shaneson discovered a counterexample for representations of the cyclic group of order eight. Shortly thereafter, Hsiang and Bill Pardon and Madsen and Mel Rothenberg independently and with quite different arguments showed that de Rham's inkling was correct, as far as groups of odd order are concerned. Subsequently, Shmuel Weinberger and Mark Steinberger and others investigated the problem for a number of groups of even order. In 2005, Erik and Hambleton published the definitive solution to the problem in a paper published in Ann. of Math. For cyclic groups of even order, the answer to de Rham’s question depends on delicate number theoretical considerations, and the relevant number theory was examined in a later paper. For both of these results, controlled

surgery was a key tool.

In the mid-1950's, Armand Borel raised the question as to whether two closed and connected aspherical manifolds of the same dimension and with isomorphic fundamental groups are homeomorphic. Such a manifold is homotopy equivalent to the classifying space $BG$ of its fundamental group $G$, which is necessarily an infinite group. In its modern formulation, the question asks whether the assembly map in $L$-theory for $BG$ is an equivalence. By comparison, Sergei Novikov's 1970 conjecture on the homotopy invariance of higher signatures is equivalent to the statement that the same assemply map is rationally injective. There is a similar assembly map in algebraic $K$-theory, for which Hsiang has promoted the analogous questions. A large number of mathematicians have contributed to the understanding of assembly mapping, including Erik.

In 1981, Farrell and Hsiang launched an attack on the Novikov conjecture for groups $G$ that appear as fundamental groups of closed and connected Riemannian manifolds of everywhere negative sectional curvature. The method was essentially geometrical with the geodesic flow as its main tool. Later, in the 1980's, Farrell and Lowell Jones developed the geometric method in a number of spectacular papers. In 1991, Erik and Gunnar Carlsson published a paper proving the Novikov conjecture for a number of spaces $BG$ with a property that can be construed as a translation of the curvature condition into homotopy theory. Over the last 20 years, the topology group in Münster around Wolfgang Lück has affirmed the Novikov conjecture and the Borel conjecture for a very long list of groups, which, however, still are required to satisfy certain group theoretical curvature conditions. It remains an open question whether it suffices for $BG$ to be homotopy equivalent to a finite $CW$-complex in order for the assembly map to be an equivalence. This question continued to occupy Erik until his untimely passing.

Erik published a total of 61 mathematical research papers.

## The starting point was Vivild

Erik grew up in the small town of Vivild in Northern Djursland, where his father was the manager of the local co-op. He was fond of comparing his life in Vivild with his later life and said, among many other things, that it was when he left Vivild that his life became complicated, rather than later when he moved abroad. While in the US, he and his family spent many of their summers in their vacation house in nearby Fjellerup, and Erik was a frequent visitor at the Department of Mathematics in Aarhus.

Everyone who has known Erik will remember his straightforward way of being. He always spoke his mind plainly and unambiguously, but at the same time, he would listen to and carefully considered what others had to say. Important decisions were contemplated several times with inputs from trusted colleagues. As head of department, he announced that he would be available every day from 12:30 in the common room to discuss various department issues. The commute by train to his home, first in Snekkersten and later in Fredensborg, was made without luggage of any kind and was spent on strategic thinking. In the last few years, however, he would bring his Kindle full of good literature.

Erik leaves behind his wife Inger and three children Anna, Jesper, and Emil, their spouses and several grandchildren.

*Lars Hesselholt, Ib Madsen, Michael Sørensen*