jsrc-sri.qxp 9/24/98 8:56 AM Page 1435 Conferences All requests will be forwarded to the appropriate orga- Joint Summer nizing committee for consideration. In late April all ap- plicants will receive formal invitations (including specific Research Conferences offers of support if applicable), a brochure of conference information, program information known to date, along with information on travel and dormitories and other local in the Mathematical housing. All participants will be required to pay a nomi- nal conference fee. Sciences Questions concerning the scientific program should be University of Colorado addressed to the organizers. Questions of a nonscientific Boulder, Colorado nature should be directed to the Summer Research Con- June 13–July 1, 1999 ferences coordinator at the address provided above. Please watch http://www.ams.org/meetings/ for future de- The 1999 Joint Summer Research Conferences will be held velopments about these conferences. at the University of Colorado, Boulder, Colorado, from Lectures begin on Sunday morning and run through June 13–July 1, 1999. The topics and organizers for the Thursday. Check-in for housing begins on Saturday. No seven conferences were selected by a committee repre- lectures are held on Saturday. senting the AMS, the Institute of Mathematical Sciences (IMS), and the Society for Industrial and Applied Math- ematics (SIAM), whose members at the time were Alejan- From Manifolds to Singular Varieties dro Adem, David Brydges, Percy Deift, James W. Demmel, Sunday, June 13–Thursday, June 17, 1999 Dipak Dey, Tom Diciccio, Steven Hurder, Alan F. Karr, Bar- bara Keyfitz, W. Brent Lindquist, Andre Manitius, and Bart Sylvain E. Cappell, Courant Institute Ng. Ronnie Lee, Yale University It is anticipated that the conferences will be partially Wolfgang Lück, Westfälische Wilhelms-Universität funded by a grant from the National Science Foundation Münster and perhaps others. Special encouragement is extended to junior scientists to apply. A special pool of funds expected Recently, researchers in topology, geometry and global from grant agencies has been earmarked for this group. analysis have been encountering some related issues in at- Other participants who wish to apply for support funds tempts to extend classical methods and results from man- should so indicate; however, available funds are limited, ifolds to more general settings of singular varieties. These and individuals who can obtain support from other sources are needed for applications because many of the natural are encouraged to do so. spaces that are the focus of current investigations are usu- All persons who are interested in participating in one ally singular. Examples include: representation spaces, of the conferences (women and minorities are especially such as those of fundamental groups of Riemann surfaces encouraged) should request an invitation by sending the arising in the study of low dimensional manifolds; mod- following information to: Summer Research Conferences uli space constructions in algebraic geometry; the singu- Coordinator, American Mathematical Society, P.O. Box lar spaces which arise in trying to study and classify gen- 6887, Providence, RI 02940, or by e-mail to [email protected] eral finite or compact group actions on manifolds; and no later than March 3, 1999. spaces produced in a variety of contexts by natural com- Please type or print the following: pactification procedures. 1. Title and dates of conference. The object of this conference will be to describe some 2. Full name. recent advances in this general area and to discuss and com- 3. Mailing address. pare questions, methods, and applications from a variety 4. Phone numbers (including area code) for office, home, of perspectives. Among other subjects, the conference will and fax. include talks on the following topics: 5. E-mail address. 1. Topological invariants and classifications of singular 6. Your anticipated arrival/departure dates. varieties. A toplogical motivation for studying singular va- 7. Scientific background relevant to the Institute topics; rieties is the study of group actions, because their orbit please indicate if you are a student or if you received your spaces are stratified spaces. A goal is to make some pow- Ph.D. on or after 7/1/93. erful new classification methods more accessible and ap- 8. The amount of financial assistance requested (or in- plicable by considering spaces arising from natural con- dicate if no support is required). texts. A natural question, which can be approached from NOVEMBER 1998 NOTICES OF THE AMS 1435 jsrc-sri.qxp 9/24/98 8:56 AM Page 1436 Conferences several viewpoints, is: what can be said about topological Atiyah conjecture, the Singer Conjecture and the zero-in- classification of algebraic varieties? the-spectrum conjecture which are related to algebraic K- Important recent developments in the theory of singu- theory, global analysis, and topological rigidity conjec- lar varieties are related to the study of their characteris- tures. tic classes. There are, in fact, several closely related theo- ries of characteristic classes developed by a number of workers. Among questions to be addressed are: What are Computability Theory and Applications the relations between different theories of characteristic Sunday, June 13–Thursday, June 17, 1999 classes and what do they reveal about geometric and an- alytic structures of the varieties? When can such classes, Peter Cholak, University of Notre Dame which for singular varieties generally take values in ho- Steffen Lempp, University of Wisconsin (co-chair) mology, be lifted back at least partially towards cohomol- Manuel Lerman, University of Connecticut (co-chair) ogy or to other theories; and what additional structures are Richard Shore, Cornell University (co-chair) necessary for such refinements? Intersection homology theory has been a powerful tool Computability theory (or recursion theory) is an area of for the investigation of intersection theories and numbers mathematical logic dealing with the theoretical bounds on in singular varieties. But its functoriality, analytical inter- and structure of computability and with the interplay be- pretation, e.g., its relation to L2 cohomolgy, as well as other tween computability and definability in mathematical lan- issues are still not fully understood. Moreover, it doesn't guages and structures. The field started in the 1930s with cover all the now needed settings, e.g., in symplectic geom- ground-breaking work of Gödel and Turing and has de- etry. This meeting will be a forum for such new issues in veloped into a rich theory with applications and connec- intersection theory. tions to areas ranging from computer science to descrip- 2. 3-manifold invariants and moduli spaces. Some sub- tive set theory as well as more traditional branches of tle invariants of 3-manifolds are related to representation mathematics, including algebra, analysis, and combina- spaces. But extensions of Casson's SU(2) ideas to more gen- torics. The meeting will focus on classical computability eral Lie groups encounter difficulties due to the more sin- theory, an area in which many recent advances have been gular nature of the representation varieties and require in- made, and those applications which currently seem most vestigations of Lagrangian subvarieties in singular directly connected to and most likely to benefit from these symplectic varieties. These invariants shoud be compared advances. In particular, applications in algebra, model the- with combinatorial invariants, e.g., those related to Vas- ory, and proof theory will be highlighted. Lectures will siliev's perspective on knot theory and to “finite type in- stress open problems, their relationship to some of the re- variants” of 3-manifolds. cent advances, and further obstacles which need to be Representation varieties have been investigated by al- overcome to solve the problems. Problems, primarily from gebraic geometers as moduli spaces of holomorphic G-bun- the following areas, will be discussed. dles over a Riemann surface. Related moduli objects (e.g., 1. Classical computability theory. There have been a of Higgs bundles, of the moduli spaces of stable k-pairs, number of major advances in the understanding of sub- etc.) arose through interactions with physics. Such mod- structures of the Turing degrees in the past few years, in- uli spaces exhibit similar symplectic and Kahler structures cluding a phenomenal number of solutions to diverse prob- as well as gauge theory interpretations. It will be desirable to compare different treatments of singularities. lems that had been open for decades and had always been 3. L2-Betti numbers and L2-torsions. L2-invariants such considered very hard. In addition, substantial progress as L2-Betti numbers, Novikov-Shubin invariants, L2-tor- has been made towards the solution of other problems. Re- sions can be defined analytically in terms of the heat ker- sults have been obtained about automorphisms of these nel of the universal covering as well as topologically. This structures, characterizing definable sets and relations, and gives fruitful links between analysis and topology with decidability and undecidability of fragments of elementary applications in differential geometry, topology, group the- theories of the structures. These results will be discussed ory and algebraic K-theory. with an eye towards the limitations of the methods and the Connections to the first topic come from the relation obstacles which need to be overcome