curriculum vitaemath.usask.ca/~skuhlman/cv2009rd.pdf · 2009-05-20 · salma kuhlmann: curriculum...

1

Dr.Salma Kuhlmann

Professor

University of SaskatchewanResearch Center for

Algebra, Logic and Computation

McLean Hall, 106 Wiggins Road

Saskatoon, SK S7N 5E6

Telephone:

(306) 966 6129 (office)

(306) 665 7849 (home)

homepage:

http://math.usask.ca/skuhlman/index.htmlemail: [email protected]

20/ 05/ 2009

CURRICULUM VITAE

EDUCATION.• Mathematische Fakultat, Universitat Heidelberg, GermanyHabilitation and Venia Legendi in Mathematics (June 1999)Habilitation Thesis: Ordered Exponential Fields.Habilitation talk: On Shelling E8-Quasicrystals (after Moody and Weiss) followed by aColloquium talk: Quasicrystals and Icosians (after Moody and Patera).• Universite Paris 7, FranceDoctorat de Mathematiques (July 1991)Thesis: Quelques Proprietes des Espaces Vectoriels Values en Theorie des ModelesAdvisor: Professor Daniel Lascar• Diplome d’Etudes Approfondies en Mathematiques (June 1984)Advisor: Professor Daniel Lascar• McGill University, Montreal, CanadaBachelor of Sciences: First Class Honours in Mathematics (June 1981).

POSITIONS.• Professor (with tenure), University of Saskatchewan (since July 2003)• Associate Professor (with tenure), University of Saskatchewan (appointed July 1997)• Visiting Member at the Fields Institute, Toronto (academic year 1996-97)• Habilitation Fellow of the German Research Association (1995-97)• Visiting Professor at the Punjab University, Chandigarh, India (1995)• Post-Doctoral Fellow of the University of Heidelberg (1992-94)• Graduate Assistant, Mathematics Institute, University of Heidelberg (1988-89)• Doctoral Fellow of the French Government, University of Paris 7 (1984-87).

Salma Kuhlmann: Curriculum Vitae 2008 (14 pages) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

HONOURS and GRANTS.• Natural Sciences and Engineering Research Council of Canada (NSERC)Discovery Grant 2005–2010 (18,000 $/year)• Natural Sciences and Engineering Research Council of Canada (NSERC)individual research grant 2001–2005 (17,000 $/year)• Natural Sciences and Engineering Research Council of Canada (NSERC)individual research grant 1998–2001 (16,000 $/year)• President’s NSERC fund from the University of Saskatchewan (1997–1998)• Habilitation grant from the German Research Association (1995-97)• Research grant from the University of Heidelberg (1992-94)• Ph.D. Fellowship from the French Government (1984-87)• Graduate Grant from Logic Research Fund of Professor Makkai, McGill University.• Scholarship from Government of Quebec (1981-82)• NSERC Undergraduate Summer Award (1980)• Prize: “Morris W. Wilson Memorial Scholarship”, McGill University (1979)• Award: “University Scholar” award by McGill University Scholarships Committee,given to recognize outstanding academic achievements (1979).

RESEARCH INTERESTS.

Mathematical logic, algebra, and analysis. I am working in: model theoretic algebra (o-minimalstructures), field theory (ordered fields, valued fields, exponential fields, Hardy fields, exponential-logarithmic power series fields, transexponential functions), ordered algebraic structures (lexico-graphic orderings, ordered vector spaces, Hahn groups), models of open induction (primes andirreducibles in exponential integer parts), real algebraic geometry (Positivstellensatze, MomentProblems, the Theory of Invariants). Further interest in: models of arithmetic, differential fields,ordered permutation groups, geometric model theory.

TRAINING OF HIGHLY QUALIFIED PERSONNEL.

Post Doctoral Fellows:• Katarzyna Osiak visiting post-doctoral fellow August - October 2008.• Mickael Matusinski visiting post-doctoral fellow March - September 2008.• Andreas Fischer visiting post-doctoral fellow April 2006 - August 2008.• Antongiulio Fornasiero visiting post-doctoral fellow January-March 2005.Currently Post-doctoral fellow Universitat Freiburg 2007 - 2008.• Ulrich Hermisson visiting post-doctoral fellow October 2004 - March 2005.• Roland Auer, co-supervised 2002 - 2003.Currently holds a tenure track position at Robert-Bosch-Gesamtschule, Gottingen (Germany).• Jaka Cimpric, co-supervised. Cimpric was a NATO PDF 2001 - 2003.Currently Associate Professor, University of Ljubljana, Slovenia.• Mikhail Kotchetov, co-supervised 2002 - 2003.Currently holds a tenure track position, Memorial University, New foundland, Canada.• M. El Bachraoui, visited upon invitation April 2003.Currently Professor at al Akhawayn University in Morocco.• Matthias Aschenbrenner visiting post-doctoral fellow March 2001.Currently Associate professor at University of California, Los Angeles.• Marcus Tressl visiting post-doctoral fellow March - May 2002.Currently holds tenure track lecturer position University of Manchester, UK.


Graduate Students:• Mehdi Ghasemi Pd.D. University Scholarship, supervise September 2008-• Tim Netzer visiting pre-doctoral fellow (Universitat Konstanz) May - September 2007.• Jonathan Lee, supervised 2003 - 2005.Currently Ph.D. student in Mathematics, Stanford University.• Manuela Haias M. Sc. , 2004 - 2007. Co-supervised [58].Currently Mathematics Instructor, SIAST Woodland Campus, Prince Albert, Saskatchewan.• Wei Fan M. Sc. , 2004-2006. Co-supervised [59].Currently Ph.D student in mathematics, Michigan State University.• Pawel Gladki Ph. D. , co-supervised 2004-2007.Currently Post-doctoral fellow, University of Santa Barbara, California. 2004.• Trevor Green M. Sc. , supervised [60] 2000–2002.• Hyunku Chun M. Sc. , supervised [61] 1999–2001.Currently Professor of Mathematics at College in Seoul, South Korea.• Mahdi Zekavat Ph. D. , co-supervised [62] 1997–2000 .Currently Professor of Mathematics at Shiraz University, Iran.

NSERC Summer Undergraduate Awardees:• Michael Hancock supervised [63], Summer 2005.• Jonathan Lee supervised [64], Summer 2003.• Brian Danchilla supervised [65], Summer 2003.• Jonathan Lee supervised [66], Summer 2002.• Chira Chilliack co-supervised [67] Summer 2000.• Amy Mader co-supervised [68], Summer 1999.Currently analyst for the Treasury Board Branch, Saskatchewan Department of Finance.• Chris Arbuthnott co-supervised [69] Summer 1999.Later completed M. Sc. in Mathematics, University of Waterloo.

TEACHING RECORD AT THE UNIVERSITY OF SASKATCHEWAN.1. Linear Algebra Math 266 (for honours students) (3 cu)2. Linear Algebra Math 264 (3 cu)3. Algebra Math 360 (for honours students, 6 cu): group theory, ring theory, field theory4. Algebra Math 862 (for graduate students, 3 cu): Galois Theory5. Number Theory Math 364 (3 cu)6. Linear Algebra and Geometry Math 358 (6 cu)7. Real Analysis Math 371 (for honours students, 3 cu)8. Topics in Algebra Math 872 (graduate students, 6cu) : Ordered Exponential Fields.9. Abstract Algebra Math 363 (3 cu)10. Calculus Math 101 (3cu)11. Calculus for Pharmacy Math 115 (3cu)12. Topics in Algebra Math 872 (graduate students and faculty, 6cu) : Ordered Structures.13. Calculus Math 116; Integration Theory (3cu)14. Calculus Math 110; Differential Calculus (3cu)15. Linear Algebra II Math 366 (3 cu)


COMMITTEE WORK.• NSERC Grant Selection Committee 336 in Pure Mathematics; member 2005–2008.• Association for Symbolic Logic (http://aslonline.org/): Member of ProgramCommittee for the annual meeting held in Montreal May 17-21 2006.• European Network “ Real Algebraic and Analytic Geometry” (http:// www.ihp-raag.org):Local Coordinator for the Canadian Team application.• College of Reviewers Canada Research Chairs Program Member.• College Graduate Studies & Research, faculty representative, U of S Council, 2002-2003.• Director; University Math Help Centre 2001-2003.• Colloquium Chair 1997 - 2003, http://math.usask.ca/colloquia/index.html.• College of Arts and Science Equity Committee member 2000–2003.• University of Saskatchewan Faculty Association member Committee onWomen issues 2002-2003.

CONFERENCE ORGANIZATION.

• International Conference and Workshop on Valuation Theory (Saskatoon, 1999).For list of speakers and program see http://math.usask.ca/fvk/annconf.html Sponsors: Fields, CRMand PIMS, U of S. With the Conference and Valuation Theory Home Pages that we created, it hasput Saskatoon on the map as the center for Valuation Theory.

• First Annual Colloquiumfest (Saskatoon, March 2000). For list of speakers and program seehttp://math.usask.ca/fvk/Mb.htm.

• Second Annual Colloquiumfest in Real Algebraic Geometry and Model Theory (Saska-toon, March 2001). For list of speakers and program see http://math.usask.ca/fvk/Mb2.htm.

• CMS Summer Meeting 2001 in Saskatoon http://math.usask.ca. Organized with B. Hart andF.-V. Kuhlmann Session on Model Theoretic Algebra, plenary speaker Z. Chatzidakis.

• Third Annual Colloquiumfest in Algebraic Geometry, Real Algebraic Geometry andComputational Algebra (Saskatoon, March 2002).For list of speakers and program see http://math.usask.ca/fvk/Mb3.htm.

• Saskatchewan Algebra and Number Theory Mini-meeting (Regina, 2002). Co-organizedwith Dr. A. Herman.For list of speakers and program see http://www.math.uregina.ca/ aherman/skant02.html.

• Saskatchewan Mini-meeting, (Saskatoon, 2003).For list of speakers and program see http://math.usask.ca/fvk/skantm2.htm

• Fourth Annual Colloquiumfest on Model Theory and Ordered Algebraic Structures(Saskatoon, March 2003).For list of speakers and program see http://math.usask.ca/fvk/mb4.htm.

• Sixth Annual Colloquiumfest on Positivity, the Multi-Dimensional Moment Problemand Noncommutative Real Geometry. (Saskatoon, April 2005).For list of speakers and program see http://math.usask.ca/ marshall/collfest.html

• Mini-Conference Topics in Real Algebraic Geometry (Saskatoon October 2006). Dedicatedto Professor Danielle Gondard, on the occasion of her 60th Birthday.http://math.usask.ca/ skuhlman/confgond.htm

• Banff International Research Station (October 2006). Co-organized with S. Lall, V. Powers, F.Sottile. Workshop on Positive Polynomials and Optimization . http://www.pims.math.ca/birs


• The Fields Institute (April 2009). Co-organized with M. Matusinski. Workshop on DifferentialKaplansky Theory. http://www.fields.utoronto.ca/programs/scientific/08-09/o-minimal/miniworkshops/

RESEARCH SEMINARS and WORKSHOPS.

• Weizmann Institut, Israel (July 2008): Invited to give a mini - course on “ Positive Polynomialsand the multi-dimensional Moment Problem”.

• Universite Paris 6 (October 2007): Invited Professor to give a mini - course on “ Hilbert’s 17thProblem and Positive Polynomials”.

• Centre Emile Borel, Institut Henri Poincare (Fall 2005): Invited Professor to give a mini -course on “Positive Polynomials and the Moment Problem”.

• Groupe de Travail ”o-minimalite, Transseries et Modeles non-Archimediens”Co-organized with Professor J.-P. Ressayre, 2003–2004, resumed October -December 2007.http://www.logique.jussieu.fr/www.rambaud/GTL.html

• Seminar on Hardy Fields 2002–2004. Seminar notes available on my homepage (ps file).

• The Algebra and Logic Seminar since 1997. (http://math.usask.ca/fvk/algsem.htm). Maintopics: Model Theoretic Algebra and Real Algebraic Geometry. Speakers included: M. Aschenbren-ner, V. Astier, E. Becker, I. Bonnard, H. Brungs, Z. Chatzidakis, J. Cimpric, M. Dickmann, R.Dorrego, P. Ehrlich, M. El Bachraoui, A. Fischer, J. Funk, M. Galatos, D. Gondard, T. Green, D.Haskell, U. Hermisson, K. Keimel, S. Khanduja, H. Knaf, F.-V. Kuhlmann, S. Kuhlmann, M. Mar-shall, C. Miller, E. Mosteig, M. Otero, H. Perdry, V. Powers, A. Prestel, S. Priess, M. Putinar, P.Roquette, H. Schoutens, C. Scheiderer, N. Schwartz, M. Schweighofer, K. Schmudgen, P. Speissegger,M. Spivakovsky, C. Steinhorn, B. Teissier, M. Tressl, V. Vinnikov.

• Mathematical Sciences Group Colloquium Series 2001-2003.http://math.usask.ca/fvk/colloqg8.htm). The aim was to bring mathematicians (in particular, logi-cians) and computer scientists together, by inviting talks on topics of interest to both groups.

• Cryptography Student Seminar for students in mathematics and computer science organizedwith R. Auer, Summer 2002. http://math.usask.ca/ marles/seminar.html.

• Geometric Algebra seminar (1999-2000) (my graduate students lectured on Artin’s book).

• Seminar on Sturm Algebra (Summer 1999) for my undergraduate NSERC Students.

• Seminar on Quasicrystals (May 1999) invited Prof. Moody, Prof. Weiss, Prof. Baake.

• Algebra and Model Theory of Exponential Fields Mini-Course given at Louisiana StateUniversity, Baton Rouge, USA (Spring 1996).

• Model Theory (co-organized with Prof. Gloede), University of Heidelberg (1995).

• Exponential Fields (co-organized with Prof. Roquette), University of Heidelberg (1993-94).

RESEARCH VISITS.

• Punjab University, Chandigarh, India (March 2009).

• The Fields Institute, Invited to participate to the Thematic Program on o-minimal structuresand real analytic geometry (January–July 2009).

• Sabbatical Leave July–December 2007, University of Ljubljana, Slowenia, Equipe de Logique,CNRS, Univ. Paris 7, LAAS, Toulouse. (September–December 2007).

• Newton Institute: Invited to participate to the Model theory research semester, University ofCambridge, UK, spring 2005.


• Centre Emile Borel “Invited Professor” to the research trimester “Real Geometry”, Paris Septem-ber 12 to December 16, 2005 Institut Henri Poincare. I have been asked to give a course on “PositivePolynomials and the Moment Problem”.

• University of Norwich, UK Invited by Prof. M. Dzamonja, May 2005.

• Sabbatical Leave September 2003–July 2004. Equipe de Logique, CNRS, Univ. Paris 7.

• Universidad Autonoma Madrid One week (invited by Prof. M. Otero, November 2003).

• University of Ljubljana, Slovenia Two weeks (invited by J. Cimpric, January 2004).

• University of Regensburg, Germany One week (invited by M. Tressl, February 2004)

• University of Bonn, Germany Two weeks (invited by Prof. Koepke, May 2004)

• TU Darmstadt, Germany 1 week (invited by Prof. K. Keimel, July 2003)

• Fields Institute Set Theory and Analysis Program, 1 week (Fall 2002)

• Rutgers University 1 week (invited by Prof. S. Shelah, October 2001)

• Departement de Mathematiques, Universite du Maine Le Mans, France, 1 week (invited byProf. Giraudet, June 1998).

• Mathematical Sciences Research Institute, Berkeley, Visiting Member for Model Theory ofFields Program (May - June 1998).

• Fields Institute, Toronto, Visiting Member Emphasis Year on Algebraic Model Theory: GeometricModel Theory, and Model Theory of Analytic Functions (1996-97).

• Special Semester on Real Algebraic Geometry and Ordered Algebraic Structures LouisianaState University, Baton Rouge. Participant, (Spring 1996).

• Bowling Green University 1 week (invited by Profs. C. Holland & S. McCleary, 1995).

• Tata Institute, Bombay. 1 week (invited by Prof. Parimala , March 1995).

• Punjab University Chandigarh, India. 3 months research visit (invited by Profs. Sankaran andKhanduja).

RESEARCH VISITORS.

since 1997, at least 50 visitors (and more around the Valuation Theory Conference and the Colloqui-umfest) from abroad visited our Algebra and Logic Research Unit, for periods varying from a few daysto several weeks. In addition to these, the following mathematicians came for longer research visits:Alexander Nenashev visited upon invitation 2001–2002. He currently a tenure track position inmathematics at York University. D. Gondard (several visits). M. Otero (March 2003). N. Schwartz(several visits), Victor Vinnikov (several months on sabbatical 2007-2008).

OTHER CONTRIBUTIONS RELEVANT TO MY ACADEMIC CAREER.

• Research Unit Algebra and Logic: (http://math.usask.ca/fvk/algg.htm). The Algebra andLogic Group was founded by us in 1997. It has been approved as a Research Unit in the College ofArts and Science in August 2002.See http://www.usask.ca/communications/ocn/02-sep-06/feature06.shtml.

• Student recruitment and exchange programs We have succeeded in establishing a network ofcollaborators (in France and Germany mainly), and are actively advertising for student recruitment.We have exchange programs with the universities of Greifswald, Marburg, Oldenburg and Rostock(Germany), we established such with the T. U. Darmstadt (in collaboration with our colleague Prof.K. Keimel). We have a collaboration and joint proposal with the Algebra Group of Paris 6 (Prof.


D. Gondard), and strong contacts to the Logic Group in Paris 7. Further contacts to the groups inPassau, Leipzig, Munster, Dortmund (Germany). We have started an initiative to establish exchangeprograms with Shiraz University (Iran) and Cairo University (Egypt).• I was invited to participate in the podium of the “Canada–German Research AssociationAlumni Day” at the Canadian Embassy in Berlin, May 13th, 2005. The event was to celebratethe new building of the Canadian Embassy and highlight the future of Academic-Exchange betweenCanada and Germany.

RESEARCH CONTRIBUTIONS

[1] Kuhlmann, S. – Matusinski, M. : Hardy type derivations in pre-logarithmic series fields andEL-series fields, preprint, 22 pages (2008).

[2] Kuhlmann, S. – Tressl, M. : A note on exponential-logarithmic and logarithmic-exponential powerseries fields, preprint, 20 pages (2008).

PAPERS SUBMITTED

[3] Kuhlmann, S. – Matusinski, M. : Hardy type derivations in generalized series fields, 22 pages,(2009).

[4] Kuhlmann, S. – Rolin, J. – P. : An uncountable family of logarithmic functions of distinct growthrates, 7 pages (2009).

PAPERS TO APPEAR:

[5] Fornasiero, A.– Kuhlmann, F.–V. – Kuhlmann, S. : Towers of complements and truncation closedembeddings of valued fields, 41 pages (2008). To appear in Journal of Algebra.

[6] Kuhlmann, F.–V. – Kuhlmann, S. – Lee, J. : Valuation bases for (generalized) algebraic seriesfields, 37 pages (2008). To appear in Journal of Algebra.

[7] Cimpric, J. – Kuhlmann, S. – Marshall, M. : Positivity in power series rings, 10 pages. Toappear in Advances in Geometry.

RESEARCH MONOGRAPH

[8] Kuhlmann, S. : Ordered Exponential Fields, The Fields Institute Monograph Series, Vol. 12, AMSPublications, 164 pages (January 2000). See Review in CMS Notes vol. 33 No. 7, November 2001


REFEREED ARTICLES:

[9] Kuhlmann, S. – Putinar, M. : Positive Polynomials on Projective Limits of Real Algebraic Vari-eties, Bulletin des sciences mathematiques 133, 92-111 (2009).

[10] Cimpric, J. – Kuhlmann, S. : – Scheiderer, C. : Sums of Squares and Invariant Moment Problemsin Equivariant Situations, Transactions Amer. Math. Soc. 361, no 2, 735-765 (2009)

[11] Kuhlmann, S. – Putinar, M. : Positive Polynomials on Fibre Products, C. R. Acad. Sci. Paris,Ser. 1344, 681-684, (2007)

[12] Biljakovic, D. – Kochetov, M. – Kuhlmann, S. : Primes and Irreducibles in Truncation IntegerParts of Real Closed Fields, in Logic, Algebra and Arithmetic, Lecture Notes in Logic, Vol 26,Association for Symbolic Logic, AK Peters, 42–65 (2006).

[13] Kuhlmann, S. – Shelah, S. : κ–bounded Exponential-Logarithmic power series fields, in Annalsfor Pure and Applied Logic 136, 284-296, (2005).

[14] Kuhlmann, S. – Marshall, M. – Schwartz, N. : Positivity, sums of squares and the multi-dimensional moment problem II, in Advances in Geometry 5, 583–607 (2005).

[15] Holland, W. C. - Kuhlmann, S. - McCleary, S.: Lexicographic Exponentiation of chains, J. Symb.Logic 70, 389–409 (2005).

[16] Kuhlmann, F.-V. – Kuhlmann, S. – Shelah, S. : Functorial equations for lexicographic products,Proc. Amer. Math. Soc. . 131 2969-2976 (2003).

[17] Kuhlmann, F.-V. – Kuhlmann, S. : Valuation theory of exponential Hardy fields I, Math.Zeitschrift. 243, 671–688 (2003).

[18] Kuhlmann, S. – Marshall, M. : Positivity, sums of squares and the multi-dimensional momentproblem, Trans. Amer. Math. Society, 354, 4285–4301 (2002)

[19] Kuhlmann, F.-V. – Kuhlmann, S. – Marshall, M. – Zekavat, M. : Embedding ordered fields informal power series fields, Journal of Pure and Applied Algebra 169, 71–90 (2002)

[20] Kuhlmann, F.-V. – Kuhlmann, S. : The exponential rank of nonarchimedean exponential fields,in: Real Algebraic Geometry and Ordered Structures, Contemporary Mathematics 253, AMS,181–201 (2000)

[21] Kuhlmann, S. : Infinitary properties for valued and ordered vector spaces, J. Symb. Logic 64,216–226 (1999)

[22] Kuhlmann, F.-V. – Kuhlmann, S. – Shelah, S. : Exponentiation in power series fields, Proc.Amer. Math. Soc. 125, 3177–3183 (1997)

[23] Kuhlmann, F.-V. – Kuhlmann, S. : Ax-Kochen-Ershov principles for valued and ordered vec-tor spaces, in Ordered Algebraic Structures (ed. C. Holland & J. Martinez), Kluwer AcademicPublishers, 237–259 (1997)

[24] Kuhlmann, S. : Valuation bases for extensions of valued vector spaces, Forum Math. 8, 723–735(1996)


[25] Kuhlmann, S. : Isomorphisms of lexicographic powers of the reals, Proc. Amer. Math. Soc. 123,2657–2662 (1995)

[26] Kuhlmann, S. : On the structure of nonarchimedean exponential fields I, Archive for Math. Logic34, 145–182 (1995)

[27] Kuhlmann, F.-V. – Kuhlmann, S. : On the structure of nonarchimedean exponential fields II,Communications in Algebra 22, 5079–5103 (1994)

[28] Alling, N. L. – Kuhlmann, S. : On ηα–groups and fields, Order 11, 85–92 (1994)

[29] Kuhlmann, S. : Groupes abeliens divisibles ordonnes, in: Seminaire sur les Structures AlgebriquesOrdonnees, Selection d’exposes 1984–1987, Vol. 1, Publications mathematiques de l’UniversiteParis VII, 3–14 (1990)

OTHER REFEREED CONTRIBUTIONS:

[30] Habilitationsschrift: Ordered Exponential Fields , 128 pages (1999)

[31] Ph.D. Thesis: Quelques proprietes des espaces vectoriels values en theorie des modeles, Equipede Logique Mathematique de Paris VII, Prepublications 27, Paris (1991).

NON REFEREED CONTRIBUTIONS:

[32] Kuhlmann, F.-V. – Kuhlmann, S. : Explicit construction of exponential-logarithmic power series,Prepublications Paris 7 No 61, Seminaire Structures Algebriques Ordonnees, Seminaires 1995-1996.

[33] Kuhlmann, F.-V. – Kuhlmann, S. : Residue fields of arbitrary convex valuations on restrictedanalytic fields with exponentiation, Fields Institute Preprints, (1996)

[34] Kuhlmann, S. : A remark on the classification problem for ordered structures, PrepublicationsParis 7, Seminaire Structures Algebriques Ordonnees, Seminaires 1987–1988.

[35] Kuhlmann, S. : Chaınes uniquements transitives, Prepublications Paris 7, Seminaire StructuresAlgebriques Ordonnees, Seminaires 1985–1986.

REVIEW ARTICLES (selected):

[36] Kuhlmann, S. : Book Review of Dales, G. – Woodin, H. : Super – Real Fields, LMS Monographs,New Series 14, Clarendon Press Oxford (1996) , Review in Bull. London Math. Soc. 31, 248–256(1999).

[37] Kuhlmann, S.: 2 entries for the Encyclopaedia of Math. (ed. Hazewinkel): Model theory of thereal exponential function / o-minimal (1996)

PUBLISHED ABSTRACTS FOR INVITED TALKS:

[38] Kuhlmann, S, : Towers of complements and truncation closed embeddings of valued fields. Asso-ciation for Symbolic Logic Annual Meeting, Gainesville, Florida, USA, (2007).

Salma Kuhlmann: Curriculum Vitae 2008 (14 pages) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

[39] Kuhlmann, S, : Approximation of positive polynomials by sums of squares. IMA Annual ProgramYear, Workshop Optimization and Control, Minneapolis (2007)

[40] Kuhlmann, S, : Positive polynomials and Invariant Theory. MTNS conference, Polynomial In-equalities and Applications, Kyoto, Japan (2006)

[41] Kuhlmann, S, : Integer parts and complements to valuation rings, Canadian Math. Society Sum-mer Meeting, Calgary (2006)

[42] Kuhlmann, S, : Positivity in Real Algebraic Geometry, CARTHAPOS06, Carthage, Tunis (2006)

[43] Kuhlmann, S, : Hausdorff’s Lexicographic Chains. Keldysh International Conference and Work-shop, Moscow State University, Russia, (2004).

[44] Kuhlmann, S. : On G-Invariant Moment Problems. Workshop on Algorithmic, Combinatorialand Applicable Real Algebraic Geometry, MSRI Berkeley USA, (2004)

[45] Kuhlmann, S, : Primes and Irreducibles in Power Series Rings, Part I and Part II. Workshopand Conference on Logic, Algebra and Arithmetic, IPM Tehran, Iran (2003).

[46] Kuhlmann, S. : Lexicographic Exponentiation of Chains. Sixth Annual Conference in OrderedAlgebraic Structures Vanderbilt University, Nashville, Tennessee, USA (2003).

[47] Kuhlmann, S. : Representation of Polynomials Positive on Subsets of the Real Line, with Ap-plications to the Multidimensional Moment Problem. First Joint International Meeting AmericanMathematical Society & Real Sociedad Matem. Espanola, Seville, Spain. (2003).

[48] Kuhlmann, S. : Lexicographic Orderings. LMS Regional Meeting, Birmingham, UK. (2002).

[49] Kuhlmann, S. : The Exponential-Logarithmic Power Series Fields. CUNY Logic Workshops, NewYork City, USA (2001).

[50] Kuhlmann, S. : Exponential Fields, Symposium in Mathematics in honor of Professor SaharonShelah, winner of the Bolyai and Wolf Prizes, Ben Gurion University , Israel (2001).

[51] Kuhlmann, S. : Ordered Exponential Fields Association for Symbolic Logic Annual Meeting,University of Pennsylvania, USA (2001).

[52] Kuhlmann, S. : Lexicographic Orderings, Canadian Math. Society Summer Meeting (2001).

[53] Kuhlmann, S. : A maximality property for the Hardy field of the expansion by real powers andrestricted analytic functions, Canadian Mathematical Society Summer Meeting (2000)

[54] Kuhlmann, S. : Hardy fields of arbitrary models of polynomially bounded plus exp expansions,American Mathematical Society Winter Meeting (1999)

[55] Kuhlmann, S. : The Exponential-Logarithmic power series field, Meeting on Model Theory ofFields Oberwolfach (1998)


[56] Kuhlmann, S.: The Growth rates of exponentials in models of real exponentiation, Association forSymbolic Logic Annual Meeting (1998)

[57] Kuhlmann, S.: Ordered Algebraic Structures: Some current research directions, Meeting CanadianWomen in Mathematics (1998)

PAPERS AND THESIS FROM RESEARCH THAT I SUPERVISED:

[58] Haias, Manuela: On the value group of exponential and differential ordered fields M.Sc. Thesis,Saskatoon (August 2007),

[59] Fan, Wei: Non-negative polynomials on compact semi-algebraic sets in one variable case M.Sc.Thesis, Saskatoon (December 2006),

[60] Green, Trevor: Properties of chain products and Ehrenfeucht–Fraısse Games on Chains, M.Sc.Thesis, Saskatoon (August 2002),

[61] Chun, Hyunku: The orthogonal group of an elliptic space, M.Sc. Thesis, Saskatoon (2001),

[62] Zekavat, Mahdi: Orderings, cuts and formal power series, Ph.D. Thesis, Saskatoon (2000)

NSERC undergraduate papers:

[63] Hancock, M. : Algorithms in Invariant Theory (2005).

[64] Lee, J. : Exponentiation on Algebraic Puiseux Series Fields (2003).

[65] Danchilla, B. : Current Status of some famous open conjectures in Number Theory (2003).

[66] Lee, J. : Elliptic Curves and Cryptography ( 2002).

[67] Chilliack, C. : The Orthogonal Group of a Euclidean Space (2000).

[68] Mader, A. : Sturm’s Algorithm Computer Program (1999).

[69] Arbuthnott, C. : A proof of Rosenlicht’s Lemma (1999).

EDITORIAL:

• Proceedings:Kuhlmann, F.-V. – Kuhlmann, S. – Marshall, M. : (Editors): Valuation Theory and its Applications,Volume I & II, Fields Institute Communications Series, Vol. 32&33 AMS Publications (2002–2003).

• Hausdorff-Edition (http://www.aic.uni-wuppertal.de/fb7/hausdorff/): I have accepted an invita-tion to join the editorial board for the publication of Volume 1.

Reviews: I have written approx. 20 reviews for the Zentralblatt, and approx. 10 reviews for theMathematical Reviews.


SELECTED INVITED TALKS (selected from past 5 years only):

Academic Year 2008-2009

• Logic Seminar, University of Illinois at Chicago (March 2009)

• Special Session on Concrete Aspects of Real Positive Polynomials, AMS Central Meeting09, Urbana, Illinois (March 2009)

• Special Session on Real Algebraic Geometry, Applications and Related Topics, MTNS08, Blacksburg, Virginia (July 2008)


• Special Session on Model Theory and Applications to Geometry, CMS-SMF joint Meeting,University of Montreal ; invited speaker (June 2008)

• Special Session on Real Algebra and its interactions with Functional Analysis, LAW’08,Kranjska Gora, Slovenia (May 2008)

• Seminaire Structures Algebriques Ordonnees, Paris 7; invited speaker (May 2008)

• Logic Seminar Notre Dame University, South Bend (November 2007)

• Logic Seminar Universite de Mons-Hainaut, Belgique 3 talks (November 2007)

• Seminaire General de Logique, Paris 7; invited speaker (November 2007).


• Journees en l’Honneur de Max Dickmann, Univ. Paris 7; invited speaker (June 2007)

• Seminaire Equations Differentielles, Universite de Bourgogne; invited speaker (June 2007).

• Special Session on Contemporary Model Theory and its Applications, CMS SummerMeeting University of Manitoba, Winnipeg ; invited speaker (June 2007)

• Prairies Mini-Meeting, Univ. of Regina; invited speaker (May 2007).

• Math Colloquium, Ben Gurion Univ; invited speaker (April 2007).

• Set Theory Seminar, Ben Gurion Univ.; invited speaker (April 2007).

• Algebraic Geometry Seminar, Ben Gurion Univ.; invited speaker (April 2007).

• Set Theory Seminar, Hebrew Univ.; invited speaker (April 2007).

• Algebra Seminar, Univ. of Haifa invited speaker (two talks) (April 2007).

• Seminar Reelle Algebraische Geometrie, Univ. Regensburg; invited speaker (March 2007).

• Seminaire General de Logique, Paris 7; invited speaker (March 2007).

• Seminaire Structures Algebriques Ordonnees, Paris 7; invited speaker (March 2007).

• Workshop on Real Algebraic Geomerty, Oberwolfach; invited participant (March 2007).

• Association for Symbolic Logic, Annual Meeting, University of Florida, Gainesville; plenaryspeaker (March 2007).

• Workshop on Optimization and Control, IMA, University of Minnesota, Minneapolis; plenaryspeaker (January 2007).


• Workshop “Positive Polynomials and Optimization”, BIRS; organizer and participant (Oc-tober 2006)

• Colloque “o-minimalite”, Equipe de Logique, CNRS Paris 7; participant (September 2006)

• Mini-Symposium on Polynomial Inequalities and Applications, Kyoto, Japan, organizors:Bill Helton, Pablo Parrilo, Mihai Putinar; invited participant (July 2006).


• CARTHAPOS06 A Workshop on Positivity, Tunis; invited participant (June 2006).

• Special Session on Model Theory, Annual Meeting of the Canadian Mathematical Society,Calgary; invited speaker (June 2006).

• Workshop on Extreme Forms of real algebraic varieties, AIM, Palo Alto, organizors: IliaItenberg, Grigory Mikhalkin, Oleg Viro ; invited participant (April 2006).

• Model Theory Seminar, McMaster, Hamilton Differentiel fields of exponential-logarithmic powerseries (February 2006)

• Seminaire general de Logique, Universite Paris 7, Paris: Corps Differentiels de series exponentielles-logarithmiques (November 2005)

• S’eminaire Algebre et Geometrie, Universite de Versailles - Saint-Quentin, Versailles: Polynomespositifs et probleme des moments (3 talks) and Corps de Transseries (November-December 2005).


• Model Theory Applications to Algebra and Analysis, Newton Institute, Cambridge , UK:Truncation Integer Parts of Valued Fields (May 2005).

• Conference on Positive Polynomials, CIRM Luminy, France: The Invariant Moment Problem(March 2005).


• Workshop and Conference on Logic, Algebra and Arithmetic, IPM Tehran, Iran: Primesand Irreducibles in Power Series Rings. 2 Talks (October 2003).

• Colloquium, Shiraz University, Iran: A talk on Model Theory. (October 2003).

• Seminaire general de Logique, Universite Paris 7, France: Lexicographic Exponentiation ofChains. (November 2003).

• Algebra Seminar Universitad Autonoma Madrid ,Spain: Primes and Irreducibles in Exponen-tial Integer Parts of Ordered Exponential Fields.(December 2003).

• Colloquium Cairo University, Egypt: A talk on Real Closed Fields, (January 2004).

• Seminaire general de Logique, Universite Paris 7 , France: Corps de series ExponentiellesLogarithmiques κ–bornees, (February 2004).

• Seminar Reelle Algebraische Geometrie, Universitat Regensburg , Germany: Primes andIrreducibles in Power Series Rings, (February 2004).

• Seminaire Algebre et Geometrie, Universite de Versailles Saint-Quentin, France: Primes andIrreducibles in Ordered Group and Power Series Rings, 3 Talks (March, May, June 2004).


• Algebra Seminar, University of Pisa, Italy: Exponential Integer Parts of Exponential Fields,(March 2004).

• Seminaire de Structures Algebriques Ordonnees, Universite Paris 7, France: Positivstellensatzet le Probleme des Moments, (March 2004).

• Workshop on Algorithmic, Combinatorial and Applicable Real Algebraic Geometry,MSRI Berkeley USA: On G-Invariant Moment Problems, (2004).

• Colloquium, Universite d’Angers, France: Le Probleme des Moments, (2004).

• Kolloquium, Universitat Bonn, Germany: Positivstellensatz & Moment Problem, (2004).

• Kolloquium Logikum, Universitat Bonn, Germany: κ–bounded Exponential–Logarithmic powerseries fields. (2004).

• Colloquium, Zagreb University: Hilbert’s 17th problem & moment problem, (2004).

• Keldysh International Conference and Workshop, Moscow State University, Russia: Haus-dorff’s Lexicographic Chains, ( 2004).

Salma Kuhlmann: Research Description and Plans (9 pages) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Page 1

Salma Kuhlmann: Research Description and Plans 1

MOST SIGNIFICANT RESEARCH CONTRIBUTIONS. 2

I. Hausdorff’s Lexicographic Operations on Ordered Sets. Hausdorff [Ha1], [Ha2] introducedlexicographic exponentiation of chains, studied its arithmetic properties, generalizing Cantor’s ordinalarithmetic. In [25], I studied lexicographic powers with base chain R: The main result of this paper,which had many consequences, initiated collaborations [15], [13], [16]. I gave many presentations,reviving this topic (1998–2004). A Seminar Series (Saskatoon 2000) attracted student T. Green,resulting in a comprehensive M.Sc. Thesis [60]. The Appendix of [60] contains a translation of [Ha1](which had never been translated before). In [15], results of Hausdorff are extended and generalized.Following my 2003 Paris seminar on this topic, Pitteloud (PDF; Paris 7) studied lexicographic powerswith ordinal base and exponent; we met regularly to discuss his progress, the outcome is a 54 pagespaper by Pitteloud Lexicographic Exponentiation of Ordinals, in which my input is acknowledged. Igot feedback on my talk from M. Dzamonja, who had studied related issues. She invited me for collab-oration (Spring 2005 at Norwich). Following invitation of Koepke, I lectured in Bonn, headquartersof the German based project “Hausdorff Edition” (partially supported by Max-Planck Institut). Thisattracted recognition from the editors, who invited me to co-edit Volume 1. My lecture in Moscow(2004) received strong feedback from specialists Kanovei and Hrbacek.

II. Positive Polynomials and Sums of Squares. The Moment Problem originates in functionalanalysis (applications in [La]). In [H], the connection between the Moment Problem and the represen-tation of positive polynomials by sums of squares (Real Algebra) is worked out. In [Sc1] a completesolution for the case of compact K is given. I became interested in this problem in Spring 1999, when Ipresented it as one of the 3 topics for my Habilitationsvortrag. In 2000, I attended Marshall’s lecturesand proof–read [Ma1]. I invited Schmudgen in March 2000, who gave a series of talks in Saskatoon,in which he formulated several conjectures concerning the solution to the Moment Problem when K

is not compact. I worked with Marshall, solving these conjectures in [18]. I continued collaborationwith Marshall, Summer 2001; my contributions are acknowledged in the resulting paper [Ma2] (I fellseriously ill towards mid-summer and withdrew my name from the planned joint paper). The results of[18] have nice applications to Optimization [Ma3]. Powers attended Schmudgen’s Saskatoon lectures;in [P-S] Schmudgen’s conjectures were considered independently. In [18] a list of open problems isgiven, which had an important impact on the target community: it was announced in [P–S] that openproblems 4, 6 and 7 can be answered based on unpublished work of Scheiderer. This work was pub-lished later in [S2]. The solution to problems 4 and 7 appear in [S3], [S4]. Discussions with Schwartzled to solving problems 1 and 2, and initiated the collaboration resulting in [14]. A special case solu-tion to problem 1 is given in [P]. Results of [14] concerning compact subsets of the real line generalize[P–R]. [14] gives a partial answer to a question posed in [Sc2]. In [14], a list of 6 open problems ispresented, which again motivated considerable activity: problems 4 and 6 of [14] are answered in [S4].To this end, Scheiderer makes essential use of [14, Basic Lemma]. Wei Fan’s M.Sc. Thesis [59] isdedicated to [14, open problem 6] and its solution by Scheiderer.

1The number citations refer to my numbered list of Contributions, and the letter citations to my list of References.All contributions were supported by my NSERC Individual Research Grant.

2The descriptions below focus on the aspect of influence on the direction of thought and activity of other researchers.For more description of content, please refer to my publications and to my Research Proposal.


III. Integer Parts and Models of Open Induction. Berarducci [B] studied irreducible elementsof the power series ring k((G<0)) ⊕ Z. Pitteloud [Pit] proved that some of the irreducible elementsconstructed in [B] are prime. Both authors concentrated on the case of archimedean G. Addressingan open question in [B] concerning integer parts of exponential fields, we generalize in [12] results of[B] and [Pit] to the case of nonarchimedean G. Consideration of this case is essential for applicationsto exponential fields. I proposed this as joint research project to my PDF Kotchetov. Discussionswith Biljakovic in Paris 2003-2004 (who had been working on this problem independently) led to ajoint paper. Open questions presented in my IPM conference lectures led F.-V. Kuhlmann to studythem, now partially presented in [KF2]. A conjecture stated in my 2004 seminar talk in Pisa (togetherwith a suggestion for a proof) was studied by A. Fornasiero (PDF; Pisa). We had regular discussionsby email and during my 2 visits to Pisa. This resulted in [F] (in which my input is acknowledged).

IV. Exponential-Logarithmic Power Series Fields. In [22], We showed that power series fieldscannot be endowed with an exponential function. As a consequence, other methods had to be developedto construct non-archimedean models of Texp, the elementary theory of the reals with exponentiation.Several constructions were given [8], [D-M-M], [Schm]. In [13] we give a new method, based onconstructing the “lexicographic iterate” of a chain. As application, for uncountable regular κ, weconstruct 2κ pairwise non-isomorphic models of Texp all isomorphic as real closed fields. This answersa question of D. Marker, and establishes an exponential analogue to a result with Alling [28]. Incollaboration with Tressl [2]: we show that models constructed in [8] are not isomorphic to those of[D-M-M].

V. Valuation Theory of Exponential Fields. In [8] I develop the valuation theory of orderedfields which admit an exponential function. I describe their structure and that of their value groups.Significance and novelty of my approach: 1) use valuation theoretic methods, 2) develop and exploit theexponential analogues of notions and methods from Real Algebra, 3) apply this machinery to describeexplicitly the structure of the models, and to give concrete constructions using power series fields. [8]settles several questions and aspects discussed in meetings. The structure induced by the exponentialon the value group and its model theory were studied in [KF1], [D] and [A–D2]. The exponential rankrelates naturally to exponential boundedness [D–M–M] and levels [M–M]. The limitations of powerseries constructions (Chap. 4), combined with [M], imply that if an o-minimal expansion of the realsadmits a power series field as a model, then it is necessarily polynomially bounded. These results alsoimply the existence of ηα - fields which are not hyper-real. Chap. 5 describes very special models, whichcarry exponentials of distinct growth rates (answering a question B. Zilber). Similar constructions aredone in [D-M-M] and [Schm]. Chap. 6 presents a structure theorem for Hardy fields of polynomiallybounded + (exp) o-minimal expansions, showing existence of levels (generalizing [M–M]). The openquestions arising from [8] provide projects for my students (see Research Proposal).

RESEARCH PROPOSAL.

(I). Hausdorff’s Lexicographic Operations on Ordered Sets. Let Γ and ∆ be chains (totallyordered sets), and 0 a distinguished element (base element or base point) of ∆. The lexicographic power∆Γ is the set of all maps from Γ into ∆ which have wellordered support in Γ. Ordered lexicographically,∆Γ is a chain. In [16], we considered the following question: When is Γ embeddable as a convex subsetin ∆Γ? A main result is that such an embedding cannot exist if Γ and ∆ have no last element. Letk((G)) denote the ordered field of generalized power series with exponents in a divisible totally orderedabelian group G and coefficients in a totally ordered field k. We deduce from our main result that it isnot possible to define an exponential function on k((G)). (This fact has important implications, thatwe discuss in (IV) below). My interest in lexicographic chains originally developed when I studied


lexicographic powers with base chain R: The challenging question that I considered in [25] is thefollowing: Does isomorphism of lexicographic powers imply isomorphism of the exponents? In thatpaper, I considered particular lexicographic chains, where the exponent Γ is an ordinal, and answerthe question positively in this case. The search for a more comprehensive answer led us to study in[15] nonisomorphic exponents for which the corresponding lexicographic powers are isomorphic. Forexample, we show that for a countable infinite ordinal α, Rα∗+α and Rα are isomorphic (here, α∗

denotes the reverse of the ordinal α). We show that RR and RQ are nonisomorphic. The second maintheme of [15] is the investigation of the automorphism group of a lexicographic power. Hausdorff[Ha2] was interested in 2-transitive automorphism groups (or 2-homogeneous lexicographic powers).We generalize results of [Ha2], and show that ∆R has a 2-transitive automorphism group, where ∆ isa countable ordinal ≥ 2 [15, Theorem 6.1]. The approach in [15] is based on two powerful tools: (i)a deep understanding of the arithmetic rules of lexicographic operations, (ii) exploiting properties ofchains of countable coterminalities. T. Green’s M.Sc. Thesis [60] (186 pages) is a comprehensive,systematic survey of lexicographically ordered products of chains.The theory offers a variety of open problems that I propose to continue studying. Lexicographicorderings appear naturally in Descriptive Set Theory, Real Algebra, Valuation Theory, Grobner Bases(monomial orders). My long term objectives are to (i) use my expertise on lexicographic orderingsto make contributions in one of these potential areas of application, (ii) systematically generalize,complement and revive this relatively unknown chapter of Hausdorff’s mathematical legacy. I nowdescribe my short term objectives.(a) Open Problems arising from [15]. Dependence on the choice of base points: The lexicographicpower ∆Γ depends in general on the choice of the base element 0 of ∆. For example if ∆ is the chain{0, 1} and Γ an anti well-ordered set (the reverse of an ordinal α), then 2Γ is the ordinal 2α if thedistinguished element is 0 ∈ {0, 1}, and is the reverse of the ordinal 2α if the distinguished element is1 ∈ {0, 1}. This introduces naturally an equivalence relation on ∆. I want to study the fascinatingand challenging problem of classifying (up to isomorphism) the corresponding lexicographic powers.In [Ha1; p.468-469], a very general formula is given, that might provide the right tool to tackle thisproblem. In [15, p.21-22] we list four questions concerning the chains appearing in [15, Theorem 6.1].These four problems will be easier to answer once we understand the issue of dependence on chosenbase points.

(b) Hausdorff Project. I was asked by Professor Purkert (Bonn), one of the main editors of the“Hausdorff Edition”, to co-edit Volume 1, which is to appear in 2008. To this end, I will: (i) Writea research essay commenting on the algebraic aspects in Hausdorff’s 1909 paper (Die Graduierungnach dem Endverlauf, Abhandlungen der Konigl. Sachs. Ges. der Wiss. zu Leipzig. Math.- Phys.Klasse 31 (1909), 295-334). In this paper, the foundational ideas for (what was later called) “Hardyfields” are laid. (ii) “Referee” an eight pages handwritten manuscript from Hausdorff’s posthumousworks entitled “Algebra und finale Rangordnung fur Functionen”. The content is closely related toHardy fields. I shall decide whether it should be published, in which case it should appear in Volume1, accompanied by my commentary.(c) Related Work. [A-B], [A-H], and [D-T] seem to have interesting connections to my work. I planto work out these exciting connections.(II). Positive Polynomials and the Moment Problem. The K-moment Problem deals with therepresentation of linear functionals by integrals: For a linear functional L on the polynomial ring inn-variables R[X], one asks whether there exists a positive Borel measure µ on Rn supported by somegiven closed set K in Rn such that ∀f ∈ R[X], L(f) =

∫Rn fdµ. By Haviland’s theorem [H], this holds if

and only if for every f ≥ 0 on K (we write f ∈ Psd(K)) we have L(f) ≥ 0, (we write L(Psd(K)) ≥ 0)).


Although Haviland’s theorem provides a nice theoretical solution, it is hard to implement in practice:the cone Psd(K) is in general not finitely generated [S1], so checking L(Psd(K)) ≥ 0 is seldom possible.For this reason, one studies the problem in the following form. Let K ⊂ Rn be a basic closed semi-algebraic set, and S = {g1, · · · , gs} ⊂ R[X]. We let TS ⊂ R[X] be the corresponding finitely generatedpreordering3 We say that S solves KMP if K = KS = {x ∈ Rn ; g1(x) ≥ 0, . . . , gs(x) ≥ 0} and for anylinear functional: L(Ts) ≥ 0 implies L( Psd (K)) ≥ 0 . If such a finite S exists, we say that the KMPis finitely solvable. In [Sc1], Schmudgen shows the striking result that if K is compact, then S solvesKMP, for any finite description S of K = KS . In fact for K = KS compact, f ∈ Psd(K) has a nicerepresentation by sums of squares:

∀f ∈ R[X], f ≥ 0 on K ⇒ ∀ real ǫ > 0 , f + ǫ ∈ TS . (1)

(Note: It can be easily verified that (1) implies that S solves KMP). In [18], we consider the situationwhen K = KS is not compact. We study an intermediate property:

∀f ∈ R[X] , f ≥ 0 on K ⇒ ∃q ∈ TS such that ∀ real ǫ > 0 , f + ǫq ∈ TS (2)

which we prove to be strictly weaker than (1) but to imply KMP. We study in detail the case n = 1;we introduce the crucial notion of natural generators (a key notion that settles completely this case).We use results of [S1] to prove that KMP is not finitely solvable if K contains a cone of dimension 2(a stronger result was obtained independently in [P–S]). We prove that (2) holds for cylinders withcompact cross-section. We list 8 open problems. We address these in [14]. We reconsider the casen = 1, this time analyzing the relationship between the preordering TS and the quadratic module MS

(one is naturally interested in a module representation, linear in the generators, of a non-negativepolynomial). We prove an important lemma, from which we deduce that if K is compact then thequadratic module generated by the natural generators of K is a preordering. We generalize the resultof [18] on cylinders with compact cross-section to subsets of cylinders; we solve open problems 1 and2 of [18]. We list 6 open problems.Representation of positive polynomials by sums of squares has many beautiful applications to opti-mization and semi-definite programming. It is my long-term goal to acquire expertise in theseareas of applications. Here are my short term objectives:(a) Further Research arising from [14] and [18]. Problems [18, prob 4 and prob 7] and [14, prob4 and prob 6] were recently solved by Scheiderer in a series of papers. We studied these papers withour M.Sc. student Wei Fan. In [59] Fan focused on the solution to [14, prob 6], which asked whetherthe quadratic module MS associated to a compact KS ⊂ R is always a preordering. Some problemsare still open. Very recently, Tim Netzer found a counterexample, answering [18]; prob 3, which askswhether KMP and (2) are equivalent. Problem 6 [18] is still open, we do not know whether thequadratic preordering associated to the two dimensional strip is saturated. I want to achieve progresson this problem.(b) Equivariant Moment Problems. The inspiration for [10] originates from the paper [G–P]. Inthis paper, the authors investigate the representation of symmetric positive polynomials as sums ofsquares, using semi-definite programming tools. They exploit the symmetry to significantly simplifythe procedure. Let G be a finite group with a linear representation in GLn(R) (or more generally, alocally compact group). Our idea is to investigate the representation (by integrals) of invariant linearfunctionals (supported by an invariant basic closed semi-algebraic subset of Rn) along an invariantmeasure. To this end, we must analyse the associated preordering in the ring R[X]G of invariant

3For the definition, see e.g., [18]; page 2.


polynomials. We show that the cone of invariant sums of squares is in general not finitely generatedover the preordering of sums of squares of invariant polynomials. We study the relationship betweenthe solvability of the KMP on the one hand, and that of the G-invariant KMP on the other. Theidea is to describe a class of invariant non-compact semialgebraic sets for which KMP is not finitelysolvable but the G-invariant KMP is. The tool that we use is the orbit map: For example, considerthe set K defined by −1 ≤ (x2 − 1)(y2 − 1) ≤ 0. This set is invariant under the action of the dihedralgroup D4, and its image under the orbit map is a cylinder with compact cross-section, so themoment problem is solvable by [18]. “Pulling back” the orbit map, we get that the G- invariantKMP is solvable. An exciting fact is that the KMP for this K is not solvable. In [7] we continuedthe study of preorderings which are saturated for invariant polynomials by generalizing Scheiderer’slocal-global principles for saturation in rings of power series in 2 variables. We plan in the future tostudy Scheiderer’s method to produce many new interesting examples. We expect a strong feedbackfrom the experts community.(c) Positive Sparse Polynomials and Fibre Products of Algebraic Varieties. Recent investi-gations in optimization theory are concerned with the structure of positive polynomials with a sparsitypattern, see [Las], [W-K-K-M]. In [11] we interpret these patterns in the more invariant language of(iterated) fibre products of real algebraic varieties. Our result cover and extend the existing resultsin the literature. Moreover, the algebraic geometric approach opens the perspective of treating on aunifying basis positivity of sparse polynomials on unbounded supports. In a follow up paper to [11],we are currently investigating sparse polynomials on fibre products of unbounded cylindres.

(III). Integer Parts and Models of Open induction. In joint work with Biljakovic and Kotch-etov [12], we study truncation integer part (IP) of a real closed field: a truncation IP Z of F is adiscretely ordered subring which decomposes as Z = A⊕ Z (lexicographic sum), where the summandA is a k-algebra and an additive complement to the valuation ring OF . It was shown that every realclosed field admits a truncation closed embedding in a field of power series k((G)) (Mourgues, M.- H.– Ressayre, J.- P.: Every real closed field has an Integer Part, Journal of Symbolic Logic, 58 (1993),641–647), so without loss of generality, we assume that F is a truncation closed subfield such thatk(G) ⊂ F ⊂ k((G)) . Berarducci [B] studies irreducible elements in the ring k((G<0)) ⊕ Z, that is,in the truncation IP of k((G)) , focusing mainly on the case when G is archimedean. Pitteloud [Pit]shows that some of the irreducible series constructed in [B] are prime, in the case when G contains amaximal proper convex subgroup. In [12], we extend the results of [B] and [Pit] to the truncation IP ZF

of an arbitrary truncation closed F ⊂ k((G)), for an arbitrary divisible ordered abelian group G 6= 0.Let k[G<0] be the semigroup ring consisting of power series with negative and finite support. [B;Theorem 11.2] says that, in the case of archimedean G, all irreducible elements of k[G<0] ⊕ Z remainirreducible in k((G<0)) ⊕ Z. We study, for an arbitrary G, the behaviour of primes and irreduciblesunder the ring extensions k[G<0] ⊕ Z ⊂ ZF ⊂ k((G<0)) ⊕ Z. In [B], the author asks whether everyIP of an exponential field contains a cofinal set of irreducible elements. We give a partial answer:applying our results to truncation IP’s of non-archimedean exponential fields, we show that theseIP’s indeed contain a cofinal set of irreducible elements. (Note that in the case of a non-archimedeanexponential field, the value group G cannot be archimedean, nor can it contain a maximal properconvex subgroup).My long term objective here is to refine the methods developed in [12] to answer Berarducci’squestion in all generality. To this end, we need to analyse IP’s which are not truncation IP’s. Thefirst short term objective is to understand normal IP’s (that is, IP’s integrally closed in their fieldof fractions). These are, so to say, at the other end of the spectrum; indeed, truncation IP’s are


never normal. An open question, formulated in [12], which I intend to attack shortly is whether everyreal closed field has a normal IP. Fornasiero [F] verified (following my suggestion) the existence oftruncation closed embeddings for Henselian fields (which in turn implies the existence of IP). In [5] wegive an intrinsic valuation theoretic interpretation of truncation closed embeddings of valued fields infields of generalized power series. It turns out that the existence of such an embedding is equivalentto that of a “tower of complements” to the valuation ring satisfying some overlap conditions. We nowprove the existence of an IP for Henselian fields directly, by constructing such tower of complementsto the valuation ring (see [5] for more details).(IV). Exponential-Logarithmic Power Series Fields. The Logarithmic-Exponential or Exponential-Logarithmic models of Texp are obtained as countable increasing unions of fields of generalized powerseries. In both cases, a partial exponential (logarithm) is constructed on every member of this union,and the exponential on the union is given by an inductive definition. In [13], we describe a differentconstruction: We start with any non-empty chain Γ0. For a given regular uncountable cardinal κ,we form the (uniquely determined) κ-th iterated lexicographic power (Γκ, ικ) of Γ0. We take Gκ andR((Gκ))κ to be the corresponding κ-bounded Hahn group and κ-bounded power series field respec-tively (see [13, Section 2]). The logarithm on the positive elements of R((Gκ))κ is now defined by auniform formula [13, page 9 (15)].My short term objective is to exploit the simplicity of the new construction to define compositionand derivation operators on these models. These operators are defined on [D-M-M] models, but thedefinitions (as the construction of the models themselves) are rather complicated. I believe that thenew models offer an interesting alternative, as computations are rather straightforward there. Theseideas can be extended to define the operators on Conway’s Field of Surreal Numbers, which we believeto be a universal domain for all Exponential - Logarithmic models. With J.-P. Rolin, we plan toapply the methods of [13] to find generalized power series representations of solutions to differentialequations. I also want to complete a joint project with M. Tressl [2]: we show that the Logarithmic-Exponential field embeds in the Exponential-Logarithmic one, but not vice-versa.(V). Valuation Theory of Exponential Fields.(a) Restricted Exponentials in Real Closed Fields: We consider the following: Does everyreal closed field admit a restricted exponential function? That is, is the ordered additive group of in-finitesimals (the maximal ideal Iv of the valuation ring) order isomorphic to the ordered multiplicativegroup of 1-units of the valuation ring (i.e., to 1 + Iv)? I answered the question positively only forcountable and power series fields [8] (but do not know the answer in general). The key idea was totranslate the meaning of “pseudo-convergence” from the additive to the multiplicative case: I show in[8] that the natural valuation on the multiplicative group of 1-units is (equivalent to) v(a − 1) (i.e.,to composing the valuation on the additive ideal of infinitesimals with the translation map). This hastwo important consequences: 1) pseudo – Cauchy sequences and limits correspond to each other viathe translation map from the ideal (additive) onto the 1-units (multiplicative), 2) the translation mappreserves valuation independence.

Short term objective: Until recently, we did not even know whether the answer to the abovequestion is positive for the field of algebraic Puiseux series (the relative algebraic closure of R(t) inR((Q))). This was J. Lee’s Summer 2003 project [64]. He came up with a clever idea (for a positiveanswer in this case), which we worked out in detail [6]. In this paper, we investigate valued fields whichadmit a valuation basis (as valued vector spaces over a given countable ground field K). We isolatea property (called TDRP in our paper) for a valued subfield L of a field of generalized power seriesF ((G)) (where G is a countable ordered abelian group and F is a real closed, or algebraically closedfield). We show that this property implies the existence of a K–valuation basis for L. In particular,


we deduce that the field of rational functions F (G) (the quotient field of the group ring F [G]) andthe field F (G)∼ of power series in F ((G)) algebraic over F (G) admit K–valuation bases. If moreoverF is archimedean and G is divisible, we conclude that the real closed field F (G)∼ admits a restrictedexponential function.(b) Realizing a contraction group as the value group of an exponential field. We considerthe following: Is every contraction group the value group of an exponential field? 4 I have solved thisproblem positively for countable contraction groups [8], but do not know yet the answer in general.The analogous question for differential fields has been studied in:Rosenlicht, M. : On the value group of a differential valuation I & II, Amer. Journal Math. 101 (1979),258–266 & 103 (1981), 977–996.(There, the question was whether every asymptotic couple is the value group of a differential field). Wealso found closely related considerations in [A–D1; Section 11 p. 35]: the authors construct an H-fieldfor a given H-couple (G,Ψ), under the assumption that the group G admits a valuation basis. Byanalogy, this suggests that my above stated question should be first considered under this assumptionon G. This was partially addressed in M. Haias M.Sc. research project[58].

(d) Miller’s Dichotomy for Transexponentials. In Boshernitzan, M.: Hardy fields and existenceof transexponential functions, Aequationes Mathematicae 30 (1986), 258–280a Hardy field is constructed containing a transexponential function T on the reals, satisfying the func-tional equation T (x + 1) = exp (T (x)). Little is known about the expansion of the reals by this tran-sexponential; it is unknown whether such an expansion can be o-minimal. My short term objectiveis to investigate the following: Is there an analogous to Miller’s dichotomy5 for the transexponentialcase? This is a joint project with Kotchetov. We recently found an important transexponentialA in: Kuczma, M.: Functional Equations in a single variable, Polska Akademia Nauk. Monografiematematyczne, Warszawa (1968) 48. (A is a uniquely determined analytic solution to Abel’s equa-tion). We conjecture that if an o-minimal expansion of the reals is not exponentially bounded, thenthe transexponential A is definable.(e) Ordered abelian groups with the “ lifting property”. In [8, chap. 3], we show howto improve a given exponential (defined on a non-archimedean real closed field) to an exponentialsatisfying the growth axiom scheme provided that the value group has the lifting property (i. e. everyautomorphism of its value set lifts to an automorphism of the value group itself). I propose to workout a characterization of ordered abelian groups having the lifting property, and to find out whetherthis property is satisfied by the value group of an exponential field.(VI). Orthogonal Groups. In [61], an expository study of the normal subgroups of the rotationgroup of an elliptic space is given. If the orthogonal geometry V (of Witt index 0) is based on anonarchimedean ordered field then the valuation of the field plays a crucial role in determining thestructure of the orthogonal group O(V ): the ideals of the valuation ring define non-trivial normalsubgroups of O(V ). A natural question is to ask whether every normal subgroup arises in this way.This is related to the congruence subgroup problem (which has been studied in the last 30 years) forelliptic spaces over valued fields. I am not aware yet of what is known in the particular case of anonarchimedean field. It is one of my long term objectives to find this out. To which extent does thevaluation determine the structure of the group? If the underlying field is exponential, what can wesay about the corresponding group? Does it inherit additional structure from the field?

4For the definition and basic properties of contraction groups, see [8].5See [M].


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