K. İLHAN İKEDA

 

Department of Mathematics

Yeditepe University

26 Ağustos Yerleşimi

İnönü Mah., Kayışdağı Cad.,

34755 Kadıköy, İstanbul,

Turkey

 

Tel. +90-216-578-1782 (office)

Fax. +90-216-578-0671

e-mail. ilhan.ikeda@yeditepe.edu.tr

 

 

 

 

 

 

FIELDS OF SPECIALIZATION

 

Algebraic number theory, automorphic forms and representation theory.

 

RESEARCH

 

(1) K-theoretic class field theory, arithmetic and geometric Langlands functoriality principles and non-abelian class field theory.

(2) The theory of Eisenstein series and Arthur trace formula.

(3) Conformal field theory and topological quantum field theory.

 

I am especially interested in the local-global Langlands functoriality and in the non-abelian local-global class field theory (local case: Fesenko-Laubie theory, global case: Fesenko-Laubie adèles), their relationship with each other via higher ramification theory and Howe theory. Their multi-dimensional generalization to the K-theoretic setting (that is, the non-abelianization of the class field theory of Kato and Parshin, where the 1-dimensional theory is the Langlands functoriality and the Fesenko-Laubie theory), i.e., to n-local fields, to semi-global fields and to schemes (that is, Kapranov-Kontsevich functoriality and the K-theoretic Fesenko theory), their relationship with each other via Zhukov's and Abbes-Saito’s higher ramification theories and higher-dimensional Howe theory. Geometrization of Kapranov-Kontsevich functoriality (that is, muti-dimensional Beilinson-Drinfel'd theory). The place of Arthur trace formula, Eisenstein series and spectral decomposition in these general higher-dimensional settings following Garland and Kapranov. I am also interested in the applications and possible analogues of this very general theory in mathematical physics and in string theory (quantum Langlands program).

 

PUBLICATIONS

 

Thesis :

 

Thesis : Linear equivalence of Eisenstein series. Princeton University, Department of Mathematics (November 1993).

Thesis supervisor: Goro Shimura.

 

Refereed Journal Papers :

 

(1) A formal theory of Eisenstein series, C. R. Acad. Sci. Paris Sér. I Math. 321, (1995), pp. 45-50. (MR1340080/ 96h:11037 and Zbl 0840.11022) (pdf)

 

(2) On the standard L-functions of central-simple algebras, Bull. Sci. Math. 124, (2000), pp. 147-157. (MR1752107/ 2001h:11065 and Zbl 1017.11024) (pdf)

 

(3) Two lemmas on formal power series, (with Masatoshi Ikeda),  Turk. J. of Math. 23, (1999), pp. 435-440. (MR1773548/ 2001f:13032 and Zbl 0999.11072) (pdf)

 

(4) On the local Artin conductor ƒArtin(χ) of a character χ of Gal(E/K) 1: Basic definitions, Turk. J. of Math. 23, (1999), pp. 519-530. (MR1780938/ 2001i:11137 and Zbl 1006.11067). (pdf)

 

(5) On the metabelian local Artin map 1: Galois conjugation law, Turk. J. of Math. 24, (2000), pp. 25-58. (MR1792243/ 2001g:11185 and Zbl 0976.11054) (pdf)

 

(6) On the local Artin conductor fArtin(χ) of a character χ of Gal(E/K) 2: Main results for the metabelian case, Proc. Indian Acad. Sci. (Math. Sci.) 113 (2003), pp. 99-137. (MR1983746/ 2004c:11224 and Zbl 0204.0287) (pdf)

 

(7) Genelleştirilmiş Fesenko karşılıklılık ilkesi : Abelyen-olmayan lokal sınıf cisim kuramı (Erol Serbest ile), 18. Ulusal Matematik Sempozyumu Bildiri Kitabı, Istanbul Kültür Üniversitesi Yayınları, 2005, pp. 147-187 (English : Generalized Fesenko reciprocity law : Non-abelian local class field theory, with Erol Serbest). (pdf)

 

(8) Langlands karşılıklılık ilkesi (Sevan Bedikyan ile çağrılı makale), İTÜ/C Dergisi 5 (2007), pp. 3-18 (English : Langlands reciprocity principle, with Sevan Bedikyan, invited paper). (pdf)

 

(9) Fesenko reciprocity map (with Erol Serbest), Algebra i Analiz : St. Petersburg Math. Journal 20(3) (2008), pp. 112-162. (pdf)

 

(10) Generalized Fesenko reciprocity map (with Erol Serbest), Algebra i Analiz : St. Petersburg Math. Journal 20(4) (2008), pp. 118-159. (pdf)

 

(11) Non-abelian local reciprocity law (with Erol Serbest), Manuscripta Math. 132 (2010), pp. 19-49. (pdf)

 

(12) Ramification theory in non-abelian local class field theory (with Erol Serbest), Acta Arithmetica, 144 (2010), pp.373-393 . (pdf)

(13) 2-boyutlu Langlands karsiliklilik ilkesi : M. M. Kapranov'un çalismasi üzerine notlar, Fizik Dünyasi Dergisi, Ankara Üniversitesi, Cilt 1 (1), 2012, 1-7. (pdf)

(14) On the non-abelian global class field theory, accepted by Annales mathématiques du Québec, 37 (2) in 2013,pp. 129-172. (pdf).

(15) ''Basic properties of the non-abelian global reciprocity map'' , Mathematics in the 21st Century, Springer Proceedings in Mathematics& Statistics Volume 98, 2015, pp 45-92. (pdf)

Submitted Papers :

 

(16) Local class field theory in the sense of Tate-Sen theory (with Sevan Bedikyan and Erol Serbest), submitted.

 

(17) Problems in non-abelian local class field theory (with Erol Serbest), submitted.

 

(18) Kapranov reciprocity principle for GL(n), submitted.

 

(19) Basic functorial properties of the Kapranov reciprocity principle for GL(n), submitted.

 

(20) On the relationship between the generalized Fesenko and the Laubie reciprocity maps (with Serdar Kazancioglu and Erol Serbest), preprint (pdf).

 

(21) On the epsilon-factor of Weil- Deligne representations, submitted. (pdf)

 

 

 

Books :

 

Multi-dimensional Langlands functoriality, to appear in Series on Number Theory and Applications, World Scientific, in progress.

 

Kisa Ders : Fonktörsellik ve Esleme (30 Mart - 27 Nisan 2011 Galatasaray Üniversitesi)

 

Conferences :

 

Turing çalistayi 4-5 Mayis 2012

 

Recent Trends in Algebraic Number Theory, Istanbul Sayilar Teorisi Bulusmalari II, Galatasaray Üniversitesi , 26 Mayis 2012

 

IWASAWA 2012, Univ Heidelberg, July 23-Aug 03 2012

 

XXV. Ulusal Matematik Sempozyumu, Nigde Üniversitesi 5-8 Eylül 2012, Nigde

 

Summer School: Higgs Bundles on p-adic curves and representation theory, Univ Mainz, Sept. 24-28 2012

 

6th World Conference of 21st Century Mathematics held by Abdus Salam School of Mathematical Sciences (ASSMS) at Lahore on March 2013.

 

Representation Theory and Applications, Yeditepe University, Istanbul, June 11-17 2013.

 

Istanbul Summer School in Algebraic Geometry, Istanbul Center for Mathematical Sciences, June 17-21 2013 

 

26. Ulusal Matematik Sempozyumu, Dicle Universitesi, Diyarbakir, 4-7 Eylül 2013.

 

Japanese Turkish Joint Geometry Meeting, Galatasaray University, Istanbul, November 21-24 2013