A density theorem for parameterized differential Galois theory
|
|
- Jemima Jordan
- 6 years ago
- Views:
Transcription
1 A density theorem for parameterized differential Galois theory Thomas Dreyfus University Paris 7 The Kolchin Seminar in Differential Algebra, 31/01/2014, New York.
2 In this talk, we are interested in the differential equations on the form: z Y (z, t) = A(z, t)y (z, t), (1) where t = t 1,..., t n is a parameter, A M m (O U ({z})) and O U ({z}) is a ring we will define later. We want to define a parameterized differential Galois group for (1) and, analogously to the density theorem of Ramis, give a set of topological generators.
3 In this talk, we are interested in the differential equations on the form: z Y (z, t) = A(z, t)y (z, t), (1) where t = t 1,..., t n is a parameter, A M m (O U ({z})) and O U ({z}) is a ring we will define later. We want to define a parameterized differential Galois group for (1) and, analogously to the density theorem of Ramis, give a set of topological generators.
4 1 Parameterized Hukuhara-Turrittin theorem
5 Let U be a polydisc of C n. Let M U be the field fo meromorphic function on U. ˆK U := M U ((z)). O U ({z}) := { f i (t)z i ˆK U t U, z fi (t)z i is a germ of meromorphic function}. K U := Frac(O U ({z})).
6 Proposition There exist U U and ν N, such that we have an invertible matrix solution of (1) of the form: where Remark F (z, t) := Ĥ(z, t)el(t) log(z) e Q(z,t), Ĥ(z, t) GL m ( ˆK U [z 1/ν ]). L(t) M m (M U ). Q(z, t) := Diag(q i (z, t)), with q i (z, t) z 1/ν M U [z 1/ν ]. If we restrict U, we may assume that U = U.
7 Proposition There exist U U and ν N, such that we have an invertible matrix solution of (1) of the form: where Remark F (z, t) := Ĥ(z, t)el(t) log(z) e Q(z,t), Ĥ(z, t) GL m ( ˆK U [z 1/ν ]). L(t) M m (M U ). Q(z, t) := Diag(q i (z, t)), with q i (z, t) z 1/ν M U [z 1/ν ]. If we restrict U, we may assume that U = U.
8 Let t := { t1,..., tn }. We still consider (1) with the fundamental solution F (z, t) given by the parameterized Hukuhara-Turrittin theorem. Let K U be the ( z, t )-differential field generated by K U and the entries of F (z, t). Proposition K U K U is a parameterized Picard-Vessiot extension of (1), i.e, the field of the z -constants of K U is M U.
9 ( ) Let Aut t KU z K U be the group of field automorphism of K U that commutes with all the derivations and that let K U invariant. Let us consider the representation: ρ F : ( ) Aut t KU z K U GL m (M U ) ϕ F 1 ϕ(f ). Theorem The image of ρ F is a differential subgroup of GL m (M U ), i.e, there exist P 1,..., P k, t -differential polynomials in coefficients in M U, such that: (A i,j ) ρ F (Aut t z ) P 1 (A i,j ) = = P k (A i,j ) = 0.
10 We define the Kolchin topology as the topology of GL m (M U ) in which closed sets are zero sets of differential algebraic polynomials in coefficients in M U. Proposition Let G be a subgroup of Aut t z ( KU K U ). If K U G = KU, then G is dense for Kolchin topology in Aut t z ( KU K U ).
11 We still consider (1) with parameterized Picard-Vessiot extension K U K U and with Galois group Aut t z ( KU K U ). Definition We define ˆm Aut t z ( KU K U ) by: ˆm ˆKU = Id. ˆm(z α ) = e 2iπα z α. ˆm(log(z)) = log(z) + 2iπ. q z 1/ν M U [z 1/ν ], ˆm(e q ) = e ˆm(q).
12 Definition We define ( the) exponential torus as the subgroup of Aut t KU z K U of elements τ that satisfies: τ ˆKU = Id. τ(z α ) = z α. τ(log(z)) = log(z). There exists α τ, a character of z 1/ν M U [z 1/ν ], such that q z 1/ν M U [z 1/ν ], τ(e q ) = α τ (q)e q.
13 Example Let us consider z 2 z Y (z) + Y (z) = z. (2)
14 Example Let us consider z 2 z Y (z) + Y (z) = z. (2) The formal power series h(z) = n 0( 1) n n!z n+1 is solution of (2).
15 Example Let us consider z 2 z Y (z) + Y (z) = z. (2) Let d (2Z + 1)π. The following function is solution of (2) where f (ζ) = 1 1+ζ S d (h) = e id 0 ( f (ζ)e ζ ) dζ, z
16 Example Let us consider z 2 z Y (z) + Y (z) = z. (2) S d (h) is 1-Gevrey asymptotic to h: N Sd (h) ( 1) n n!z n+1 AN+1 (N + 1)! z N+1, n 0 where A > 0 is a constant.
17 Let d R and let k Q >0. ˆB k ( a n z n ) = L d 1 (f ) (z) = e id L d k := ρ k L d 1 ρ 1/k. 0 a n Γ(n/k) ζn. f (ζ) ( )dζ. ze ζ z
18 Let us consider h(z) C[[z]] solution of a linear differential equation in coefficients germs of meromorphic functions. There exist Σ R (set of singular directions), (κ 1,..., κ r ) ( Q >0) r, such that if d / Σ, S d (h) = L d κ r L d κ 1 ˆB κ1 ˆB κr (h), is a germ of analytic solution on the sector arg(z) ]d π/2κ r, d + π/2κ r [. Σ is finite modulo 2πZ. The map h S d (h) induces a morphism of differential field.
19 Let us consider h(z) C[[z]] solution of a linear differential equation in coefficients germs of meromorphic functions. There exist Σ R (set of singular directions), (κ 1,..., κ r ) ( Q >0) r, such that if d / Σ, S d (h) = L d κ r L d κ 1 ˆB κ1 ˆB κr (h), is a germ of analytic solution on the sector arg(z) ]d π/2κ r, d + π/2κ r [. Σ is finite modulo 2πZ. The map h S d (h) induces a morphism of differential field.
20 Let us consider h(z) C[[z]] solution of a linear differential equation in coefficients germs of meromorphic functions. There exist Σ R (set of singular directions), (κ 1,..., κ r ) ( Q >0) r, such that if d / Σ, S d (h) = L d κ r L d κ 1 ˆB κ1 ˆB κr (h), is a germ of analytic solution on the sector arg(z) ]d π/2κ r, d + π/2κ r [. Σ is finite modulo 2πZ. The map h S d (h) induces a morphism of differential field.
21 Let us consider h(z) C[[z]] solution of a linear differential equation in coefficients germs of meromorphic functions. There exist Σ R (set of singular directions), (κ 1,..., κ r ) ( Q >0) r, such that if d / Σ, S d (h) = L d κ r L d κ 1 ˆB κ1 ˆB κr (h), is a germ of analytic solution on the sector arg(z) ]d π/2κ r, d + π/2κ r [. Σ is finite modulo 2πZ. The map h S d (h) induces a morphism of differential field.
22 Let us consider system z Y (z) = A(z)Y (z) with coefficients that are germs of meromorphic functions. Let Ĥ(z)eL log(z) e Q(z) be the Hukuhara-Turrittin solution of z Y (z) = A(z)Y (z). There exist Σ R finite modulo 2π, ε > 0, such that if d / Σ, S d (Ĥ(z) ) e L log(z) e Q(z) is solution of z Y (z) = A(z)Y (z) with entries that are germs of meromorphic functions on the sector arg(z) ]d π/2ε, d + π/2ε[.
23 Let us consider system z Y (z) = A(z)Y (z) with coefficients that are germs of meromorphic functions. Let Ĥ(z)eL log(z) e Q(z) be the Hukuhara-Turrittin solution of z Y (z) = A(z)Y (z). There exist Σ R finite modulo 2π, ε > 0, such that if d / Σ, S d (Ĥ(z) ) e L log(z) e Q(z) is solution of z Y (z) = A(z)Y (z) with entries that are germs of meromorphic functions on the sector arg(z) ]d π/2ε, d + π/2ε[.
24 For d R, let St d GL m (C) such that: ( ) ( ) S d Ĥ(z) e L log(z) e Q(z) = S d + Ĥ(z) e L log(z) e Q(z) St d, where d π/2ε < d < d < d + < d + π/2ε and [d, d[ ]d, d + ] Σ =.
25 Let us consider (1). Let Ĥ(z, t)el(t) log(z) e Q(z,t) be parameterized Hukuhara-Turrittin solution. If we restrict U, we may assume that: There exist (d i (t)) continuous in t and finite modulo 2πZ, that satisfies d i (t) < d i+1 (t). (κ 1,i,j,..., κ r,i,j ) ( Q >0) r, such that for all d(t) continuous in t that does not interect Σ t := d i (t), we have an analytic solution of (1): ) S (Ĥ(z, d(t) t) e L(t) log(z) e Q(z,t) := ) L d(t) κ r,i,j L d(t) κ 1,i,j ˆB κ1,i,j ˆB κr,i,j (Ĥi,j (z, t) e L(t) log(z) e Q(z,t),.
26 Let us consider (1). Let Ĥ(z, t)el(t) log(z) e Q(z,t) be parameterized Hukuhara-Turrittin solution. If we restrict U, we may assume that: There exist (d i (t)) continuous in t and finite modulo 2πZ, that satisfies d i (t) < d i+1 (t). (κ 1,i,j,..., κ r,i,j ) ( Q >0) r, such that for all d(t) continuous in t that does not interect Σ t := d i (t), we have an analytic solution of (1): ) S (Ĥ(z, d(t) t) e L(t) log(z) e Q(z,t) := ) L d(t) κ r,i,j L d(t) κ 1,i,j ˆB κ1,i,j ˆB κr,i,j (Ĥi,j (z, t) e L(t) log(z) e Q(z,t),.
27 For d(t) Σ t continuous in t, let t St d(t) GL m (M U ) such that for all t 0 U, St d(t 0) GL m (C), is the Stokes matrix in the direction t 0 of z Y (z, t 0 ) = A(z, t 0 )Y (z, t 0 ). Proposition St d(t) Aut t z ( KU K U ).
28 For d(t) Σ t continuous in t, let t St d(t) GL m (M U ) such that for all t 0 U, St d(t 0) GL m (C), is the Stokes matrix in the direction t 0 of z Y (z, t 0 ) = A(z, t 0 )Y (z, t 0 ). Proposition St d(t) Aut t z ( KU K U ).
29 Let us consider (1). Let K U K U be the ( parameterized ) Picard-Vessiot extension and Aut t KU z K U be the Galois group. Theorem (D) The group generated by the (parameterized) monodromy matrix, the exponential torus and the Stokes matrices is dense for Kolchin topology in Aut t z ( KU K U ).
30 Let us consider z Y (z, t) = A(z, t)y (z, t) with A M m (M U (z)). We can define, MU (z) M ( U (z), parameterized ) Picard-Vessiot extension and Aut t MU z (z) M U (z) the Galois group. Theorem (D) The group generated by the (parameterized) monodromy matrix, the exponential torus and the Stokes matrices of all the singularities is dense for Kolchin topology in Aut t z ( ) MU (z) M U (z).
31 Definition ) Let t0 := z. Let A 0 M m (M U (z). We say that the linear differential equation t0 Y = A 0 Y ) is completely integrable if there exist A 1,..., A n M m (M U (z) such that, for all 0 i, j n, ti A j tj A i = A i A j A j A i.
32 Theorem (Cassidy/Singer 1 2, D 1 3) We have the equivalence: 1 t0 Y = A 0 Y is completely integrable. 2 The parameterized differential Galois group of t0 Y = A 0 Y is conjugated to an algebraic subgroup of GL m (C). 3 The topological generator of the parameterized differential Galois group of t0 Y = A 0 Y are conjugated to constant matrices.
33 Theorem (Cassidy/Singer 1 2, D 1 3) We have the equivalence: 1 t0 Y = A 0 Y is completely integrable. 2 The parameterized differential Galois group of t0 Y = A 0 Y is conjugated to an algebraic subgroup of GL m (C). 3 The topological generator of the parameterized differential Galois group of t0 Y = A 0 Y are conjugated to constant matrices.
34 Theorem (Cassidy/Singer 1 2, D 1 3) We have the equivalence: 1 t0 Y = A 0 Y is completely integrable. 2 The parameterized differential Galois group of t0 Y = A 0 Y is conjugated to an algebraic subgroup of GL m (C). 3 The topological generator of the parameterized differential Galois group of t0 Y = A 0 Y are conjugated to constant matrices.
35 Definition We say that k is a so-called universal t -field, if it has characteristic 0 and for any t -field k 0 k, t -finitely generated over Q, and any t -finitely generated extension k 1 of k 0, there is a t -differential k 0 -isomorphism of k 1 into k.
36 Theorem ( D, Mitschi/Singer) Let G be a differential subgroup of GL m (k). Then, G is the global parameterized differential Galois group of some equation having coefficients in k(z) if and only if G contains a finitely generated subgroup that is Kolchin-dense in G.
37 Cassidy, Phyllis J.; Singer, Michael F., Galois theory of parameterized differential equations and linear differential algebraic groups. Differential equations and quantum groups, , IRMA Lect. Math. Theor. Phys., 9, Eur. Math. Soc., Zürich, Dreyfus, Thomas, A density theorem for parameterized differential Galois theory. To appear in Pacific Journal of Mathematics. Mitschi, Claude; Singer, Michael F., Monodromy groups of parameterized linear differential equations with regular singularities. Bull. Lond. Math. Soc. 44 (2012), no. 5,
Generic Picard-Vessiot extensions for connected by finite groups Kolchin Seminar in Differential Algebra October 22nd, 2005
Generic Picard-Vessiot extensions for connected by finite groups Kolchin Seminar in Differential Algebra October 22nd, 2005 Polynomial Galois Theory case Noether first introduced generic equations in connection
More informationTutorial on Differential Galois Theory III
Tutorial on Differential Galois Theory III T. Dyckerhoff Department of Mathematics University of Pennsylvania 02/14/08 / Oberflockenbach Outline Today s plan Monodromy and singularities Riemann-Hilbert
More informationOn the structure of Picard-Vessiot extensions - Joint work with Arne Ledet - Kolchin Seminar in Differential Algebra May 06, 2006
On the structure of Picard-Vessiot extensions - Joint work with Arne Ledet - Kolchin Seminar in Differential Algebra May 06, 2006 (Sixth Visit Since March 17, 2001) 1 K is assumed to be a differential
More informationLinear Algebraic Groups as Parameterized Picard-Vessiot Galois Groups
See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/51928543 Linear Algebraic Groups as Parameterized Picard-Vessiot Galois Groups Article in Journal
More informationPacific Journal of Mathematics
Pacific Journal of Mathematics PICARD VESSIOT EXTENSIONS WITH SPECIFIED GALOIS GROUP TED CHINBURG, LOURDES JUAN AND ANDY R. MAGID Volume 243 No. 2 December 2009 PACIFIC JOURNAL OF MATHEMATICS Vol. 243,
More informationSymmetries of meromorphic connections over Riemann Surfaces
Symmetries of meromorphic connections over Riemann Surfaces Camilo Sanabria The CUNY Graduate Center February 13 th, 2009 Example Consider: y + 3(3z2 1) z(z 1)(z + 1) y + 221z4 206z 2 + 5 12z 2 (z 1) 2
More informationProfinite Groups. Hendrik Lenstra. 1. Introduction
Profinite Groups Hendrik Lenstra 1. Introduction We begin informally with a motivation, relating profinite groups to the p-adic numbers. Let p be a prime number, and let Z p denote the ring of p-adic integers,
More informationDifferential Groups and Differential Relations. Michael F. Singer Department of Mathematics North Carolina State University Raleigh, NC USA
Differential Groups and Differential Relations Michael F. Singer Department of Mathematics North Carolina State University Raleigh, NC 27695-8205 USA http://www.math.ncsu.edu/ singer Theorem: (Hölder,
More informationBackground on Chevalley Groups Constructed from a Root System
Background on Chevalley Groups Constructed from a Root System Paul Tokorcheck Department of Mathematics University of California, Santa Cruz 10 October 2011 Abstract In 1955, Claude Chevalley described
More informationLiouvillian solutions of third order differential equations
Article Submitted to Journal of Symbolic Computation Liouvillian solutions of third order differential equations Felix Ulmer IRMAR, Université de Rennes, 0 Rennes Cedex, France felix.ulmer@univ-rennes.fr
More informationInverse Galois Problem for C(t)
Inverse Galois Problem for C(t) Padmavathi Srinivasan PuMaGraSS March 2, 2012 Outline 1 The problem 2 Compact Riemann Surfaces 3 Covering Spaces 4 Connection to field theory 5 Proof of the Main theorem
More informationON THE CONSTRUCTIVE INVERSE PROBLEM IN DIFFERENTIAL GALOIS THEORY #
Communications in Algebra, 33: 3651 3677, 2005 Copyright Taylor & Francis, Inc. ISSN: 0092-7872 print/1532-4125 online DOI: 10.1080/00927870500243304 ON THE CONSTRUCTIVE INVERSE PROBLEM IN DIFFERENTIAL
More informationPicard-Vessiot theory, algebraic groups and group schemes
Picard-Vessiot theory, algebraic groups and group schemes Jerald J. Kovacic Department of Mathematics The City College of The City University of New York New York, NY 003 email: jkovacic@verizon.net URL:
More informationHyperelliptic Jacobians in differential Galois theory (preliminary report)
Hyperelliptic Jacobians in differential Galois theory (preliminary report) Jerald J. Kovacic Department of Mathematics The City College of The City University of New York New York, NY 10031 jkovacic@member.ams.org
More informationTrace fields of knots
JT Lyczak, February 2016 Trace fields of knots These are the knotes from the seminar on knot theory in Leiden in the spring of 2016 The website and all the knotes for this seminar can be found at http://pubmathleidenunivnl/
More informationTwists and residual modular Galois representations
Twists and residual modular Galois representations Samuele Anni University of Warwick Building Bridges, Bristol 10 th July 2014 Modular curves and Modular Forms 1 Modular curves and Modular Forms 2 Residual
More informationON GENERALIZED HYPERGEOMETRIC EQUATIONS AND MIRROR MAPS
ON GENERALIZED HYPERGEOMETRIC EQUATIONS AND MIRROR MAPS JULIEN ROQUES Abstract This paper deals with generalized hypergeometric differential equations of order n 3 having maximal unipotent monodromy at
More informationz, w = z 1 w 1 + z 2 w 2 z, w 2 z 2 w 2. d([z], [w]) = 2 φ : P(C 2 ) \ [1 : 0] C ; [z 1 : z 2 ] z 1 z 2 ψ : P(C 2 ) \ [0 : 1] C ; [z 1 : z 2 ] z 2 z 1
3 3 THE RIEMANN SPHERE 31 Models for the Riemann Sphere One dimensional projective complex space P(C ) is the set of all one-dimensional subspaces of C If z = (z 1, z ) C \ 0 then we will denote by [z]
More informationGalois Correspondence Theorem for Picard-Vessiot Extensions
Arnold Math J. (2016) 2:21 27 DOI 10.1007/s40598-015-0029-z RESEARCH CONTRIBUTION Galois Correspondence Theorem for Picard-Vessiot Extensions Teresa Crespo 1 Zbigniew Hajto 2 Elżbieta Sowa-Adamus 3 Received:
More informationMathematical Research Letters 3, (1996) EQUIVARIANT AFFINE LINE BUNDLES AND LINEARIZATION. Hanspeter Kraft and Frank Kutzschebauch
Mathematical Research Letters 3, 619 627 (1996) EQUIVARIANT AFFINE LINE BUNDLES AND LINEARIZATION Hanspeter Kraft and Frank Kutzschebauch Abstract. We show that every algebraic action of a linearly reductive
More informationCurtis Heberle MTH 189 Final Paper 12/14/2010. Algebraic Groups
Algebraic Groups Curtis Heberle MTH 189 Final Paper 12/14/2010 The primary objects of study in algebraic geometry are varieties. Having become acquainted with these objects, it is interesting to consider
More informationCorrection to On Satake parameters for representations with parahoric fixed vectors
T. J. Haines (2017) Correction to On Satake parameters for representations with parahoric fixed vectors, International Mathematics Research Notices, Vol. 2017, No. 00, pp. 1 11 doi: 10.1093/imrn/rnx088
More informationREMARKS ON THE INTRINSIC INVERSE PROBLEM. Daniel BERTRAND Institut de Mathématiques de Jussieu
BANACH CENTER PUBLICATIONS, VOLUME ** REMARKS ON THE INTRINSIC INVERSE PROBLEM Daniel BERTRAND Institut de Mathématiques de Jussieu E-mail: bertrand@math.jussieu.fr Abstract : The intrinsic differential
More informationCover Page. The handle holds various files of this Leiden University dissertation.
Cover Page The handle http://hdl.handle.net/1887/22043 holds various files of this Leiden University dissertation. Author: Anni, Samuele Title: Images of Galois representations Issue Date: 2013-10-24 Chapter
More informationMetabelian Galois Representations
Metabelian Galois Representations Edray Herber Goins AMS 2018 Spring Western Sectional Meeting Special Session on Automorphisms of Riemann Surfaces and Related Topics Portland State University Department
More informationDemushkin s Theorem in Codimension One
Universität Konstanz Demushkin s Theorem in Codimension One Florian Berchtold Jürgen Hausen Konstanzer Schriften in Mathematik und Informatik Nr. 176, Juni 22 ISSN 143 3558 c Fachbereich Mathematik und
More informationVariations on a Theme: Fields of Definition, Fields of Moduli, Automorphisms, and Twists
Variations on a Theme: Fields of Definition, Fields of Moduli, Automorphisms, and Twists Michelle Manes (mmanes@math.hawaii.edu) ICERM Workshop Moduli Spaces Associated to Dynamical Systems 17 April, 2012
More informationTOPOLOGICAL GALOIS THEORY. A. Khovanskii
TOPOLOGICAL GALOIS THEORY A. Khovanskii METHODS 1. ABEL AND LIOUVILLE (1833). 2.GALOIS AND PICARD- VESSIO (1910). 3. TOPOLOGICAL VERSION: ONE DIMENSIONAL CASE (1972), MULTI- DIMENSIONAL CASE (2003). 2
More informationFORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS
Sairaiji, F. Osaka J. Math. 39 (00), 3 43 FORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS FUMIO SAIRAIJI (Received March 4, 000) 1. Introduction Let be an elliptic curve over Q. We denote by ˆ
More informationTopological approach to 13th Hilbert problem
Topological approach to 13th Hilbert problem Workshop Braids, Resolvent Degree and Hilberts 13th Problem Askold Khovanskii Department of Mathematics, University of Toronto February 19, 2019 1.13-th HILBERT
More informationQuantizations and classical non-commutative non-associative algebras
Journal of Generalized Lie Theory and Applications Vol. (008), No., 35 44 Quantizations and classical non-commutative non-associative algebras Hilja Lisa HURU and Valentin LYCHAGIN Department of Mathematics,
More informationTHE QUANTUM CONNECTION
THE QUANTUM CONNECTION MICHAEL VISCARDI Review of quantum cohomology Genus 0 Gromov-Witten invariants Let X be a smooth projective variety over C, and H 2 (X, Z) an effective curve class Let M 0,n (X,
More informationMath 210C. A non-closed commutator subgroup
Math 210C. A non-closed commutator subgroup 1. Introduction In Exercise 3(i) of HW7 we saw that every element of SU(2) is a commutator (i.e., has the form xyx 1 y 1 for x, y SU(2)), so the same holds for
More informationProblem 4 (Wed Jan 29) Let G be a finite abelian group. Prove that the following are equivalent
Last revised: May 16, 2014 A.Miller M542 www.math.wisc.edu/ miller/ Problem 1 (Fri Jan 24) (a) Find an integer x such that x = 6 mod 10 and x = 15 mod 21 and 0 x 210. (b) Find the smallest positive integer
More informationOn the renormalization problem of multiple zeta values
On the renormalization problem of multiple zeta values joint work with K. Ebrahimi-Fard, D. Manchon and J. Zhao Johannes Singer Universität Erlangen Paths to, from and in renormalization Universität Potsdam
More informationOn the Definitions of Difference Galois Groups
On the Definitions of Difference Galois Groups Zoé Chatzidakis CNRS - Université Paris 7 Charlotte Hardouin Universität Heidelberg IWR INF 368 Michael F. Singer University of North Carolina Summary We
More informationWhat is the Langlands program all about?
What is the Langlands program all about? Laurent Lafforgue November 13, 2013 Hua Loo-Keng Distinguished Lecture Academy of Mathematics and Systems Science, Chinese Academy of Sciences This talk is mainly
More informationLinear-fractionally-induced Composition. Operators
by Linear-fractionally-induced Spurrier James Madison University March 19, 2011 by Notation Let D denote the open unit disk and T the unit circle. H 2 (D) is the space of all functions f = n=0 a nz n that
More informationPart II. Riemann Surfaces. Year
Part II Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 96 Paper 2, Section II 23F State the uniformisation theorem. List without proof the Riemann surfaces which are uniformised
More informationStructure of Compact Quantum Groups A u (Q) and B u (Q) and their Isomorphism Classification
Structure of Compact Quantum Groups A u (Q) and B u (Q) and their Isomorphism Classification Shuzhou Wang Department of Mathematics University of Georgia 1. The Notion of Quantum Groups G = Simple compact
More informationDi erentially Closed Fields
Di erentially Closed Fields Phyllis Cassidy City College of CUNY Kolchin Seminar in Di erential Algebra Graduate Center of CUNY November 16, 2007 Some Preliminary Language = fδ 1,..., δ m g, commuting
More informationDifferential Central Simple Algebras and Picard Vessiot Representations
Differential Central Simple Algebras and Picard Vessiot Representations Lourdes Juan Department of Mathematics Texas Tech University Lubbock TX 79409 Andy R. Magid Department of Mathematics University
More informationl-adic Representations
l-adic Representations S. M.-C. 26 October 2016 Our goal today is to understand l-adic Galois representations a bit better, mostly by relating them to representations appearing in geometry. First we ll
More informationAn algorithm for solving second order linear homogeneous differential equations
An algorithm for solving second order linear homogeneous differential equations Jerald J. Kovacic Department of Mathematics The City College of The City University of New York New York, NY 10031 email:
More informationResidual modular Galois representations: images and applications
Residual modular Galois representations: images and applications Samuele Anni University of Warwick London Number Theory Seminar King s College London, 20 th May 2015 Mod l modular forms 1 Mod l modular
More informationHIGHER HOLONOMY OF FORMAL HOMOLOGY CONNECTIONS AND BRAID COBORDISMS
HIGHER HOLONOMY OF FORMAL HOMOLOGY CONNECTIONS AND BRAID COBORDISMS TOSHITAKE KOHNO Abstract We construct a representation of the homotopy 2-groupoid of a manifold by means of K-T Chen s formal homology
More informationClassification of Algebraic Subgroups of Lower Order Unipotent Algebraic Groups
Classification of Algebraic Subgroups of Lower Order Unipotent Algebraic Groups AMS Special Session, Boston, MA; Differential Algebraic Geometry and Galois Theory (in memory of J. Kovacic) V. Ravi Srinivasan
More informationCommutative nilpotent rings and Hopf Galois structures
Commutative nilpotent rings and Hopf Galois structures Lindsay Childs Exeter, June, 2015 Lindsay Childs Commutative nilpotent rings and Hopf Galois structures Exeter, June, 2015 1 / 1 Hopf Galois structures
More information8 8 THE RIEMANN MAPPING THEOREM. 8.1 Simply Connected Surfaces
8 8 THE RIEMANN MAPPING THEOREM 8.1 Simply Connected Surfaces Our aim is to prove the Riemann Mapping Theorem which states that every simply connected Riemann surface R is conformally equivalent to D,
More informationLimits in differential fields of holomorphic germs
Limits in differential fields of holomorphic germs D. Gokhman Division of Mathematics, Computer Science and Statistics University of Texas at San Antonio Complex Variables 28:27 36 (1995) c 1995 Gordon
More informationSan Francisco State University
AN INTRODUCTION TO DIFFERENTIAL GALOIS THEORY BRUCE SIMON San Francisco State University Abstract. Differential Galois theory takes the approach of algebraic Galois theory and applies it to differential
More informationOn values of Modular Forms at Algebraic Points
On values of Modular Forms at Algebraic Points Jing Yu National Taiwan University, Taipei, Taiwan August 14, 2010, 18th ICFIDCAA, Macau Hermite-Lindemann-Weierstrass In value distribution theory the exponential
More informationMATRIX LIE GROUPS AND LIE GROUPS
MATRIX LIE GROUPS AND LIE GROUPS Steven Sy December 7, 2005 I MATRIX LIE GROUPS Definition: A matrix Lie group is a closed subgroup of Thus if is any sequence of matrices in, and for some, then either
More informationTHE GALOIS CORRESPONDENCE FOR LINEAR HOMOGENEOUS DIFFERENCE EQUATIONS
THE GALOIS CORRESPONDENCE FOR LINEAR HOMOGENEOUS DIFFERENCE EQUATIONS CHARLES H. FRANKE 1. Summary. In general the notation and terminology are as in [l ]. We assume the following throughout. All fields
More informationThe Grothendieck-Katz Conjecture for certain locally symmetric varieties
The Grothendieck-Katz Conjecture for certain locally symmetric varieties Benson Farb and Mark Kisin August 20, 2008 Abstract Using Margulis results on lattices in semi-simple Lie groups, we prove the Grothendieck-
More informationExistence of a Picard-Vessiot extension
Existence of a Picard-Vessiot extension Jerald Kovacic The City College of CUNY jkovacic@verizon.net http:/mysite.verizon.net/jkovacic November 3, 2006 1 Throughout, K is an ordinary -field of characteristic
More informationSpring 2018 CIS 610. Advanced Geometric Methods in Computer Science Jean Gallier Homework 3
Spring 2018 CIS 610 Advanced Geometric Methods in Computer Science Jean Gallier Homework 3 March 20; Due April 5, 2018 Problem B1 (80). This problem is from Knapp, Lie Groups Beyond an Introduction, Introduction,
More informationRiemann-Hilbert problems from Donaldson-Thomas theory
Riemann-Hilbert problems from Donaldson-Thomas theory Tom Bridgeland University of Sheffield Preprints: 1611.03697 and 1703.02776. 1 / 25 Motivation 2 / 25 Motivation Two types of parameters in string
More informationENDOMORPHISMS AND BIJECTIONS OF THE CHARACTER
ENDOMORPHISMS AND BIJECTIONS OF THE CHARACTER VARIETY χ(f 2,SL 2 (C)) SERGE CANTAT ABSTRACT. We answer a question of Gelander and Souto in the special case of the free group of rank 2. The result may be
More informationUniversal Covers and Category Theory in Polynomial and Differential Galois Theory
Fields Institute Communications Volume 00, 0000 Universal Covers and Category Theory in Polynomial and Differential Galois Theory Andy R. Magid Department of Mathematics University of Oklahoma Norman OK
More informationOn combinatorial zeta functions
On combinatorial zeta functions Christian Kassel Institut de Recherche Mathématique Avancée CNRS - Université de Strasbourg Strasbourg, France Colloquium Potsdam 13 May 2015 Generating functions To a sequence
More informationµ INVARIANT OF NANOPHRASES YUKA KOTORII TOKYO INSTITUTE OF TECHNOLOGY GRADUATE SCHOOL OF SCIENCE AND ENGINEERING 1. Introduction A word will be a sequ
µ INVARIANT OF NANOPHRASES YUKA KOTORII TOKYO INSTITUTE OF TECHNOLOGY GRADUATE SCHOOL OF SCIENCE AND ENGINEERING 1. Introduction A word will be a sequence of symbols, called letters, belonging to a given
More informationTHE HITCHIN FIBRATION
THE HITCHIN FIBRATION Seminar talk based on part of Ngô Bao Châu s preprint: Le lemme fondamental pour les algèbres de Lie [2]. Here X is a smooth connected projective curve over a field k whose genus
More informationCOMPLEX ANALYSIS-II HOMEWORK
COMPLEX ANALYSIS-II HOMEWORK M. LYUBICH Homework (due by Thu Sep 7). Cross-ratios and symmetries of the four-punctured spheres The cross-ratio of four distinct (ordered) points (z,z 2,z 3,z 4 ) Ĉ4 on the
More informationENTRY GROUP THEORY. [ENTRY GROUP THEORY] Authors: started Mark Lezama: October 2003 Literature: Algebra by Michael Artin, Mathworld.
ENTRY GROUP THEORY [ENTRY GROUP THEORY] Authors: started Mark Lezama: October 2003 Literature: Algebra by Michael Artin, Mathworld Group theory [Group theory] is studies algebraic objects called groups.
More informationLocal Langlands correspondence and examples of ABPS conjecture
Local Langlands correspondence and examples of ABPS conjecture Ahmed Moussaoui UPMC Paris VI - IMJ 23/08/203 Notation F non-archimedean local field : finite extension of Q p or F p ((t)) O F = {x F, v(x)
More informationReal representations
Real representations 1 Definition of a real representation Definition 1.1. Let V R be a finite dimensional real vector space. A real representation of a group G is a homomorphism ρ VR : G Aut V R, where
More informationA COMPUTATION OF MINIMAL POLYNOMIALS OF SPECIAL VALUES OF SIEGEL MODULAR FUNCTIONS
MATHEMATICS OF COMPUTATION Volume 7 Number 4 Pages 969 97 S 5-571814-8 Article electronically published on March A COMPUTATION OF MINIMAL POLYNOMIALS OF SPECIAL VALUES OF SIEGEL MODULAR FUNCTIONS TSUYOSHI
More informationLecture 4.1: Homomorphisms and isomorphisms
Lecture 4.: Homomorphisms and isomorphisms Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4, Modern Algebra M. Macauley (Clemson) Lecture
More informationp-divisible Groups and the Chromatic Filtration
p-divisible Groups and the Chromatic Filtration January 20, 2010 1 Chromatic Homotopy Theory Some problems in homotopy theory involve studying the interaction between generalized cohomology theories. This
More informationLecture III. Five Lie Algebras
Lecture III Five Lie Algebras 35 Introduction The aim of this talk is to connect five different Lie algebras which arise naturally in different theories. Various conjectures, put together, state that they
More informationReductive group actions and some problems concerning their quotients
Reductive group actions and some problems concerning their quotients Brandeis University January 2014 Linear Algebraic Groups A complex linear algebraic group G is an affine variety such that the mappings
More informationDifferential Algebra and Related Topics
Differential Algebra and Related Topics Workshop Abstracts November 2 3, 2000, Rutgers University at Newark Differential Algebra and Symbolic Integration Manuel Bronstein Abstract: An elementary function
More informationArboreal Cantor Actions
Arboreal Cantor Actions Olga Lukina University of Illinois at Chicago March 15, 2018 1 / 19 Group actions on Cantor sets Let X be a Cantor set. Let G be a countably generated discrete group. The action
More informationTHE TATE MODULE. Seminar: Elliptic curves and the Weil conjecture. Yassin Mousa. Z p
THE TATE MODULE Seminar: Elliptic curves and the Weil conjecture Yassin Mousa Abstract This paper refers to the 10th talk in the seminar Elliptic curves and the Weil conjecture supervised by Prof. Dr.
More informationWeil-étale Cohomology
Weil-étale Cohomology Igor Minevich March 13, 2012 Abstract We will be talking about a subject, almost no part of which is yet completely defined. I will introduce the Weil group, Grothendieck topologies
More informationLanglands parameters of quivers in the Sato Grassmannian
Langlands parameters of quivers in the Sato Grassmannian Martin T. Luu, Matej enciak Abstract Motivated by quantum field theoretic partition functions that can be expressed as products of tau functions
More informationIIT Mumbai 2015 MA 419, Basic Algebra Tutorial Sheet-1
IIT Mumbai 2015 MA 419, Basic Algebra Tutorial Sheet-1 Let Σ be the set of all symmetries of the plane Π. 1. Give examples of s, t Σ such that st ts. 2. If s, t Σ agree on three non-collinear points, then
More informationSTRUCTURE THEORY OF UNIMODULAR LATTICES
STRUCTURE THEORY OF UNIMODULAR LATTICES TONY FENG 1 Unimodular Lattices 11 Definitions Let E be a lattice, by which we mean a free abelian group equipped with a symmetric bilinear form, : E E Z Definition
More informationON THE MATRIX EQUATION XA AX = τ(x)
Applicable Analysis and Discrete Mathematics, 1 (2007), 257 264. Available electronically at http://pefmath.etf.bg.ac.yu Presented at the conference: Topics in Mathematical Analysis and Graph Theory, Belgrade,
More informationRes + X F F + is defined below in (1.3). According to [Je-Ki2, Definition 3.3 and Proposition 3.4], the value of Res + X
Theorem 1.2. For any η HH (N) we have1 (1.1) κ S (η)[n red ] = c η F. Here HH (F) denotes the H-equivariant Euler class of the normal bundle ν(f), c is a non-zero constant 2, and is defined below in (1.3).
More informationThe Hilbert-Mumford Criterion
The Hilbert-Mumford Criterion Klaus Pommerening Johannes-Gutenberg-Universität Mainz, Germany January 1987 Last change: April 4, 2017 The notions of stability and related notions apply for actions of algebraic
More informationThe Yang-Mills equations over Klein surfaces
The Yang-Mills equations over Klein surfaces Chiu-Chu Melissa Liu & Florent Schaffhauser Columbia University (New York City) & Universidad de Los Andes (Bogotá) Seoul ICM 2014 Outline 1 Moduli of real
More informationRepresentations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III
Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III Lie algebras. Let K be again an algebraically closed field. For the moment let G be an arbitrary algebraic group
More informationCover Page. The handle holds various files of this Leiden University dissertation.
Cover Page The handle http://hdl.handle.net/1887/20310 holds various files of this Leiden University dissertation. Author: Jansen, Bas Title: Mersenne primes and class field theory Date: 2012-12-18 Chapter
More informationMath 249B. Nilpotence of connected solvable groups
Math 249B. Nilpotence of connected solvable groups 1. Motivation and examples In abstract group theory, the descending central series {C i (G)} of a group G is defined recursively by C 0 (G) = G and C
More informationReal and p-adic Picard-Vessiot fields
Spring Central Sectional Meeting Texas Tech University, Lubbock, Texas Special Session on Differential Algebra and Galois Theory April 11th 2014 Real and p-adic Picard-Vessiot fields Teresa Crespo, Universitat
More informationNoetherian property of infinite EI categories
Noetherian property of infinite EI categories Wee Liang Gan and Liping Li Abstract. It is known that finitely generated FI-modules over a field of characteristic 0 are Noetherian. We generalize this result
More informationGalois fields in quantum mechanics A. Vourdas University of Bradford
Galois fields in quantum mechanics A. Vourdas University of Bradford Phase space methods: position and momentum in the ring Z(d) and the field GF(p l ) Finite quantum systems (eg spin) phase space: toroidal
More informationThe 3-local cohomology of the Mathieu group M 24
The 3-local cohomology of the Mathieu group M 24 David John Green Institut für Experimentelle Mathematik Universität GHS Essen Ellernstraße 29 D 45326 Essen Germany Email: david@exp-math.uni-essen.de 11
More informationProjective Quantum Spaces
DAMTP/94-81 Projective Quantum Spaces U. MEYER D.A.M.T.P., University of Cambridge um102@amtp.cam.ac.uk September 1994 arxiv:hep-th/9410039v1 6 Oct 1994 Abstract. Associated to the standard SU (n) R-matrices,
More information1 Notations and Statement of the Main Results
An introduction to algebraic fundamental groups 1 Notations and Statement of the Main Results Throughout the talk, all schemes are locally Noetherian. All maps are of locally finite type. There two main
More informationDescent for differential Galois theory of difference equations Confluence and q-dependency
Descent for differential Galois theory of difference equations Confluence and q-dependency Lucia Di Vizio and Charlotte Hardouin November 8, 207 arxiv:03.5067v2 [math.qa] 30 Nov 20 Abstract The present
More informationCLASS FIELD THEORY WEEK Motivation
CLASS FIELD THEORY WEEK 1 JAVIER FRESÁN 1. Motivation In a 1640 letter to Mersenne, Fermat proved the following: Theorem 1.1 (Fermat). A prime number p distinct from 2 is a sum of two squares if and only
More informationComputing modular polynomials in dimension 2 ECC 2015, Bordeaux
Computing modular polynomials in dimension 2 ECC 2015, Bordeaux Enea Milio 29/09/2015 Enea Milio Computing modular polynomials 29/09/2015 1 / 49 Computing modular polynomials 1 Dimension 1 : elliptic curves
More informationOn orderability of fibred knot groups
Under consideration for publication in Math. Proc. Camb. Phil. Soc. 1 On orderability of fibred knot groups By BERNARD PERRON Laboratoire de Topologie, Université de Bourgogne, BP 47870 21078 - Dijon Cedex,
More informationFields and Galois Theory
Fields and Galois Theory Rachel Epstein September 12, 2006 All proofs are omitted here. They may be found in Fraleigh s A First Course in Abstract Algebra as well as many other algebra and Galois theory
More informationSylow subgroups of GL(3,q)
Jack Schmidt We describe the Sylow p-subgroups of GL(n, q) for n 4. These were described in (Carter & Fong, 1964) and (Weir, 1955). 1 Overview The groups GL(n, q) have three types of Sylow p-subgroups:
More informationRiemann surfaces with extra automorphisms and endomorphism rings of their Jacobians
Riemann surfaces with extra automorphisms and endomorphism rings of their Jacobians T. Shaska Oakland University Rochester, MI, 48309 April 14, 2018 Problem Let X be an algebraic curve defined over a field
More informationRepresentations and Linear Actions
Representations and Linear Actions Definition 0.1. Let G be an S-group. A representation of G is a morphism of S-groups φ G GL(n, S) for some n. We say φ is faithful if it is a monomorphism (in the category
More information