Kähler configurations of points


 Hillary Fletcher
 2 years ago
 Views:
Transcription
1 Kähler configurations of points Simon Salamon Oxford, 22 May 2017
2 The Hesse configuration 1/24 Let ω = e 2πi/3. Consider the nine points [0, 1, 1] [0, 1, ω] [0, 1, ω 2 ] [1, 0, 1] [1, 0, ω] [1, 0, ω 2 ] [1, 1, 0] [1, ω, 0] [1, ω 2, 0] of CP 2. They are the flexes of the nonsingular cubic for any c C \ {1, ω, ω 3 } and belong in triples on 12 lines: x 3 + y 3 + z 3 3c xyz = 0
3 A 1parameter family of nine points 2/24 The representative vectors all have norm 1/ 2, and any two satisfy w, z = 1. It follows that the 9 points are mutually equidistant. The same is true of the set M θ consisting of [0, 1, 1] [0, 1, ω] [0, 1, ω 2 ] [1, 0, 1] [1, 0, ω] [1, 0, ω 2 ] [e iθ, 1, 0] [e iθ, ω, 0] [e iθ, ω 2, 0] Theorem. Any unordered set of nine mutually equidistant points in (CP 2, g) is isometric to M θ for some angle θ. L. Hughston, S. Salamon, in Advances Math. 2016
4 FubiniStudy distance 3/24 Complex projective space CP n 1 is a compact Kähler manifold. Its Riemannian metric g arises from the standard Hermitian form w, z = w z = n w i z i, i=1 on C n, which is invariant by the unitary group U(n). The associated distance d satisfies cos 2 ( 1 2 d([w], [z])) = w, z 2 w 2 z 2 = w, z z, w w, w z, z. The RHS is really a cross ratio of four points [w], [z], [w ], [z ].
5 Moment mapping 4/24 The 2form ω 0 = i dz i dz i on C n induces a symplectic form on CP n 1, regarded as a symplectic quotient of C n by U(1). Let H = {A C n,n : A = A, tr A = 1} su(n). The mapping S 2n 1 H for which [z] z z = z 1 2 z 1 z 2 z 1 z 3 z 2 z 1 z 2 2 z 2 z 3 z 3 z 1 z 3 z 2 z 3 2 is U(n)equivariant and defines an isometric embedding CP n 1 H R n2 1.
6 SICPOVM 5/24 is shorthand for Symmetric Informationally Complete Positive Operator Valued Measure. We shall consider such objects in finitedimensional Hilbert space C n. Definition 1. A SICPOVM is a set of n 2 points [z α ] in CP n 1 such that (with the normalization z α = 1) z α, z β 2 = λ, α β, for some fixed λ (0, 1). This defines a subset {P α = z αz α } of H and n 2 { 1 α = β P α = n I, tr(p α P β ) = λ α β. α=1 E. B. Davies: The Quantum Theory of Open Systems, 1976
7 SIC sets 6/24 Definition 2. A SIC set consists of a regular simplex in su(n) whose n 2 vertices lie in the adjoint orbit CP n 1. In fact, n 2 is the maximum number of mutually equidistant points possible in CP n 1, and in this case λ must equal 1 n+1. Proof. Given an equidistant set {P α }, set Q β = P β λi. Then tr(p α Q β ) = { 1 λ α = β 0 α β So {P α } is linearly independent in i u(n). Applying tr(q β ) to I = c α P α, gives 1 λn = c α (1 λ) α. Then n = n 2 c α, and so n(1 λn) = 1 λ.
8 Example: the 2sphere 7/24 The embedding CP 1 = S 2 R 3 is given by setting ( ) ( ) z1 2 z 1 z a b + ic z 1 z 2 z 2 2 = 1 2. b ic 1 a One SIC set consists of the points [1 + 3, 1 + i] [1 + i, 1 + 3] [1 + 3, 1 i] [ 1 i, 1 + 3] This is an orbit of A, B where A = ( ) ( ) , B = Any SIC set determines an inscribed tetrahedron and any two are congruent by SO(3) = SU(2)/Z 2. If n 3, two SIC sets are not in general congruent by SU(n).
9 Arbitrary dimensions 8/24 Conjecture 1. CP n 1 possesses a SIC set for all n. Algebraic solutions are known for n = 2, 3, 4,..., 15, 19, 24, 35, 48. Extensive numerical verification has been carried out for n 151. Conjecture 2. For all n, there is a SIC set that is an orbit of the (Weyl)Heisenberg group H (Z/nZ) 2. A vector z or point [z] whose orbit H [z] is a SIC set is called fiducial. An example for CP 3 is [ ] [z] = s i(r +s), 1 r + i, s + i(s r), 1+r + i. where r = 2 and s = G. Zauner, PhD thesis, Vienna, 1999
10 Example: a SIC set in CP 7 9/24 Using the identification C 8 = H 4, consider the groups V 1, right multiplication by 1, i, j, k Sp(1) V 2, double sign changes in H 4 V 3, double transpositions of the coordinates. Then G = V 1 V 2 V 3 = (Z2 ) 6 is a subgroup of Sp(4) SU(8). Fix unit quaternions p = 1 2 (1 + i + j k), q = 1 2 (1 + i j k). Proposition. G [0, p, q, j] is a SIC set in CP 7. This orbit cannot project neatly to HP 2 because no S 2 fibre can contain 4 points. S. G. Hoggar in Geometria Dedicata, 1998
11 Analogue: a Gosset polytope 10/24 Its 56 vertices form the orbit under S 8 Z 2 of (1, 1, 1, 3, 3, 1, 1, 1) R 8. They lie in a subspace R 7 and are the weights for the action of E 7 on C 28. All ( ) 56 2 inner products equal 8 or 8, defining 28 mutually equidistant points on RP 6.
12 A 2torus acting on CP 2 11/24 Fix T = { diag(e iθ 1, e iθ 2, e iθ 3 ) : θ i = 0 } in SU(3). Consider the moment map for the action of T : µ([z]) = 1 z 2 ( z 1 2, z 2 2, z 3 2 ) = (a 2, b 2, c 2 ). The image of µ is a 2simplex whose inscribed circle C will play a vital role: Let m 1, m 2, m 3 be the midpoints, and set C i = µ 1 (m i ) CP 2. Note that M θ C 1 C 2 C 3.
13 Correct separation 12/24 We set a= z 1 / z, b = z 2 / z, c = z 3 / z. Then C is the intersection of the plane a 2 + b 2 + c 2 = 1 and the sphere a 4 + b 4 + c 4 = 1 2. Lemma. The following are equivalent: µ([z]) C (a+b+c)( a+b+c)(a b+c)(a+b c) = 0 the three points z=[z 1, z 2, z 3 ], [z 1, ωz 2, ω 2 z 3 ], [z 1, ω 2 z 2, ωz 3 ] are the correct distance (d = 2 arccos 1 2 = 2π/3) apart to form part of a SIC set.
14 The generic point in a SIC set 13/24 Let S be a SIC set in CP 2. Up to the action of SU(3), we are free to assume that S contains the two points of C 1 given by z 1 = 1 2 (0, 1, ω), z 2 = 1 2 (0, 1, ω 2 ). Any other point [z] of S satisfies z, z j 2 = 1 4 z 2 for j = 0, 1. Lemma. µ([z]) C and we can take ] [z] = z[σ, φ] = [e iσ cos φ, cos(φ + 2π 3 ), cos(φ + 4π 3 ) for some π < σ π and π 2 < φ π 2. Thus, [z] lies in a 2torus, pinched to a point where φ = π/2.
15 The pinched torus T 14/24 containing z[0, φ + kπ 3 ] with k = 0, 1, 2, and two points in both C 2, C 3, forming a SIC set L φ when [z 1 ], [z 2 ] are added: C 2 C 3
16 The same thing in a rectangle 15/24 Recall that z[σ, φ] = [ e iσ cos φ, cos(φ + 2π 3 ), cos(φ + 4π 3 )]. Lemma. There exists a SIC set L φ containing [z 1 ], [z 2 ] and z[0, φ] for any φ with π 2 < φ π 2 :
17 Equivalent SIC sets 16/24 The remaining six points are then z[0, φ + π 3 ] z[ 2π 3, π 6 ] z[ 2π 3, π 6 ] C 2 z[ 2π 3, π 6 ] z[ 2π 3, π 6 ] C 3 z[0, φ π 3 ] Lemma. X L φ = M 2φ+π C 1 C 2 C 3, where ( ω 2 ) ω 1 X = 1 1 ω ω 2 3 U(3)
18 The Clifford group 17/24 is the normalizer G of H in U(n), isomorphic (modulo phase) to SL(2, Z n ) (Z n ) 2. For n = 3, G has order 216 and contains X, since Moreover, X 3 = iω 2 I. XAX 1 = ωb, XBX 1 = ω 2 A 1 B 1. Conjecture 3. A fiducial vector z can always be found in an eigenspace of the unitary matrix X where X ij = 1 n τ 2ij+j2, τ = e (n+1)πi/n. For n = 5 see: G. Horrocks, D. Mumford, in Topology 1973
19 Trigonometry 18/24 Let S be a SIC set containing [z 1 ], [z 2 ], and [z 3 ] = z[σ, φ], [z 4 ] = z[τ, ψ], [z 5 ] = z[υ, χ] T. Set x = tan φ, y = tan ψ, z = tan χ. Key lemma. [z 3 ] and [z 4 ] are distance 2π/3 apart iff cos(σ τ) = 1 If A + B + C = 0 then 9(1 + 2 cos 2(φ ψ)) 4(4 cos 2 φ cos 2 ψ + 3 sin 2φ sin 2ψ) = x 2 + 9y 2 27x 2 y 2 24xy, 16(1 + 3xy) cos 2 A + cos 2 B + cos 2 C = cos A cos B cos C.
20 Equilateral triangles in T 19/24 Specify their vertices by vertical coordinates x, y, z and set p = x + y + z, q = yz + zx + xy, r = xyz. Corollary. If z[σ, φ], z[τ, ψ], z[υ, χ] are all 2π/3 apart then where f (x, y, z) = F (p, q, r) = 0, F = 9 22p 2 + 9p q 126p 2 q + 27p 4 q + 298q 2 226p 2 q 2 +24p 4 q q 3 138p 2 q q q 5 3pr 50p 3 r 15p 5 r +88pqr 48p 3 qr + 234pq 2 r + 18p 3 q 2 r 144pq 3 r + 81pq 4 r + 189r 2 480p 2 r 2 153p 4 r qr 2 306p 2 qr q 2 r 2 486p 2 q 2 r q 3 r q 4 r 2 558pr 3 486p 3 r pqr 3 810pq 2 r r 4 162p 2 r qr q 2 r pr r 6. cf. p 2 3q = 3 4 in the Euclidean case.
21 Hexagons in CP 2 20/24 Let S be a SIC set containing [z 1 ], [z 2 ] and [z i ] = z[σ i, φ i ] for i = 3, 4, 5, 6. Set t = tan φ 1, x = tan φ 2, y = tan φ 3, z = tan φ 4, giving 4 equations in 4 unknowns: f (x, y, z) = f (t, y, z) = f (t, x, z) = f (t, x, y) = 0. If a, b, c, d are the elementary symmetric polynomials in t, x, y, z, we can convert this into a system F 1 (a, b, c, d) = F 2 (a, b, c, d) = F 3 (a, b, c, d) = F 4 (a, b, c, d) = 0. For example, F 1 = f (x, y, z) + f (t, y, z) + f (t, x, z) + f (t, x, y).
22 A quotient ideal 21/24 Solutions for which φ i =±π/6 give rise to L φ, and we may assume G = 1 3a 2 + 6b + 9b 2 18ac 27c d + 54bd + 81d 2 is nonzero. Consider I = F 1, F 2, F 3, F 4, and compute I : G = {r R[a, b, c, d] : r G I }, by finding a Gröbner basis for uf 1, uf 2, uf 3, uf 4, (1 u)g. With approriate orderings, its first element (/G ) is (d 1) 3 (3d 1) 3 (3+b+3d)(9d 1) 3 (1+3b+9d) 3 (19+9b+27d). Corollary. If S is not isometric to L φ, one of the following holds: 1 d = 1, 3, 1 9, b = (3d + 3), 1 3 (9d + 1), 1 9 (27d + 19).
23 Cross field passes 22/24 The case 9b + 27d + 19 = 0 gives rise to a 1parameter set of solutions to the system F 1 = F 2 = F 3 = F 4 = 0, such as (a, b, c, d) = (0, 22 9, c, 1 9 ). However, this gives rise to 9 points of which only 27 of the ( 9 2) distances equal 2π/3:
24 Conclusion 23/24 Theorem. Any SIC set in CP 2 is isometric to L φ for some φ and therefore to the set we started with. [0, 1, 1] [0, 1, ω] [0, 1, ω 2 ] [1, 0, 1] [1, 0, ω] [1, 0, ω 2 ] [e 2iφ, 1, 0] [e 2iφ, ω, 0] [e 2iφ, ω 2, 0] Classifying SICPOVM s in higher dimensions appears beyond the scope of present methods. For n = 4, the analogue of C consists of two parallel circles, isolated points of which generate 64 SIC sets in CP 3.
25 More references 24/24 M. Appleby: J. Math. Phys M. Appleby, S. Flammia, G. McConnell, J. Yard: arxiv: S, T. Flammia: J. Phys. A, 2006 J. M. Renes et al: J. Math. Phys A. J. Scott, M. Grassl: J. Math. Phys A. J. Scott: arxiv: G. Zauner: J. Quant. Inf H. Zhu: J. Phys. A 2010
26 Translationinvariant metrics 25/24 Here we see two different fibres µ 1 (p i ) = R 2 /Z 2, for p 1, p 2 C. The red curves are points of distance d from the origin:
27 The pinched torus T 26/24 We may assume that a third (black) point of S is given by z 3 =z[0, φ], so [z 1 ], [z 2 ], [z 3 ] form an equilateral triangle in CP 2. Most points the correct distance from [z 3 ] cannot belong to S :
arxiv: v2 [math.dg] 30 Sep 2015
SURVEYING POINTS IN THE COMPLEX PROJECTIVE PLANE LANE HUGHSTON AND SIMON SALAMON arxiv:1410.5862v2 [math.dg] 30 Sep 2015 We classify SICPOVMs of rank one in CP 2, or equivalently sets of nine equallyspaced
More informationA remarkable representation of the Clifford group
A remarkable representation of the Clifford group Steve Brierley University of Bristol March 2012 Work with Marcus Appleby, Ingemar Bengtsson, Markus Grassl, David Gross and JanAke Larsson Outline Useful
More informationarxiv: v1 [math.sg] 6 Nov 2015
A CHIANGTYPE LAGRANGIAN IN CP ANA CANNAS DA SILVA Abstract. We analyse a simple Chiangtype lagrangian in CP which is topologically an RP but exhibits a distinguishing behaviour under reduction by one
More informationSYMPLECTIC GEOMETRY: LECTURE 5
SYMPLECTIC GEOMETRY: LECTURE 5 LIAT KESSLER Let (M, ω) be a connected compact symplectic manifold, T a torus, T M M a Hamiltonian action of T on M, and Φ: M t the assoaciated moment map. Theorem 0.1 (The
More informationB 1 = {B(x, r) x = (x 1, x 2 ) H, 0 < r < x 2 }. (a) Show that B = B 1 B 2 is a basis for a topology on X.
Math 6342/7350: Topology and Geometry Sample Preliminary Exam Questions 1. For each of the following topological spaces X i, determine whether X i and X i X i are homeomorphic. (a) X 1 = [0, 1] (b) X 2
More informationLECTURE 10: THE ATIYAHGUILLEMINSTERNBERG CONVEXITY THEOREM
LECTURE 10: THE ATIYAHGUILLEMINSTERNBERG CONVEXITY THEOREM Contents 1. The AtiyahGuilleminSternberg Convexity Theorem 1 2. Proof of the AtiyahGuilleminSternberg Convexity theorem 3 3. Morse theory
More informationNotes 10: Consequences of Eli Cartan s theorem.
Notes 10: Consequences of Eli Cartan s theorem. Version 0.00 with misprints, The are a few obvious, but important consequences of the theorem of Eli Cartan on the maximal tori. The first one is the observation
More informationLECTURE 11: SYMPLECTIC TORIC MANIFOLDS. Contents 1. Symplectic toric manifolds 1 2. Delzant s theorem 4 3. Symplectic cut 8
LECTURE 11: SYMPLECTIC TORIC MANIFOLDS Contents 1. Symplectic toric manifolds 1 2. Delzant s theorem 4 3. Symplectic cut 8 1. Symplectic toric manifolds Orbit of torus actions. Recall that in lecture 9
More informationAlgebra I Fall 2007
MIT OpenCourseWare http://ocw.mit.edu 18.701 Algebra I Fall 007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.701 007 Geometry of the Special Unitary
More informationOrthogonal Complex Structures and Twistor Surfaces
Orthogonal Complex Structures and Twistor Surfaces Simon Salamon joint work with Jeff Viaclovsky arxiv:0704.3422 Bayreuth, May 2007 Definitions and examples Let Ω be an open set of R 4. An OCS on an open
More informationGeometric Structures in Mathematical Physics Nonexistence of almost complex structures on quaternionkähler manifolds of positive type
Geometric Structures in Mathematical Physics Nonexistence of almost complex structures on quaternionkähler manifolds of positive type Paul Gauduchon Golden Sands, Bulgaria September, 19 26, 2011 1 Joint
More informationHYPERKÄHLER MANIFOLDS
HYPERKÄHLER MANIFOLDS PAVEL SAFRONOV, TALK AT 2011 TALBOT WORKSHOP 1.1. Basic definitions. 1. Hyperkähler manifolds Definition. A hyperkähler manifold is a C Riemannian manifold together with three covariantly
More informationREGULAR TRIPLETS IN COMPACT SYMMETRIC SPACES
REGULAR TRIPLETS IN COMPACT SYMMETRIC SPACES MAKIKO SUMI TANAKA 1. Introduction This article is based on the collaboration with Tadashi Nagano. In the first part of this article we briefly review basic
More informationREPRESENTATION THEORY WEEK 7
REPRESENTATION THEORY WEEK 7 1. Characters of L k and S n A character of an irreducible representation of L k is a polynomial function constant on every conjugacy class. Since the set of diagonalizable
More informationMinimal surfaces in quaternionic symmetric spaces
From: "Geometry of lowdimensional manifolds: 1", C.U.P. (1990), pp. 231235 Minimal surfaces in quaternionic symmetric spaces F.E. BURSTALL University of Bath We describe some birational correspondences
More informationLECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY
LECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY WEIMIN CHEN, UMASS, SPRING 07 1. Blowing up and symplectic cutting In complex geometry the blowingup operation amounts to replace a point in
More informationHyperkähler geometry lecture 3
Hyperkähler geometry lecture 3 Misha Verbitsky Cohomology in Mathematics and Physics Euler Institute, September 25, 2013, St. Petersburg 1 Broom Bridge Here as he walked by on the 16th of October 1843
More informationExample 2 (new version): the generators are. Example 4: the generators are
First, let us quickly dismiss Example 3 (and Example 6): the second generator may be removed (being the square of the third or, respectively, the fourth one), and then the quotient is clearly the simplest
More informationThe Geometrization Theorem
The Geometrization Theorem Matthew D. Brown Wednesday, December 19, 2012 In this paper, we discuss the Geometrization Theorem, formerly Thurston s Geometrization Conjecture, which is essentially the statement
More informationKstability and Kähler metrics, I
Kstability and Kähler metrics, I Gang Tian Beijing University and Princeton University Let M be a Kähler manifold. This means that M be a complex manifold together with a Kähler metric ω. In local coordinates
More informationMarsdenWeinstein Reductions for Kähler, Hyperkähler and Quaternionic Kähler Manifolds
MarsdenWeinstein Reductions for Kähler, Hyperkähler and Quaternionic Kähler Manifolds Chenchang Zhu Nov, 29th 2000 1 Introduction If a Lie group G acts on a symplectic manifold (M, ω) and preserves the
More informationLECTURE 2526: CARTAN S THEOREM OF MAXIMAL TORI. 1. Maximal Tori
LECTURE 2526: CARTAN S THEOREM OF MAXIMAL TORI 1. Maximal Tori By a torus we mean a compact connected abelian Lie group, so a torus is a Lie group that is isomorphic to T n = R n /Z n. Definition 1.1.
More informationFrom Bernstein approximation to Zauner s conjecture
From Bernstein approximation to Zauner s conjecture Shayne Waldron Mathematics Department, University of Auckland December 5, 2017 Shayne Waldron (University of Auckland) Workshop on Spline Approximation
More informationMATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators.
MATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators. Adjoint operator and adjoint matrix Given a linear operator L on an inner product space V, the adjoint of L is a transformation
More information12 Geometric quantization
12 Geometric quantization 12.1 Remarks on quantization and representation theory Definition 12.1 Let M be a symplectic manifold. A prequantum line bundle with connection on M is a line bundle L M equipped
More informationzi z i, zi+1 z i,, zn z i. z j, zj+1 z j,, zj 1 z j,, zn
The Complex Projective Space Definition. Complex projective nspace, denoted by CP n, is defined to be the set of 1dimensional complexlinear subspaces of C n+1, with the quotient topology inherited from
More informationCHAPTER 6. Representations of compact groups
CHAPTER 6 Representations of compact groups Throughout this chapter, denotes a compact group. 6.1. Examples of compact groups A standard theorem in elementary analysis says that a subset of C m (m a positive
More informationIn a Ddimensional Hilbert space, a symmetric informationally complete POVM is a
To: C. A. Fuchs and P. Rungta From: C. M. Caves Subject: Symmetric informationally complete POVMs 1999 September 9; modified 00 June 18 to include material on G In a dimensional Hilbert space, a symmetric
More informationLarge automorphism groups of 16dimensional planes are Lie groups
Journal of Lie Theory Volume 8 (1998) 83 93 C 1998 Heldermann Verlag Large automorphism groups of 16dimensional planes are Lie groups Barbara Priwitzer, Helmut Salzmann Communicated by Karl H. Hofmann
More informationA LITTLE TASTE OF SYMPLECTIC GEOMETRY: THE SCHURHORN THEOREM CONTENTS
A LITTLE TASTE OF SYMPLECTIC GEOMETRY: THE SCHURHORN THEOREM TIMOTHY E. GOLDBERG ABSTRACT. This is a handout for a talk given at Bard College on Tuesday, 1 May 2007 by the author. It gives careful versions
More informationLECTURE 2: SYMPLECTIC VECTOR BUNDLES
LECTURE 2: SYMPLECTIC VECTOR BUNDLES WEIMIN CHEN, UMASS, SPRING 07 1. Symplectic Vector Spaces Definition 1.1. A symplectic vector space is a pair (V, ω) where V is a finite dimensional vector space (over
More informationHamiltonian Toric Manifolds
Hamiltonian Toric Manifolds JWR (following Guillemin) August 26, 2001 1 Notation Throughout T is a torus, T C is its complexification, V = L(T ) is its Lie algebra, and Λ V is the kernel of the exponential
More informationLECTURE 6: JHOLOMORPHIC CURVES AND APPLICATIONS
LECTURE 6: JHOLOMORPHIC CURVES AND APPLICATIONS WEIMIN CHEN, UMASS, SPRING 07 1. Basic elements of Jholomorphic curve theory Let (M, ω) be a symplectic manifold of dimension 2n, and let J J (M, ω) be
More informationKilling Vector Fields of Constant Length on Riemannian Normal Homogeneous Spaces
Killing Vector Fields of Constant Length on Riemannian Normal Homogeneous Spaces Ming Xu & Joseph A. Wolf Abstract Killing vector fields of constant length correspond to isometries of constant displacement.
More informationLECTURE 4: SYMPLECTIC GROUP ACTIONS
LECTURE 4: SYMPLECTIC GROUP ACTIONS WEIMIN CHEN, UMASS, SPRING 07 1. Symplectic circle actions We set S 1 = R/2πZ throughout. Let (M, ω) be a symplectic manifold. A symplectic S 1 action on (M, ω) is
More information1: Lie groups Matix groups, Lie algebras
Lie Groups and Bundles 2014/15 BGSMath 1: Lie groups Matix groups, Lie algebras 1. Prove that O(n) is Lie group and that its tangent space at I O(n) is isomorphic to the space so(n) of skewsymmetric matrices
More informationClifford Algebras and Spin Groups
Clifford Algebras and Spin Groups Math G4344, Spring 2012 We ll now turn from the general theory to examine a specific class class of groups: the orthogonal groups. Recall that O(n, R) is the group of
More informationEvgeniy V. Martyushev RESEARCH STATEMENT
Evgeniy V. Martyushev RESEARCH STATEMENT My research interests lie in the fields of topology of manifolds, algebraic topology, representation theory, and geometry. Specifically, my work explores various
More informationNumerical range and random matrices
Numerical range and random matrices Karol Życzkowski in collaboration with P. Gawron, J. Miszczak, Z. Pucha la (Gliwice), C. Dunkl (Virginia), J. Holbrook (Guelph), B. Collins (Ottawa) and A. Litvak (Edmonton)
More informationSimplices, frames and questions about reality. Helena Granström
Simplices, frames and questions about reality Helena Granström July 30, 2009 Simplices, frames and questions about reality Filosofie licentiatavhandling Helena Granström Att presenteras den 14 september
More informationA Numerical Criterion for Lower bounds on Kenergy maps of Algebraic manifolds
A Numerical Criterion for Lower bounds on Kenergy maps of Algebraic manifolds Sean Timothy Paul University of Wisconsin, Madison stpaul@math.wisc.edu Outline Formulation of the problem: To bound the Mabuchi
More informationThe Real Grassmannian Gr(2, 4)
The Real Grassmannian Gr(2, 4) We discuss the topology of the real Grassmannian Gr(2, 4) of 2planes in R 4 and its double cover Gr + (2, 4) by the Grassmannian of oriented 2planes They are compact fourmanifolds
More information7.3 Singular Homology Groups
184 CHAPTER 7. HOMOLOGY THEORY 7.3 Singular Homology Groups 7.3.1 Cycles, Boundaries and Homology Groups We can define the singular pchains with coefficients in a field K. Furthermore, we can define the
More informationarxiv: v1 [math.dg] 2 Oct 2015
An estimate for the Singer invariant via the Jet Isomorphism Theorem Tillmann Jentsch October 5, 015 arxiv:1510.00631v1 [math.dg] Oct 015 Abstract Recently examples of Riemannian homogeneous spaces with
More informationHandlebody Decomposition of a Manifold
Handlebody Decomposition of a Manifold Mahuya Datta Statistics and Mathematics Unit Indian Statistical Institute, Kolkata mahuya@isical.ac.in January 12, 2012 contents Introduction What is a handlebody
More informationarxiv: v1 [quantph] 11 Mar 2017
SICs: Extending the list of solutions A. J. Scott, arxiv:1703.03993v1 [quantph] 11 Mar 2017 Zauner s conjecture asserts that d 2 equiangular lines exist in all d complex dimensions. In quantum theory,
More information1. If 1, ω, ω 2, , ω 9 are the 10 th roots of unity, then (1 + ω) (1 + ω 2 ) (1 + ω 9 ) is A) 1 B) 1 C) 10 D) 0
4 INUTES. If, ω, ω, , ω 9 are the th roots of unity, then ( + ω) ( + ω )  ( + ω 9 ) is B) D) 5. i If  i = a + ib, then a =, b = B) a =, b = a =, b = D) a =, b= 3. Find the integral values for
More informationAn Invitation to Geometric Quantization
An Invitation to Geometric Quantization Alex Fok Department of Mathematics, Cornell University April 2012 What is quantization? Quantization is a process of associating a classical mechanical system to
More informationStable bundles on CP 3 and special holonomies
Stable bundles on CP 3 and special holonomies Misha Verbitsky Géométrie des variétés complexes IV CIRM, Luminy, Oct 26, 2010 1 Hyperkähler manifolds DEFINITION: A hyperkähler structure on a manifold M
More informationRepresentation theory and quantum mechanics tutorial Spin and the hydrogen atom
Representation theory and quantum mechanics tutorial Spin and the hydrogen atom Justin Campbell August 3, 2017 1 Representations of SU 2 and SO 3 (R) 1.1 The following observation is long overdue. Proposition
More informationAn eightfold path to E 8
An eightfold path to E 8 Robert A. Wilson First draft 17th November 2008; this version 29th April 2012 Introduction Finitedimensional real reflection groups were classified by Coxeter [2]. In two dimensions,
More informationDelzant s Garden. A onehour tour to symplectic toric geometry
Delzant s Garden A onehour tour to symplectic toric geometry Tour Guide: Zuoqin Wang Travel Plan: The earth America MIT Main building Math. dept. The moon Toric world Symplectic toric Delzant s theorem
More informationManifolds with holonomy. Sp(n)Sp(1) SC in SHGAP Simon Salamon Stony Brook, 9 Sep 2016
Manifolds with holonomy Sp(n)Sp(1) SC in SHGAP Simon Salamon Stony Brook, 9 Sep 2016 The list 1.1 SO(N) U( N 2 ) Sp( N 4 )Sp(1) SU( N 2 ) Sp( N 4 ) G 2 (N =7) Spin(7) (N =8) All act transitively on S N
More informationInfinitesimal Einstein Deformations. Kähler Manifolds
on Nearly Kähler Manifolds (joint work with P.A. Nagy and U. Semmelmann) Gemeinsame Jahrestagung DMV GDM Berlin, March 30, 2007 Nearly Kähler manifolds Definition and first properties Examples of NK manifolds
More informationHermitian Symmetric Spaces
Hermitian Symmetric Spaces Maria Beatrice Pozzetti Notes by Serena Yuan 1. Symmetric Spaces Definition 1.1. A Riemannian manifold X is a symmetric space if each point p X is the fixed point set of an involutive
More informationTHE EULER CHARACTERISTIC OF A LIE GROUP
THE EULER CHARACTERISTIC OF A LIE GROUP JAY TAYLOR 1 Examples of Lie Groups The following is adapted from [2] We begin with the basic definition and some core examples Definition A Lie group is a smooth
More informationQUATERNIONS AND ROTATIONS
QUATERNIONS AND ROTATIONS SVANTE JANSON 1. Introduction The purpose of this note is to show some wellknown relations between quaternions and the Lie groups SO(3) and SO(4) (rotations in R 3 and R 4 )
More informationHolonomy groups. Thomas Leistner. Mathematics Colloquium School of Mathematics and Physics The University of Queensland. October 31, 2011 May 28, 2012
Holonomy groups Thomas Leistner Mathematics Colloquium School of Mathematics and Physics The University of Queensland October 31, 2011 May 28, 2012 1/17 The notion of holonomy groups is based on Parallel
More informationCourse 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra
Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra D. R. Wilkins Contents 3 Topics in Commutative Algebra 2 3.1 Rings and Fields......................... 2 3.2 Ideals...............................
More informationUniform Kstability of pairs
Uniform Kstability of pairs Gang Tian Peking University Let G = SL(N + 1, C) with two representations V, W over Q. For any v V\{0} and any oneparameter subgroup λ of G, we can associate a weight w λ
More informationLinear connections on Lie groups
Linear connections on Lie groups The affine space of linear connections on a compact Lie group G contains a distinguished line segment with endpoints the connections L and R which make left (resp. right)
More informationSYMPLECTIC MANIFOLDS, GEOMETRIC QUANTIZATION, AND UNITARY REPRESENTATIONS OF LIE GROUPS. 1. Introduction
SYMPLECTIC MANIFOLDS, GEOMETRIC QUANTIZATION, AND UNITARY REPRESENTATIONS OF LIE GROUPS CRAIG JACKSON 1. Introduction Generally speaking, geometric quantization is a scheme for associating Hilbert spaces
More informationMath 215B: Solutions 1
Math 15B: Solutions 1 Due Thursday, January 18, 018 (1) Let π : X X be a covering space. Let Φ be a smooth structure on X. Prove that there is a smooth structure Φ on X so that π : ( X, Φ) (X, Φ) is an
More informationTorus actions and Ricciflat metrics
Department of Mathematics, University of Aarhus November 2016 / Trondheim To Eldar Straume on his 70th birthday DFF  610800358 Delzant HyperKähler G2 http://mscand.dk https://doi.org/10.7146/math.scand.a12294
More information1 Hermitian symmetric spaces: examples and basic properties
Contents 1 Hermitian symmetric spaces: examples and basic properties 1 1.1 Almost complex manifolds............................................ 1 1.2 Hermitian manifolds................................................
More informationGLOBALIZING LOCALLY COMPACT LOCAL GROUPS
GLOBALIZING LOCALLY COMPACT LOCAL GROUPS LOU VAN DEN DRIES AND ISAAC GOLDBRING Abstract. Every locally compact local group is locally isomorphic to a topological group. 1. Introduction In this paper a
More informationLinear Algebra and Dirac Notation, Pt. 3
Linear Algebra and Dirac Notation, Pt. 3 PHYS 500  Southern Illinois University February 1, 2017 PHYS 500  Southern Illinois University Linear Algebra and Dirac Notation, Pt. 3 February 1, 2017 1 / 16
More information4.2. ORTHOGONALITY 161
4.2. ORTHOGONALITY 161 Definition 4.2.9 An affine space (E, E ) is a Euclidean affine space iff its underlying vector space E is a Euclidean vector space. Given any two points a, b E, we define the distance
More informationThe Strominger Yau Zaslow conjecture
The Strominger Yau Zaslow conjecture Paul Hacking 10/16/09 1 Background 1.1 Kähler metrics Let X be a complex manifold of dimension n, and M the underlying smooth manifold with (integrable) almost complex
More informationDIFFERENTIAL FORMS AND COHOMOLOGY
DIFFERENIAL FORMS AND COHOMOLOGY ONY PERKINS Goals 1. Differential forms We want to be able to integrate (holomorphic functions) on manifolds. Obtain a version of Stokes heorem  a generalization of the
More informationRestricted Numerical Range and some its applications
Restricted Numerical Range and some its applications Karol Życzkowski in collaboration with P. Gawron, J. Miszczak, Z. Pucha la (Gliwice), L. Skowronek (Kraków), M.D. Choi (Toronto), C. Dunkl (Virginia)
More informationON NEARLY SEMIFREE CIRCLE ACTIONS
ON NEARLY SEMIFREE CIRCLE ACTIONS DUSA MCDUFF AND SUSAN TOLMAN Abstract. Recall that an effective circle action is semifree if the stabilizer subgroup of each point is connected. We show that if (M, ω)
More informationMatrix Lie groups. and their Lie algebras. Mahmood Alaghmandan. A project in fulfillment of the requirement for the Lie algebra course
Matrix Lie groups and their Lie algebras Mahmood Alaghmandan A project in fulfillment of the requirement for the Lie algebra course Department of Mathematics and Statistics University of Saskatchewan March
More informationTopics in Representation Theory: Cultural Background
Topics in Representation Theory: Cultural Background This semester we will be covering various topics in representation theory, see the separate syllabus for a detailed list of topics, including some that
More informationLecture 4: Harmonic forms
Lecture 4: Harmonic forms Jonathan Evans 29th September 2010 Jonathan Evans () Lecture 4: Harmonic forms 29th September 2010 1 / 15 Jonathan Evans () Lecture 4: Harmonic forms 29th September 2010 2 / 15
More informationMathematical Research Letters 2, (1995) A VANISHING THEOREM FOR SEIBERGWITTEN INVARIANTS. Shuguang Wang
Mathematical Research Letters 2, 305 310 (1995) A VANISHING THEOREM FOR SEIBERGWITTEN INVARIANTS Shuguang Wang Abstract. It is shown that the quotients of Kähler surfaces under free antiholomorphic involutions
More informationSimon Salamon. Turin, 24 April 2004
G 2 METRICS AND M THEORY Simon Salamon Turin, 24 April 2004 I Weak holonomy and supergravity II S 1 actions and triality in six dimensions III G 2 and SU(3) structures from each other 1 PART I The exceptional
More informationA CHIANGTYPE LAGRANGIAN IN CP 2
A CHIANGTYPE LAGRANGIAN IN CP 2 ANA CANNAS DA SILVA Abstract. We analyse a monotone lagrangian in CP 2 that is hamiltonian isotopic to the standard lagrangian RP 2, yet exhibits a distinguishing behaviour
More informationNilBott Tower of Aspherical Manifolds and Torus Actions
NilBott Tower of Aspherical Manifolds and Torus Actions Tokyo Metropolitan University November 29, 2011 (Tokyo NilBottMetropolitan Tower of Aspherical University) Manifolds and Torus ActionsNovember 29,
More informationHomogeneous parakähler Einstein manifolds. Dmitri V. Alekseevsky
Homogeneous parakähler Einstein manifolds Dmitri V. Alekseevsky Hamburg,1418 July 2008 1 The talk is based on a joint work with C.Medori and A.Tomassini (Parma) See ArXiv 0806.2272, where also a survey
More informationAn Introduction to Kuga Fiber Varieties
An Introduction to Kuga Fiber Varieties Dylan AttwellDuval Department of Mathematics and Statistics McGill University Montreal, Quebec attwellduval@math.mcgill.ca April 28, 2012 Notation G a Qsimple
More informationIntroduction to Modern Quantum Field Theory
Department of Mathematics University of Texas at Arlington Arlington, TX USA Febuary, 2016 Recall Einstein s famous equation, E 2 = (Mc 2 ) 2 + (c p) 2, where c is the speed of light, M is the classical
More informationSubgroups of Lie groups. Definition 0.7. A Lie subgroup of a Lie group G is a subgroup which is also a submanifold.
Recollections from finite group theory. The notion of a group acting on a set is extremely useful. Indeed, the whole of group theory arose through this route. As an example of the abstract power of this
More informationA Highly Symmetric FourDimensional Quasicrystal * Veit Elser and N. J. A. Sloane AT&T Bell Laboratories Murray Hill, New Jersey
A Highly Symmetric FourDimensional Quasicrystal * Veit Elser and N. J. A. Sloane AT&T Bell Laboratories Murray Hill, New Jersey 7974 Abstract A quasiperiodic pattern (or quasicrystal) is constructed in
More informationIntroduction to Group Theory
Chapter 10 Introduction to Group Theory Since symmetries described by groups play such an important role in modern physics, we will take a little time to introduce the basic structure (as seen by a physicist)
More informationComplex manifolds, Kahler metrics, differential and harmonic forms
Complex manifolds, Kahler metrics, differential and harmonic forms Cattani June 16, 2010 1 Lecture 1 Definition 1.1 (Complex Manifold). A complex manifold is a manifold with coordinates holomorphic on
More informationarxiv:quantph/ v1 11 Nov 2005
On the SU() Parametrization of Qutrits arxiv:quantph/5v Nov 5 A. T. Bölükbaşı, T. Dereli Department of Physics, Koç University 445 Sarıyer, İstanbul, Turkey Abstract Parametrization of qutrits on the
More informationMath 550 / David Dumas / Fall Problems
Math 550 / David Dumas / Fall 2014 Problems Please note: This list was last updated on November 30, 2014. Problems marked with * are challenge problems. Some problems are adapted from the course texts;
More informationINTRINSIC MEAN ON MANIFOLDS. Abhishek Bhattacharya Project Advisor: Dr.Rabi Bhattacharya
INTRINSIC MEAN ON MANIFOLDS Abhishek Bhattacharya Project Advisor: Dr.Rabi Bhattacharya 1 Overview Properties of Intrinsic mean on Riemannian manifolds have been presented. The results have been applied
More informationHodge Structures. October 8, A few examples of symmetric spaces
Hodge Structures October 8, 2013 1 A few examples of symmetric spaces The upper halfplane H is the quotient of SL 2 (R) by its maximal compact subgroup SO(2). More generally, Siegel upperhalf space H
More informationStratification of 3 3 Matrices
Stratification of 3 3 Matrices Meesue Yoo & Clay Shonkwiler March 2, 2006 1 Warmup with 2 2 Matrices { Best matrices of rank 2} = O(2) S 3 ( 2) { Best matrices of rank 1} S 3 (1) 1.1 Viewing O(2) S 3 (
More informationSymmetries and particle physics Exercises
Symmetries and particle physics Exercises Stefan Flörchinger SS 017 1 Lecture From the lecture we know that the dihedral group of order has the presentation D = a, b a = e, b = e, bab 1 = a 1. Moreover
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 informationHolonomy groups. Thomas Leistner. School of Mathematical Sciences Colloquium University of Adelaide, May 7, /15
Holonomy groups Thomas Leistner School of Mathematical Sciences Colloquium University of Adelaide, May 7, 2010 1/15 The notion of holonomy groups is based on Parallel translation Let γ : [0, 1] R 2 be
More information2 Lie Groups. Contents
2 Lie Groups Contents 2.1 Algebraic Properties 25 2.2 Topological Properties 27 2.3 Unification of Algebra and Topology 29 2.4 Unexpected Simplification 31 2.5 Conclusion 31 2.6 Problems 32 Lie groups
More informationarxiv: v4 [math.dg] 18 Jun 2015
SMOOTHING 3DIMENSIONAL POLYHEDRAL SPACES NINA LEBEDEVA, VLADIMIR MATVEEV, ANTON PETRUNIN, AND VSEVOLOD SHEVCHISHIN arxiv:1411.0307v4 [math.dg] 18 Jun 2015 Abstract. We show that 3dimensional polyhedral
More informationarxiv: v1 [math.dg] 29 Dec 2018
KOSNIOWSKI S CONJECTURE AND WEIGHTS arxiv:1121121v1 [mathdg] 29 Dec 201 DONGHOON JANG Abstract The conjecture of Kosniowski asserts that if the circle acts on a compact unitary manifold M with a nonempty
More informationLecture 4  The Basic Examples of Collapse
Lecture 4  The Basic Examples of Collapse July 29, 2009 1 Berger Spheres Let X, Y, and Z be the leftinvariant vector fields on S 3 that restrict to i, j, and k at the identity. This is a global frame
More informationComparison of the Weyl integration formula and the equivariant localization formula for a maximal torus of Sp(1)
Comparison of the Weyl integration formula and the equivariant localization formula for a maximal torus of Sp() Jeffrey D. Carlson January 3, 4 Introduction In this note, we aim to prove the Weyl integration
More informationTheorem 2. Let n 0 3 be a given integer. is rigid in the sense of Guillemin, so are all the spaces ḠR n,n, with n n 0.
This monograph is motivated by a fundamental rigidity problem in Riemannian geometry: determine whether the metric of a given Riemannian symmetric space of compact type can be characterized by means of
More information