How to minimize without knowing how to differentiate (in Riemannian geometry)


 Maude Webb
 11 months ago
 Views:
Transcription
1 How to minimize without knowing how to differentiate (in Riemannian geometry) Spiro Karigiannis Mathematical Institute, University of Oxford Calibrated Geometries and Special Holonomy p. 1/29
2 Introduction Calibrated Geometries and Special Holonomy p. 2/29
3 Introduction In this talk I want to consider two simple ideas from calculus and show you some ways in which they arise in Riemannian geometry: Calibrated Geometries and Special Holonomy p. 2/29
4 Introduction In this talk I want to consider two simple ideas from calculus and show you some ways in which they arise in Riemannian geometry: finding some solutions to a second order differential equation by instead solving a first order equation Calibrated Geometries and Special Holonomy p. 2/29
5 Introduction In this talk I want to consider two simple ideas from calculus and show you some ways in which they arise in Riemannian geometry: finding some solutions to a second order differential equation by instead solving a first order equation in the presence of additional structure, finding the absolute minima of some function (and not just the critical points) without needing to take derivatives Calibrated Geometries and Special Holonomy p. 2/29
6 Introduction In this talk I want to consider two simple ideas from calculus and show you some ways in which they arise in Riemannian geometry: finding some solutions to a second order differential equation by instead solving a first order equation in the presence of additional structure, finding the absolute minima of some function (and not just the critical points) without needing to take derivatives Let us first consider some simple examples. Calibrated Geometries and Special Holonomy p. 2/29
7 Example. Consider the differential equation d 2 dx 2f = k2 f Calibrated Geometries and Special Holonomy p. 3/29
8 Example. Consider the differential equation d 2 dx 2f = k2 f We can rewrite this equation as ( )( ) d d dx + k dx k f = 0 Calibrated Geometries and Special Holonomy p. 3/29
9 Example. Consider the differential equation d 2 dx 2f = k2 f We can rewrite this equation as ( )( ) d d dx + k dx k f = 0 Then we see that any solution of d dxf = kf will be a solution to our original equation, but not conversely. Calibrated Geometries and Special Holonomy p. 3/29
10 We will see two situations in Riemannian geometry where we can obtain some of the solutions to a second order partial differential equation, by instead solving a first order equation. Calibrated Geometries and Special Holonomy p. 4/29
11 We will see two situations in Riemannian geometry where we can obtain some of the solutions to a second order partial differential equation, by instead solving a first order equation. It will not be the case in general that the second order equation factors into the composition of two first order differential operators. (Although in some cases this can be shown to happen.) Calibrated Geometries and Special Holonomy p. 4/29
12 We will see two situations in Riemannian geometry where we can obtain some of the solutions to a second order partial differential equation, by instead solving a first order equation. It will not be the case in general that the second order equation factors into the composition of two first order differential operators. (Although in some cases this can be shown to happen.) Now let us consider an example of the second idea. Calibrated Geometries and Special Holonomy p. 4/29
13 Example. Suppose we want to find the absolute minima of some fixed function f on a domain without boundary. Calibrated Geometries and Special Holonomy p. 5/29
14 Example. Suppose we want to find the absolute minima of some fixed function f on a domain without boundary. By solving the algebraic equation d dx f = 0 we can find all possible candidate points, but they might be maxima or inflection points. Calibrated Geometries and Special Holonomy p. 5/29
15 Example. Suppose we want to find the absolute minima of some fixed function f on a domain without boundary. By solving the algebraic equation d dx f = 0 we can find all possible candidate points, but they might be maxima or inflection points. In some situations we can completely avoid the need to differentiate and at the same time find the actual minima. Calibrated Geometries and Special Holonomy p. 5/29
16 For example, suppose that for some reason we know f(x) 1 for all x in our domain. Calibrated Geometries and Special Holonomy p. 6/29
17 For example, suppose that for some reason we know f(x) 1 for all x in our domain. Then by solving the algebraic equation f(x) = 1 we obtain the absolute minima of f, if any such points actually exist. Calibrated Geometries and Special Holonomy p. 6/29
18 For example, suppose that for some reason we know f(x) 1 for all x in our domain. Then by solving the algebraic equation f(x) = 1 we obtain the absolute minima of f, if any such points actually exist. We will see this phenomenon also occur in Riemannian geometry. However, we will be trying to minimize a functional, so the set of points on which it is defined will be a space of functions (or tensors.) Calibrated Geometries and Special Holonomy p. 6/29
19 The definition of the functionals will involve derivatives of the arguments, so we will not be able to completely avoid taking derivatives. But we will be able to take one less derivative than we would normally need. Calibrated Geometries and Special Holonomy p. 7/29
20 The definition of the functionals will involve derivatives of the arguments, so we will not be able to completely avoid taking derivatives. But we will be able to take one less derivative than we would normally need. More precisely, we will have functionals whose critical points (potential minima) are given by solutions to certain second order equations. Calibrated Geometries and Special Holonomy p. 7/29
21 The definition of the functionals will involve derivatives of the arguments, so we will not be able to completely avoid taking derivatives. But we will be able to take one less derivative than we would normally need. More precisely, we will have functionals whose critical points (potential minima) are given by solutions to certain second order equations. In special situations (in the presence of some additional structure), we can show that solutions to a first order equation are a subset of the solutions to the second order equation, and also that they are actually minima. Calibrated Geometries and Special Holonomy p. 7/29
22 Riemannian geometry Calibrated Geometries and Special Holonomy p. 8/29
23 Riemannian geometry Let M be a smooth manifold with Riemannian metric g. This is a smoothly varying, positive definite inner product g p on each tangent space T p M of M, for each point p in M. The metric g determines some curvature tensors on M: Calibrated Geometries and Special Holonomy p. 8/29
24 Riemannian geometry Let M be a smooth manifold with Riemannian metric g. This is a smoothly varying, positive definite inner product g p on each tangent space T p M of M, for each point p in M. The metric g determines some curvature tensors on M: Riemann curvature tensor R ijkl Calibrated Geometries and Special Holonomy p. 8/29
25 Riemannian geometry Let M be a smooth manifold with Riemannian metric g. This is a smoothly varying, positive definite inner product g p on each tangent space T p M of M, for each point p in M. The metric g determines some curvature tensors on M: Riemann curvature tensor R ijkl Ricci curvature tensor R ij (a symmetric 2tensor) Calibrated Geometries and Special Holonomy p. 8/29
26 Riemannian geometry Let M be a smooth manifold with Riemannian metric g. This is a smoothly varying, positive definite inner product g p on each tangent space T p M of M, for each point p in M. The metric g determines some curvature tensors on M: Riemann curvature tensor R ijkl Ricci curvature tensor R ij (a symmetric 2tensor) Scalar curvature S (a smooth function) Calibrated Geometries and Special Holonomy p. 8/29
27 A natural question to ask is when does M admit a metric g with special curvature properties? For example: Calibrated Geometries and Special Holonomy p. 9/29
28 A natural question to ask is when does M admit a metric g with special curvature properties? For example: R ijkl = 0; a flat metric, M is locally isometric to R n Calibrated Geometries and Special Holonomy p. 9/29
29 A natural question to ask is when does M admit a metric g with special curvature properties? For example: R ijkl = 0; a flat metric, M is locally isometric to R n R ij = λg ij ; an Einstein metric Calibrated Geometries and Special Holonomy p. 9/29
30 A natural question to ask is when does M admit a metric g with special curvature properties? For example: R ijkl = 0; a flat metric, M is locally isometric to R n R ij = λg ij ; an Einstein metric R ij = 0; a Ricciflat metric Calibrated Geometries and Special Holonomy p. 9/29
31 A natural question to ask is when does M admit a metric g with special curvature properties? For example: R ijkl = 0; a flat metric, M is locally isometric to R n R ij = λg ij ; an Einstein metric R ij = 0; a Ricciflat metric S = constant; a constant scalar curvature metric Calibrated Geometries and Special Holonomy p. 9/29
32 A natural question to ask is when does M admit a metric g with special curvature properties? For example: R ijkl = 0; a flat metric, M is locally isometric to R n R ij = λg ij ; an Einstein metric R ij = 0; a Ricciflat metric S = constant; a constant scalar curvature metric These are all second order, nonlinear partial differential equations on M which are in general very difficult to solve. Especially when M is compact, we only expect to find exact solutions in situations with a high degree of symmetry. Calibrated Geometries and Special Holonomy p. 9/29
33 Sometimes these equations arise naturally as the EulerLagrange equations for the critical points of some functional. Calibrated Geometries and Special Holonomy p. 10/29
34 Sometimes these equations arise naturally as the EulerLagrange equations for the critical points of some functional. For example, the Einstein metrics are critical points of F[g] = S g Vol g This is called the Total Scalar Curvature functional. M Calibrated Geometries and Special Holonomy p. 10/29
35 Sometimes these equations arise naturally as the EulerLagrange equations for the critical points of some functional. For example, the Einstein metrics are critical points of F[g] = S g Vol g This is called the Total Scalar Curvature functional. Thus a minimum of F[g] will be an Einstein metric, but the converse need not hold (it could also be a maximum or an inflection point.) M Calibrated Geometries and Special Holonomy p. 10/29
36 Now let N be a kdimensional immersed submanifold of M. By restriction, g induces a metric g N on N. This determines the second fundamental form h of the immersion, which is a symmetric 2tensor on N with values in the normal bundle of N in M. Calibrated Geometries and Special Holonomy p. 11/29
37 Now let N be a kdimensional immersed submanifold of M. By restriction, g induces a metric g N on N. This determines the second fundamental form h of the immersion, which is a symmetric 2tensor on N with values in the normal bundle of N in M. One again we can consider immersions with special curvature properties. For example: Calibrated Geometries and Special Holonomy p. 11/29
38 Now let N be a kdimensional immersed submanifold of M. By restriction, g induces a metric g N on N. This determines the second fundamental form h of the immersion, which is a symmetric 2tensor on N with values in the normal bundle of N in M. One again we can consider immersions with special curvature properties. For example: h ij = 0; a totally geodesic immersion Calibrated Geometries and Special Holonomy p. 11/29
39 Now let N be a kdimensional immersed submanifold of M. By restriction, g induces a metric g N on N. This determines the second fundamental form h of the immersion, which is a symmetric 2tensor on N with values in the normal bundle of N in M. One again we can consider immersions with special curvature properties. For example: h ij = 0; a totally geodesic immersion Trace(h ij ) = 0; a zero mean curvature immersion (also called a minimal immersion) Calibrated Geometries and Special Holonomy p. 11/29
40 Now let N be a kdimensional immersed submanifold of M. By restriction, g induces a metric g N on N. This determines the second fundamental form h of the immersion, which is a symmetric 2tensor on N with values in the normal bundle of N in M. One again we can consider immersions with special curvature properties. For example: h ij = 0; a totally geodesic immersion Trace(h ij ) = 0; a zero mean curvature immersion (also called a minimal immersion) Trace(h ij ) = constant; a CMC immersion Calibrated Geometries and Special Holonomy p. 11/29
41 These are second order, nonlinear partial differential equations on the immersion map. Calibrated Geometries and Special Holonomy p. 12/29
42 These are second order, nonlinear partial differential equations on the immersion map. Again, they can be critical points of a functional. The minimal immersions are critical points of the Volume functional Vol N which is why they are called minimal. N Calibrated Geometries and Special Holonomy p. 12/29
43 These are second order, nonlinear partial differential equations on the immersion map. Again, they can be critical points of a functional. The minimal immersions are critical points of the Volume functional Vol N which is why they are called minimal. N Note that minimal just means critical point for volume. It could actually be a maximum or inflection point. The actual minima are called minimizing immersions. Calibrated Geometries and Special Holonomy p. 12/29
44 Special Holonomy Calibrated Geometries and Special Holonomy p. 13/29
45 Special Holonomy On a Riemannian manifold (M, g), we have a covariant derivative which allows us to differentiate any tensor in the direction of a vector field V. This allows us to define the notion of parallel transport: Calibrated Geometries and Special Holonomy p. 13/29
46 Special Holonomy On a Riemannian manifold (M, g), we have a covariant derivative which allows us to differentiate any tensor in the direction of a vector field V. This allows us to define the notion of parallel transport: Given a tangent vector X p at a point p on M, and a closed loop γ : [0, 1] M at p, define τ γ,p : T p M T p M X p X γ(1) where X γ(t) is in T γ(t) M, and γ (t)x γ(t) = 0 0 t 1 Calibrated Geometries and Special Holonomy p. 13/29
47 This construction involves solving a linear first order ODE, and hence it can always be done. It is also easy to see that the map τ γ,p is in GL(T p M). Calibrated Geometries and Special Holonomy p. 14/29
48 This construction involves solving a linear first order ODE, and hence it can always be done. It is also easy to see that the map τ γ,p is in GL(T p M). In fact, because τ γ1 γ 2 = τ γ1 τ γ2 and τ γ 1 = τ 1 γ, the set of all such τ γ s at a point p is a subgroup H p of GL(T p M), called the holonomy group at p. Calibrated Geometries and Special Holonomy p. 14/29
49 This construction involves solving a linear first order ODE, and hence it can always be done. It is also easy to see that the map τ γ,p is in GL(T p M). In fact, because τ γ1 γ 2 = τ γ1 τ γ2 and τ γ 1 = τ 1 γ, the set of all such τ γ s at a point p is a subgroup H p of GL(T p M), called the holonomy group at p. Suppose M is connected, simplyconnected, and oriented. Then one can show that H p = H q for any two points p, q in M. Thus, up to isomorphism, we can talk about the holonomy group H of (M, g), which is a subgroup of GL(n, R). Calibrated Geometries and Special Holonomy p. 14/29
50 A tensor T on M is called parallel if V T = 0 for all vector fields V. Calibrated Geometries and Special Holonomy p. 15/29
51 A tensor T on M is called parallel if V T = 0 for all vector fields V. Theorem. The holonomy group H is the subgroup of GL(n, R) preserving all parallel tensors on M. Calibrated Geometries and Special Holonomy p. 15/29
52 A tensor T on M is called parallel if V T = 0 for all vector fields V. Theorem. The holonomy group H is the subgroup of GL(n, R) preserving all parallel tensors on M. Since the metric g and volume form Vol are parallel, the holonomy group is always a subgroup of SO(n). But the holonomy can be a strictly smaller subgroup if there exist other parallel tensors. Calibrated Geometries and Special Holonomy p. 15/29
53 Berger Classification. Let M be irreducible, not locally symmetric. Then the holonomy can only be one of the following seven possibilities: Calibrated Geometries and Special Holonomy p. 16/29
54 Berger Classification. Let M be irreducible, not locally symmetric. Then the holonomy can only be one of the following seven possibilities: H dim Name Parallel Tensors SO(n) n Riemannian U(m) 2m Kähler ω SU(m) 2m CalabiYau ω, Ω Sp(m) 4m hyperkähler ω I, ω J, ω K Sp(m) Sp(1) 4m quaternionickähler σ G 2 7 G 2 manifold ϕ, ψ Spin(7) 8 Spin(7) manifold Φ Calibrated Geometries and Special Holonomy p. 16/29
55 Some examples: Calibrated Geometries and Special Holonomy p. 17/29
56 Some examples: Projective varieties are all Kähler Calibrated Geometries and Special Holonomy p. 17/29
57 Some examples: Projective varieties are all Kähler By Yau s solution of the Calabi conjecture, any compact Kähler manifold with c 1 = 0 admits a CalabiYau metric Calibrated Geometries and Special Holonomy p. 17/29
58 Some examples: Projective varieties are all Kähler By Yau s solution of the Calabi conjecture, any compact Kähler manifold with c 1 = 0 admits a CalabiYau metric (BryantSalamon, 1989) Noncompact G 2 and Spin(7) manifolds: Λ 2 (S 4 ), Λ 2 (CP 2 ), /S(S 4 ) Calibrated Geometries and Special Holonomy p. 17/29
59 Some examples: Projective varieties are all Kähler By Yau s solution of the Calabi conjecture, any compact Kähler manifold with c 1 = 0 admits a CalabiYau metric (BryantSalamon, 1989) Noncompact G 2 and Spin(7) manifolds: Λ 2 (S 4 ), Λ 2 (CP 2 ), /S(S 4 ) (Joyce, 1994) Compact G 2 and Spin(7) manifolds: obtained by resolving orbifolds Calibrated Geometries and Special Holonomy p. 17/29
60 AmbroseSinger Theorem. The Lie algebra h of H is generated by the Riemann curvature tensor R ijkl. Calibrated Geometries and Special Holonomy p. 18/29
61 AmbroseSinger Theorem. The Lie algebra h of H is generated by the Riemann curvature tensor R ijkl. Thus, if the holonomy group H is small, this puts restrictions on the Riemann curvature tensor. In fact: Calibrated Geometries and Special Holonomy p. 18/29
62 AmbroseSinger Theorem. The Lie algebra h of H is generated by the Riemann curvature tensor R ijkl. Thus, if the holonomy group H is small, this puts restrictions on the Riemann curvature tensor. In fact: CalabiYau, hyperkähler, G 2 and Spin(7) manifolds are always Ricciflat Calibrated Geometries and Special Holonomy p. 18/29
63 AmbroseSinger Theorem. The Lie algebra h of H is generated by the Riemann curvature tensor R ijkl. Thus, if the holonomy group H is small, this puts restrictions on the Riemann curvature tensor. In fact: CalabiYau, hyperkähler, G 2 and Spin(7) manifolds are always Ricciflat QuaternionicKähler manifolds are always Einstein Calibrated Geometries and Special Holonomy p. 18/29
64 AmbroseSinger Theorem. The Lie algebra h of H is generated by the Riemann curvature tensor R ijkl. Thus, if the holonomy group H is small, this puts restrictions on the Riemann curvature tensor. In fact: CalabiYau, hyperkähler, G 2 and Spin(7) manifolds are always Ricciflat QuaternionicKähler manifolds are always Einstein The holonomy condition is a first order partial differential equation. Thus we see that solutions to a first order equation are automatically solutions to a more complicated second order equation. Calibrated Geometries and Special Holonomy p. 18/29
65 Calibrated Geometry Calibrated Geometries and Special Holonomy p. 19/29
66 Calibrated Geometry (HarveyLawson, 1982) Suppose (M, g) is a Riemannian manifold of dimension n. Let 0 < k < n. Calibrated Geometries and Special Holonomy p. 19/29
67 Calibrated Geometry (HarveyLawson, 1982) Suppose (M, g) is a Riemannian manifold of dimension n. Let 0 < k < n. Definition. A smooth kform α on M is called a calibration if dα = 0 and whenever e 1,...,e k is an oriented orthonormal basis for a tangent kplane in T p M, α p (e 1,...,e k ) 1 = Vol p (e 1,...,e k ) Calibrated Geometries and Special Holonomy p. 19/29
68 Calibrated Geometry (HarveyLawson, 1982) Suppose (M, g) is a Riemannian manifold of dimension n. Let 0 < k < n. Definition. A smooth kform α on M is called a calibration if dα = 0 and whenever e 1,...,e k is an oriented orthonormal basis for a tangent kplane in T p M, α p (e 1,...,e k ) 1 = Vol p (e 1,...,e k ) An oriented kplane V p in T p M is called calibrated if α p (V p ) = 1 (that is, if the supremum is attained) Calibrated Geometries and Special Holonomy p. 19/29
69 Finally, we say an oriented kdimensional submanifold N k is a calibrated submanifold if each oriented tangent space to N is calibrated. This is equivalent to α N = Vol N which is a first order PDE for the immersion of N in M. Calibrated Geometries and Special Holonomy p. 20/29
70 Finally, we say an oriented kdimensional submanifold N k is a calibrated submanifold if each oriented tangent space to N is calibrated. This is equivalent to α N = Vol N which is a first order PDE for the immersion of N in M. Fundamental theorem of calibrated geometry: Suppose N is calibrated with respect to α. Then N is absolutely volume minimizing in its homology class. Calibrated Geometries and Special Holonomy p. 20/29
71 Proof. Suppose [N] = [N ], so N N = L. Then Vol(N) = Vol N = α = N N α + α = α + dα Vol N = Vol(N ) N L N L N Calibrated Geometries and Special Holonomy p. 21/29
72 Proof. Suppose [N] = [N ], so N N = L. Then Vol(N) = Vol N = α = N N α + α = α + dα Vol N = Vol(N ) N L N L N Hence, calibrated submanifolds are solutions to a first order differential equation which are automatically solutions to the second order equation for minimal submanifolds. That is, they have zero mean curvature. Calibrated Geometries and Special Holonomy p. 21/29
73 The main examples are: Calibrated Geometries and Special Holonomy p. 22/29
74 The main examples are: Let M be Kähler, with Kähler form ω. Then α = 1 p! ωp is a calibration. Calibrated submanifolds are the 2p dimensional complex submanifolds. Calibrated Geometries and Special Holonomy p. 22/29
75 The main examples are: Let M be Kähler, with Kähler form ω. Then α = 1 p! ωp is a calibration. Calibrated submanifolds are the 2p dimensional complex submanifolds. Let M be CalabiYau, with Kähler form ω and complex holomorphic volume form Ω. Then α = Re(Ω) is a calibration. Calibrated submanifolds are called special Lagrangian. They are Lagrangian with respect to the symplectic form ω, and satisfy an additional metric condition. Calibrated Geometries and Special Holonomy p. 22/29
76 Let M 7 be a G 2 manifold. The 3form ϕ and 4form ψ are both calibrations. Calibrated submanifolds are called associative and coassociative, respectively. Calibrated Geometries and Special Holonomy p. 23/29
77 Let M 7 be a G 2 manifold. The 3form ϕ and 4form ψ are both calibrations. Calibrated submanifolds are called associative and coassociative, respectively. Let M 8 be a Spin(7) manifold. The 4form Φ is a calibration. Calibrated submanifolds are called Cayley. Calibrated Geometries and Special Holonomy p. 23/29
78 Let M 7 be a G 2 manifold. The 3form ϕ and 4form ψ are both calibrations. Calibrated submanifolds are called associative and coassociative, respectively. Let M 8 be a Spin(7) manifold. The 4form Φ is a calibration. Calibrated submanifolds are called Cayley. Many explicit examples of such submanifolds have been found in Euclidean spaces, and some noncompact, nonflat manifolds. Calibrated Geometries and Special Holonomy p. 23/29
79 Let M 7 be a G 2 manifold. The 3form ϕ and 4form ψ are both calibrations. Calibrated submanifolds are called associative and coassociative, respectively. Let M 8 be a Spin(7) manifold. The 4form Φ is a calibration. Calibrated submanifolds are called Cayley. Many explicit examples of such submanifolds have been found in Euclidean spaces, and some noncompact, nonflat manifolds. The only compact examples which have been found so far arise in situations with a high degree of symmetry. Calibrated Geometries and Special Holonomy p. 23/29
80 Analogies between the two subjects Calibrated Geometries and Special Holonomy p. 24/29
81 Analogies between the two subjects Manifolds of special holonomy and calibrated submanifolds both provide first order examples of solutions to second order equations. Calibrated Geometries and Special Holonomy p. 24/29
82 Analogies between the two subjects Manifolds of special holonomy and calibrated submanifolds both provide first order examples of solutions to second order equations. We replace a quasilinear second order equation by a fully nonlinear first order equation. Hence it is not necessarily easier to solve. Calibrated Geometries and Special Holonomy p. 24/29
83 Analogies between the two subjects Manifolds of special holonomy and calibrated submanifolds both provide first order examples of solutions to second order equations. We replace a quasilinear second order equation by a fully nonlinear first order equation. Hence it is not necessarily easier to solve. The main examples of calibrations are all defined on manifolds with special holonomy. The reasons are still not very well understood, but seem to be related to spin geometry. (Existence of parallel spinors.) Calibrated Geometries and Special Holonomy p. 24/29
84 We are missing a classification theorem of calibrations. Is there some kind of analogue of the AmbroseSinger theorem? This appears to be much more complicated. Calibrated Geometries and Special Holonomy p. 25/29
85 We are missing a classification theorem of calibrations. Is there some kind of analogue of the AmbroseSinger theorem? This appears to be much more complicated. Is there a simple analogue to the fundamental theorem of calibrated geometry for metrics with special holonomy? Work of M.Wang and DaiWangWei has shown that special holonomy metrics are stable critical points of the total scalar curvature functional with respect to some types of deformations. Their arguments use parallel spinors. Calibrated Geometries and Special Holonomy p. 25/29
86 Mirror Symmetry Calibrated Geometries and Special Holonomy p. 26/29
87 Mirror Symmetry Modern physics, specifically string theory, 11dimensional supergravity (Mtheory), and Ftheory require compact manifolds of dimensions 6, 7, and 8, respectively for their description. Calibrated Geometries and Special Holonomy p. 26/29
88 Mirror Symmetry Modern physics, specifically string theory, 11dimensional supergravity (Mtheory), and Ftheory require compact manifolds of dimensions 6, 7, and 8, respectively for their description. Considerations of supersymmetry say that these manifolds have parallel spinors, which implies they are CalabiYau, G 2, and Spin(7) manifolds, respectively. Calibrated Geometries and Special Holonomy p. 26/29
89 Mirror Symmetry Modern physics, specifically string theory, 11dimensional supergravity (Mtheory), and Ftheory require compact manifolds of dimensions 6, 7, and 8, respectively for their description. Considerations of supersymmetry say that these manifolds have parallel spinors, which implies they are CalabiYau, G 2, and Spin(7) manifolds, respectively. Physics also predicts there sometimes exist dual manifolds which are different but actually describe the same physics. These are called mirror manifolds. Calibrated Geometries and Special Holonomy p. 26/29
90 StromingerYauZaslow Conjecture, Let M 6 be a compact CalabiYau 3fold which admits a mirror. There should exist a fibration f : M 6 L 3. The generic (nonsingular) fibres of f are special Lagrangian torii in M, and the mirror M is obtained from M by taking the dual torus over each nonsingular fibre, and suitably compactifying. Calibrated Geometries and Special Holonomy p. 27/29
91 StromingerYauZaslow Conjecture, Let M 6 be a compact CalabiYau 3fold which admits a mirror. There should exist a fibration f : M 6 L 3. The generic (nonsingular) fibres of f are special Lagrangian torii in M, and the mirror M is obtained from M by taking the dual torus over each nonsingular fibre, and suitably compactifying. This conjecture is still not formulated precisely. Some progress has been made, but much remains to be done. (D. Joyce, M. Gross, N.C. Leung, W.D. Ruan, S.T. Yau, E. Zaslow, and others.) Calibrated Geometries and Special Holonomy p. 27/29
92 We know there must exist singular fibres. Calibrated Geometries and Special Holonomy p. 28/29
93 We know there must exist singular fibres. The SYZ conjecture is probably only true near a large complex structure limit point in the moduli space of CalabiYaus. Calibrated Geometries and Special Holonomy p. 28/29
94 We know there must exist singular fibres. The SYZ conjecture is probably only true near a large complex structure limit point in the moduli space of CalabiYaus. Similar incomplete conjectures exist for G 2 and Spin(7) manifolds, which involve fibrations by calibrated torii and their dual fibrations. Calibrated Geometries and Special Holonomy p. 28/29
95 We know there must exist singular fibres. The SYZ conjecture is probably only true near a large complex structure limit point in the moduli space of CalabiYaus. Similar incomplete conjectures exist for G 2 and Spin(7) manifolds, which involve fibrations by calibrated torii and their dual fibrations. We need to understand the types of singularities that can arise in calibrated submanifolds, to understand the possible singular fibres. Predictions from physics seem to indicate that conical singularities are the most important. Calibrated Geometries and Special Holonomy p. 28/29
96 Mirror Symmetry for CalabiYau manifolds also involves GromovWitten invariants. These invariants count holomorphic curves. Calibrated Geometries and Special Holonomy p. 29/29
97 Mirror Symmetry for CalabiYau manifolds also involves GromovWitten invariants. These invariants count holomorphic curves. Similar theories in the G 2 and Spin(7) cases are still far from even being properly formulated. We expect the analytic difficulty to be much greater. Calibrated Geometries and Special Holonomy p. 29/29
98 Mirror Symmetry for CalabiYau manifolds also involves GromovWitten invariants. These invariants count holomorphic curves. Similar theories in the G 2 and Spin(7) cases are still far from even being properly formulated. We expect the analytic difficulty to be much greater. Thank you for your attention. Calibrated Geometries and Special Holonomy p. 29/29
A new viewpoint on geometry of a lightlike hypersurface in a semieuclidean space
A new viewpoint on geometry of a lightlike hypersurface in a semieuclidean space Aurel Bejancu, Angel Ferrández Pascual Lucas Saitama Math J 16 (1998), 31 38 (Partially supported by DGICYT grant PB970784
More informationCARTAN S GENERALIZATION OF LIE S THIRD THEOREM
CARTAN S GENERALIZATION OF LIE S THIRD THEOREM ROBERT L. BRYANT MATHEMATICAL SCIENCES RESEARCH INSTITUTE JUNE 13, 2011 CRM, MONTREAL In many ways, this talk (and much of the work it reports on) owes its
More informationWhy do mathematicians make things so complicated?
Why do mathematicians make things so complicated? Zhiqin Lu, The Math Department March 9, 2010 Introduction What is Mathematics? Introduction What is Mathematics? 24,100,000 answers from Google. Introduction
More informationold supersymmetry as new mathematics
old supersymmetry as new mathematics PILJIN YI Korea Institute for Advanced Study with help from Sungjay Lee AtiyahSinger Index Theorem ~ 1963 CalabiYau ~ 1978 Calibrated Geometry ~ 1982 (Harvey & Lawson)
More informationINVARIANT METRICS WITH NONNEGATIVE CURVATURE ON COMPACT LIE GROUPS
INVARIANT METRICS WITH NONNEGATIVE CURVATURE ON COMPACT LIE GROUPS NATHAN BROWN, RACHEL FINCK, MATTHEW SPENCER, KRISTOPHER TAPP, AND ZHONGTAO WU Abstract. We classify the leftinvariant metrics with nonnegative
More informationExtrinsic geometric flows
On joint work with Vladimir Rovenski from Haifa Paweł Walczak Uniwersytet Łódzki CRM, Bellaterra, July 16, 2010 Setting Throughout this talk: (M, F, g 0 ) is a (compact, complete, any) foliated, Riemannian
More informationInvariant Metrics with Nonnegative Curvature on Compact Lie Groups
Canad. Math. Bull. Vol. 50 (1), 2007 pp. 24 34 Invariant Metrics with Nonnegative Curvature on Compact Lie Groups Nathan Brown, Rachel Finck, Matthew Spencer, Kristopher Tapp and Zhongtao Wu Abstract.
More informationDIVISORS AND LINE BUNDLES
DIVISORS AND LINE BUNDLES TONY PERKINS 1. Cartier divisors An analytic hypersurface of M is a subset V M such that for each point x V there exists an open set U x M containing x and a holomorphic function
More informationtr g φ hdvol M. 2 The EulerLagrange equation for the energy functional is called the harmonic map equation:
Notes prepared by Andy Huang (Rice University) In this note, we will discuss some motivating examples to guide us to seek holomorphic objects when dealing with harmonic maps. This will lead us to a brief
More informationQuantum Field Theory and Representation Theory
Quantum Field Theory and Representation Theory Peter Woit woit@math.columbia.edu Department of Mathematics Columbia University Quantum Field Theory and Representation Theory p.1 Outline of the talk Quantum
More informationBianchi type I solutions to Einstein s vacuum equations. Oliver Lindblad Petersen,
Bianchi type I solutions to Einstein s vacuum equations. Oliver Lindblad Petersen, 9005010419 May 21, 2013 Abstract A natural question in general relativity is whether there exist singularities, like
More informationThe Topology of Fiber Bundles Lecture Notes. Ralph L. Cohen Dept. of Mathematics Stanford University
The Topology of Fiber Bundles Lecture Notes Ralph L. Cohen Dept. of Mathematics Stanford University Contents Introduction v Chapter 1. Locally Trival Fibrations 1 1. Definitions and examples 1 1.1. Vector
More informationMetric Spaces. Chapter 7. 7.1. Metrics
Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some
More informationMathematical Physics, Lecture 9
Mathematical Physics, Lecture 9 Hoshang Heydari Fysikum April 25, 2012 Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 1 / 42 Table of contents 1 Differentiable manifolds 2 Differential
More informationTHE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear
More informationTheta Functions. Lukas Lewark. Seminar on Modular Forms, 31. Januar 2007
Theta Functions Lukas Lewark Seminar on Modular Forms, 31. Januar 007 Abstract Theta functions are introduced, associated to lattices or quadratic forms. Their transformation property is proven and the
More information2 Complex Functions and the CauchyRiemann Equations
2 Complex Functions and the CauchyRiemann Equations 2.1 Complex functions In onevariable calculus, we study functions f(x) of a real variable x. Likewise, in complex analysis, we study functions f(z)
More informationBANACH AND HILBERT SPACE REVIEW
BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but
More informationAndrea Borghese The role of photons in Supergravity
Based on: A.B., A.Guarino, D.Roest, [1209.3003] A.B., G.Dibitetto, A.Guarino, D.Roest, O.Varela, [1211.5335] A.B., A.Guarino, D.Roest, [1302.5067] PLAN Describe the spin1 sector in supergravity theories
More informationLecture 18  Clifford Algebras and Spin groups
Lecture 18  Clifford Algebras and Spin groups April 5, 2013 Reference: Lawson and Michelsohn, Spin Geometry. 1 Universal Property If V is a vector space over R or C, let q be any quadratic form, meaning
More informationIncreasing for all. Convex for all. ( ) Increasing for all (remember that the log function is only defined for ). ( ) Concave for all.
1. Differentiation The first derivative of a function measures by how much changes in reaction to an infinitesimal shift in its argument. The largest the derivative (in absolute value), the faster is evolving.
More information88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a
88 CHAPTER. VECTOR FUNCTIONS.4 Curvature.4.1 Definitions and Examples The notion of curvature measures how sharply a curve bends. We would expect the curvature to be 0 for a straight line, to be very small
More informationMath 5311 Gateaux differentials and Frechet derivatives
Math 5311 Gateaux differentials and Frechet derivatives Kevin Long January 26, 2009 1 Differentiation in vector spaces Thus far, we ve developed the theory of minimization without reference to derivatives.
More informationSOME RESULTS ON COMPLEX SPACES. 1. Introduction. This paper deals with some recent results obtained in collaboration
SOME RESULTS ON COMPLEX SPACES. GIUSEPPE TOMASSINI 1. Introduction This paper deals with some recent results obtained in collaboration with Viorel Vâjâitu. The first one (cfr. [22]) corcerns a new proof
More informationIndex notation in 3D. 1 Why index notation?
Index notation in 3D 1 Why index notation? Vectors are objects that have properties that are independent of the coordinate system that they are written in. Vector notation is advantageous since it is elegant
More informationRIGIDITY OF HOLOMORPHIC MAPS BETWEEN FIBER SPACES
RIGIDITY OF HOLOMORPHIC MAPS BETWEEN FIBER SPACES GAUTAM BHARALI AND INDRANIL BISWAS Abstract. In the study of holomorphic maps, the term rigidity refers to certain types of results that give us very specific
More informationAppendix A. Appendix. A.1 Algebra. Fields and Rings
Appendix A Appendix A.1 Algebra Algebra is the foundation of algebraic geometry; here we collect some of the basic algebra on which we rely. We develop some algebraic background that is needed in the text.
More informationClass Meeting # 1: Introduction to PDEs
MATH 18.152 COURSE NOTES  CLASS MEETING # 1 18.152 Introduction to PDEs, Fall 2011 Professor: Jared Speck Class Meeting # 1: Introduction to PDEs 1. What is a PDE? We will be studying functions u = u(x
More informationTOPOLOGY OF SINGULAR FIBERS OF GENERIC MAPS
TOPOLOGY OF SINGULAR FIBERS OF GENERIC MAPS OSAMU SAEKI Dedicated to Professor Yukio Matsumoto on the occasion of his 60th birthday Abstract. We classify singular fibers of C stable maps of orientable
More informationIntroduction to N = 2 Gauge Theory
Introduction to N = 2 Gauge Theory Chris Elliott January 13th, 2015 1 Introduction and Emphasis The goal of this seminar is to understand the Nekrasov partition function introduced by Nekrasov in 2003
More informationThe Geometry of Graphs
The Geometry of Graphs Paul Horn Department of Mathematics University of Denver May 21, 2016 Graphs Ultimately, I want to understand graphs: Collections of vertices and edges. Graphs Ultimately, I want
More informationRICCI SUMMER SCHOOL COURSE PLANS AND BACKGROUND READING LISTS
RICCI SUMMER SCHOOL COURSE PLANS AND BACKGROUND READING LISTS The Summer School consists of four courses. Each course is made up of four 1hour lectures. The titles and provisional outlines are provided
More informationMatrix Norms. Tom Lyche. September 28, Centre of Mathematics for Applications, Department of Informatics, University of Oslo
Matrix Norms Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo September 28, 2009 Matrix Norms We consider matrix norms on (C m,n, C). All results holds for
More informationThe Finiteness Requirement for SixDimensional Euclidean Einstein Gravity
arxiv:hepth/9911167v1 22 Nov 1999 The Finiteness Requirement for SixDimensional Euclidean Einstein Gravity G.W. Gibbons and S. Ichinose Laboratoire de Physique Théorique de l Ecole Normale Supérieure,
More informationMath 4310 Handout  Quotient Vector Spaces
Math 4310 Handout  Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationMath 497C Sep 9, Curves and Surfaces Fall 2004, PSU
Math 497C Sep 9, 2004 1 Curves and Surfaces Fall 2004, PSU Lecture Notes 2 15 sometries of the Euclidean Space Let M 1 and M 2 be a pair of metric space and d 1 and d 2 be their respective metrics We say
More informationThe determinant of a skewsymmetric matrix is a square. This can be seen in small cases by direct calculation: 0 a. 12 a. a 13 a 24 a 14 a 23 a 14
4 Symplectic groups In this and the next two sections, we begin the study of the groups preserving reflexive sesquilinear forms or quadratic forms. We begin with the symplectic groups, associated with
More information8.1 Examples, definitions, and basic properties
8 De Rham cohomology Last updated: May 21, 211. 8.1 Examples, definitions, and basic properties A kform ω Ω k (M) is closed if dω =. It is exact if there is a (k 1)form σ Ω k 1 (M) such that dσ = ω.
More informationLorentzian Quantum Einstein Gravity
Lorentzian Quantum Einstein Gravity Stefan Rechenberger Uni Mainz 12.09.2011 Phys.Rev.Lett. 106 (2011) 251302 with Elisa Manrique and Frank Saueressig Stefan Rechenberger (Uni Mainz) Lorentzian Quantum
More informationLuminy Lecture 1: The inverse spectral problem
Luminy Lecture 1: The inverse spectral problem Steve Zelditch Northwestern University Luminy April 10, 2015 The inverse spectral problem The goal of the lectures is to introduce the ISP = the inverse spectral
More informationVector and Matrix Norms
Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a nonempty
More informationGEOMETRICAL QUANTUM MECHANICS
GEOMETRICAL QUANTUM MECHANICS Robert Geroch (University of Chicago, 1974) TEXed for posterity by a grad student from an n th generation photocopy of the original set of lecture notes. (Aug 1994) Contents
More information4.6 Null Space, Column Space, Row Space
NULL SPACE, COLUMN SPACE, ROW SPACE Null Space, Column Space, Row Space In applications of linear algebra, subspaces of R n typically arise in one of two situations: ) as the set of solutions of a linear
More informationNOTES ON LINEAR TRANSFORMATIONS
NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all
More informationThe variable λ is a dummy variable called a Lagrange multiplier ; we only really care about the values of x, y, and z.
Math a Lagrange Multipliers Spring, 009 The method of Lagrange multipliers allows us to maximize or minimize functions with the constraint that we only consider points on a certain surface To find critical
More informationGeometric discretisation of GR
Geometric discretisation of GR AMS Meeting New Orleans, January 2007 1 Motivation 2 2 Motivation GR is a geometric theory 2 2 Motivation GR is a geometric theory invariance under arbitrary diffeomorphisms
More informationManifold Learning Examples PCA, LLE and ISOMAP
Manifold Learning Examples PCA, LLE and ISOMAP Dan Ventura October 14, 28 Abstract We try to give a helpful concrete example that demonstrates how to use PCA, LLE and Isomap, attempts to provide some intuition
More informationISU Department of Mathematics. Graduate Examination Policies and Procedures
ISU Department of Mathematics Graduate Examination Policies and Procedures There are four primary criteria to be used in evaluating competence on written or oral exams. 1. Knowledge Has the student demonstrated
More informationSimilarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
More informationIntroduction to Algebraic Geometry. Bézout s Theorem and Inflection Points
Introduction to Algebraic Geometry Bézout s Theorem and Inflection Points 1. The resultant. Let K be a field. Then the polynomial ring K[x] is a unique factorisation domain (UFD). Another example of a
More informationSageManifolds. A free package for differential geometry
SageManifolds A free package for differential geometry Éric Gourgoulhon 1, Micha l Bejger 2 1 Laboratoire Univers et Théories (LUTH) CNRS / Observatoire de Paris / Université Paris Diderot 92190 Meudon,
More information1.7 Graphs of Functions
64 Relations and Functions 1.7 Graphs of Functions In Section 1.4 we defined a function as a special type of relation; one in which each xcoordinate was matched with only one ycoordinate. We spent most
More informationFinite dimensional topological vector spaces
Chapter 3 Finite dimensional topological vector spaces 3.1 Finite dimensional Hausdorff t.v.s. Let X be a vector space over the field K of real or complex numbers. We know from linear algebra that the
More informationMath 241, Exam 1 Information.
Math 241, Exam 1 Information. 9/24/12, LC 310, 11:1512:05. Exam 1 will be based on: Sections 12.112.5, 14.114.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)
More informationLecture 2: Homogeneous Coordinates, Lines and Conics
Lecture 2: Homogeneous Coordinates, Lines and Conics 1 Homogeneous Coordinates In Lecture 1 we derived the camera equations λx = P X, (1) where x = (x 1, x 2, 1), X = (X 1, X 2, X 3, 1) and P is a 3 4
More informationarxiv:1112.3556v3 [math.at] 10 May 2012
ON FIBRATIONS WITH FORMAL ELLIPTIC FIBERS MANUEL AMANN AND VITALI KAPOVITCH arxiv:1112.3556v3 [math.at] 10 May 2012 Abstract. We prove that for a fibration of simplyconnected spaces of finite type F E
More information3. INNER PRODUCT SPACES
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.
More informationAnalytic smoothing of geometric maps with applications to KAM theory
Analytic smoothing of geometric maps with applications to KAM theory A. GonzálezEnríquez R. de la Llave Abstract We prove that finitely differentiable diffeomorphisms preserving a geometric structure
More informationTHE MINIMAL GENUS PROBLEM
THE MINIMAL GENUS PROBLEM TERRY LAWSON Abstract. This paper gives a survey of recent work on the problem of finding the minimal genus of an embedded surface which represents a twodimensional homology
More informationMonotone maps of R n are quasiconformal
Monotone maps of R n are quasiconformal K. Astala, T. Iwaniec and G. Martin For Neil Trudinger Abstract We give a new and completely elementary proof showing that a δ monotone mapping of R n, n is K quasiconformal
More informationFiber sums of genus 2 Lefschetz fibrations
Proceedings of 9 th Gökova GeometryTopology Conference, pp, 1 10 Fiber sums of genus 2 Lefschetz fibrations Denis Auroux Abstract. Using the recent results of Siebert and Tian about the holomorphicity
More informationWintersemester 2015/2016. University of Heidelberg. Geometric Structures on Manifolds. Geometric Manifolds. by Stephan Schmitt
Wintersemester 2015/2016 University of Heidelberg Geometric Structures on Manifolds Geometric Manifolds by Stephan Schmitt Contents Introduction, first Definitions and Results 1 Manifolds  The Group way....................................
More informationVECTOR CALCULUS: USEFUL STUFF Revision of Basic Vectors
Prof. S.M. Tobias Jan 2009 VECTOR CALCULUS: USEFUL STUFF Revision of Basic Vectors A scalar is a physical quantity with magnitude only A vector is a physical quantity with magnitude and direction A unit
More informationAN EXPANSION FORMULA FOR FRACTIONAL DERIVATIVES AND ITS APPLICATION. Abstract
AN EXPANSION FORMULA FOR FRACTIONAL DERIVATIVES AND ITS APPLICATION T. M. Atanackovic 1 and B. Stankovic 2 Dedicated to Professor Ivan H. Dimovski on the occasion of his 7th birthday Abstract An expansion
More informationSINTESI DELLA TESI. EnriquesKodaira classification of Complex Algebraic Surfaces
Università degli Studi Roma Tre Facoltà di Scienze Matematiche, Fisiche e Naturali Corso di Laurea Magistrale in Matematica SINTESI DELLA TESI EnriquesKodaira classification of Complex Algebraic Surfaces
More informationMean value theorem, Taylors Theorem, Maxima and Minima.
MA 001 Preparatory Mathematics I. Complex numbers as ordered pairs. Argand s diagram. Triangle inequality. De Moivre s Theorem. Algebra: Quadratic equations and expressions. Permutations and Combinations.
More informationLINEAR ALGEBRA W W L CHEN
LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,
More informationby the matrix A results in a vector which is a reflection of the given
Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the yaxis We observe that
More informationA new look at NewtonCartan gravity
A new look at NewtonCartan gravity Eric Bergshoeff Groningen University Memorial Meeting for Nobel Laureate Professor Abdus Salam s 90th Birthday NTU, Singapore, January 27 2016 Einstein (1905/1915)
More informationBindel, Spring 2012 Intro to Scientific Computing (CS 3220) Week 3: Wednesday, Feb 8
Spaces and bases Week 3: Wednesday, Feb 8 I have two favorite vector spaces 1 : R n and the space P d of polynomials of degree at most d. For R n, we have a canonical basis: R n = span{e 1, e 2,..., e
More information15.062 Data Mining: Algorithms and Applications Matrix Math Review
.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop
More informationSingular fibers of stable maps and signatures of 4 manifolds
359 399 359 arxiv version: fonts, pagination and layout may vary from GT published version Singular fibers of stable maps and signatures of 4 manifolds OSAMU SAEKI TAKAHIRO YAMAMOTO We show that for a
More informationPOLYTOPES WITH MASS LINEAR FUNCTIONS, PART I
POLYTOPES WITH MASS LINEAR FUNCTIONS, PART I DUSA MCDUFF AND SUSAN TOLMAN Abstract. We analyze mass linear functions on simple polytopes, where a mass linear function is an affine function on whose value
More informationHOMEWORK 4 SOLUTIONS. All questions are from Vector Calculus, by Marsden and Tromba
HOMEWORK SOLUTIONS All questions are from Vector Calculus, by Marsden and Tromba Question :..6 Let w = f(x, y) be a function of two variables, and let x = u + v, y = u v. Show that Solution. By the chain
More informationQuantum Mechanics and Representation Theory
Quantum Mechanics and Representation Theory Peter Woit Columbia University Texas Tech, November 21 2013 Peter Woit (Columbia University) Quantum Mechanics and Representation Theory November 2013 1 / 30
More informationChapter 2. Parameterized Curves in R 3
Chapter 2. Parameterized Curves in R 3 Def. A smooth curve in R 3 is a smooth map σ : (a, b) R 3. For each t (a, b), σ(t) R 3. As t increases from a to b, σ(t) traces out a curve in R 3. In terms of components,
More informationAnalytic cohomology groups in top degrees of Zariski open sets in P n
Analytic cohomology groups in top degrees of Zariski open sets in P n Gabriel Chiriacescu, Mihnea Colţoiu, Cezar Joiţa Dedicated to Professor Cabiria Andreian Cazacu on her 80 th birthday 1 Introduction
More informationA HOPF DIFFERENTIAL FOR CONSTANT MEAN CURVATURE SURFACES IN S 2 R AND H 2 R
A HOPF DIFFERENTIAL FOR CONSTANT MEAN CURVATURE SURFACES IN S 2 R AND H 2 R UWE ABRESCH AND HAROLD ROSENBERG Dedicated to Hermann Karcher on the Occasion of his 65 th Birthday Abstract. A basic tool in
More informationLow Dimensional Representations of the Loop Braid Group LB 3
Low Dimensional Representations of the Loop Braid Group LB 3 Liang Chang Texas A&M University UT Dallas, June 1, 2015 Supported by AMS MRC Program Joint work with Paul Bruillard, Cesar Galindo, SeungMoon
More informationCONNECTIONS ON PRINCIPAL GBUNDLES
CONNECTIONS ON PRINCIPAL GBUNDLES RAHUL SHAH Abstract. We will describe connections on principal Gbundles via two perspectives: that of distributions and that of connection 1forms. We will show that
More informationA QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS
A QUIK GUIDE TO THE FOMULAS OF MULTIVAIABLE ALULUS ontents 1. Analytic Geometry 2 1.1. Definition of a Vector 2 1.2. Scalar Product 2 1.3. Properties of the Scalar Product 2 1.4. Length and Unit Vectors
More informationNotes on Orthogonal and Symmetric Matrices MENU, Winter 2013
Notes on Orthogonal and Symmetric Matrices MENU, Winter 201 These notes summarize the main properties and uses of orthogonal and symmetric matrices. We covered quite a bit of material regarding these topics,
More informationVectors, Gradient, Divergence and Curl.
Vectors, Gradient, Divergence and Curl. 1 Introduction A vector is determined by its length and direction. They are usually denoted with letters with arrows on the top a or in bold letter a. We will use
More informationTOPIC 4: DERIVATIVES
TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the
More informationMultiplicity. Chapter 6
Chapter 6 Multiplicity The fundamental theorem of algebra says that any polynomial of degree n 0 has exactly n roots in the complex numbers if we count with multiplicity. The zeros of a polynomial are
More informationNEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS
NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document
More informationInner product. Definition of inner product
Math 20F Linear Algebra Lecture 25 1 Inner product Review: Definition of inner product. Slide 1 Norm and distance. Orthogonal vectors. Orthogonal complement. Orthogonal basis. Definition of inner product
More informationLecture 5 Principal Minors and the Hessian
Lecture 5 Principal Minors and the Hessian Eivind Eriksen BI Norwegian School of Management Department of Economics October 01, 2010 Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and
More informationMA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES
MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES 2016 47 4. Diophantine Equations A Diophantine Equation is simply an equation in one or more variables for which integer (or sometimes rational) solutions
More informationThe Method of Lagrange Multipliers
The Method of Lagrange Multipliers S. Sawyer October 25, 2002 1. Lagrange s Theorem. Suppose that we want to maximize (or imize a function of n variables f(x = f(x 1, x 2,..., x n for x = (x 1, x 2,...,
More informationMetrics on SO(3) and Inverse Kinematics
Mathematical Foundations of Computer Graphics and Vision Metrics on SO(3) and Inverse Kinematics Luca Ballan Institute of Visual Computing Optimization on Manifolds Descent approach d is a ascent direction
More informationLECTURE III. BiHamiltonian chains and it projections. Maciej B laszak. Poznań University, Poland
LECTURE III BiHamiltonian chains and it projections Maciej B laszak Poznań University, Poland Maciej B laszak (Poznań University, Poland) LECTURE III 1 / 18 BiHamiltonian chains Let (M, Π) be a Poisson
More informationSurface bundles over S 1, the Thurston norm, and the Whitehead link
Surface bundles over S 1, the Thurston norm, and the Whitehead link Michael Landry August 16, 2014 The Thurston norm is a powerful tool for studying the ways a 3manifold can fiber over the circle. In
More informationSection 4.4 Inner Product Spaces
Section 4.4 Inner Product Spaces In our discussion of vector spaces the specific nature of F as a field, other than the fact that it is a field, has played virtually no role. In this section we no longer
More informationFiber Bundles. 4.1 Product Manifolds: A Visual Picture
4 Fiber Bundles In the discussion of topological manifolds, one often comes across the useful concept of starting with two manifolds M ₁ and M ₂, and building from them a new manifold, using the product
More informationMultivariable Calculus and Optimization
Multivariable Calculus and Optimization Dudley Cooke Trinity College Dublin Dudley Cooke (Trinity College Dublin) Multivariable Calculus and Optimization 1 / 51 EC2040 Topic 3  Multivariable Calculus
More informationA NOTE ON TRIVIAL FIBRATIONS
A NOTE ON TRIVIAL FIBRATIONS Petar Pavešić Fakulteta za Matematiko in Fiziko, Univerza v Ljubljani, Jadranska 19, 1111 Ljubljana, Slovenija petar.pavesic@fmf.unilj.si Abstract We study the conditions
More informationFiber bundles and nonabelian cohomology
Fiber bundles and nonabelian cohomology Nicolas Addington April 22, 2007 Abstract The transition maps of a fiber bundle are often said to satisfy the cocycle condition. If we take this terminology seriously
More informationWHICH LINEARFRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE?
WHICH LINEARFRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE? JOEL H. SHAPIRO Abstract. These notes supplement the discussion of linear fractional mappings presented in a beginning graduate course
More information