MAT5343 - Revision history

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Duynguyenvu.hoang: Created page with "MAT 5343 Differential Geometry '''Sample textbook''': Victor V. Prasolov: Differential Geometry, Springer (2022) '''Catalog entry''' ''Prerequisite'': Linear Algebra and Ca..."

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Created page with "MAT 5343 Differential Geometry '''Sample textbook''': Victor V. Prasolov: Differential Geometry, Springer (2022) '''Catalog entry''' ''Prerequisite'': Linear Algebra and Ca..."

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MAT 5343 Differential Geometry

'''Sample textbook''': Victor V. Prasolov: Differential Geometry, Springer (2022)

'''Catalog entry'''

''Prerequisite'': Linear Algebra and Calculus sequence

''Contents'':
(1) Introduction to post-quantum cryptography.
(2) Quantum computing.
(3) Hash-based Digital Signature Schemes.
(4) Code-based cryptography.
(5) Lattice-based Cryptography
(6) Multivariate Public Key Cryptography.
(7) Homomorphic encryption.

==Topics List==
{| class="wikitable sortable"
! Week !! Topic !! Chapter 1 from the Bernstein-Johannes Buchmann-Dahmen book !! MAT1313, or CS2233/2231, or instructor consent.
|-
| 1-3
|| [[Introduction to post-quantum cryptography]]
|| 1
|| MAT1313 or CS2233/2231, or equivalent.
|-
| 2-4
|| [[Quantum computing]]
|| 2
||
|-
| 5-6
|| [[Hash-based Digital Signature Schemes]]
|| 3
||
|-
| 7-8
|| [[Code-based cryptography]]
|| 4
||
|-
| 9-10
|| [[Lattice-based Cryptography]]
|| 5
||
|-
| 11-12
|| [[Multivariate Public Key Cryptography]]
|| 6
||
|-
| 13-end
|| [[Homomorphic encryption]]
||
||
|}

@@ Line 32: / Line 32: @@
 || [[Gaussian curvature]]
 || 3.5-3.7
-|| Compute and interpret Gaussian curvature
+|| Compute and interpret Gaussian curvature, understand Theorema Egregium.
 |-
 |  5
 || [[Covariant differentiation on surfaces and parallel transport]]
 || 3.8-3.11
-||
+|| Become fluent with tensor calculus and covariant differentiation
 |-
 |  6
 || [[Gauss-Codazzi equations and Riemann curvature tensor]]
 || 3.13-3.14
-||
+|| Be able to compute Riemann curvature tensor, become familiar with symmetry properties of the Riemann tensor
 |-
 |  7
 || [[Riemannian manifolds I]]
 || 5.1-5.5
-||
+|| Understand definition of Riemannian manifolds
 |-
 |  8
 || [[Riemannian manifolds II]]
 || 5.6-5.14
-||
+|| Understand properties of Riemannian manifolds
 |-
 |  9
 || [[Introduction into Lorentzian manifolds and relativity]]
 || instructor provided material
-||
+|| Understand the difference between Riemannian and Lorentzian manifolds
 |-
 |  10
 || [[Einstein manifolds]]
 || instructor provided material
-||
+|| Understand where Einstein equations come from
 |-
 |  11
@@ Line 72: / Line 73: @@
 || [[Differential forms and integral theorems]]
 || instructor provided material
-||
+|| Be able to apply Stokes' theorem
 |-
 |  13
 || [[Introduction to Lie groups]]
 || 6.1-6.5
-||
+|| Be able to compute generators of a Lie group
 |-
 |  14

@@ Line 16: / Line 16: @@
 |  1
 || [[Curves in space and on the plane]]
-|| 1
+||
 || Understand how to compute the curvature of a space/planar curve
 |-
 |  2
-|| [[]]
+|| [[Moving frames, Serret-Frenet formulas]]
-|| 2
+|| 2.1-2.5
-||
+|| Understand how apply the Serret-Frenet formulas, determine curve from given curvature and torsion
 |-
 |  3
-|| [[ ]]
+|| [[Surfaces in space, introduction to differential forms]]
-|| 3
+|| 3.1-3.4
-||
+|| Parameterize surfaces in space, compute and interpret first and second fundamental form
 |-
 |  4
-|| [[ ]]
+|| [[Gaussian curvature]]
-|| 4
+|| 3.5-3.7
-||
+|| Compute and interpret Gaussian curvature
 |-
 |  5
-|| [[ ]]
+|| [[Covariant differentiation on surfaces and parallel transport]]
-|| 5
+|| 3.8-3.11
 ||
 |-
 |  6
-|| [[ ]]
+|| [[Gauss-Codazzi equations and Riemann curvature tensor]]
-|| 6
+|| 3.13-3.14
 ||
 |-
 |  7
-|| [[ ]]
+|| [[Riemannian manifolds I]]
-|| 6
+|| 5.1-5.5
 ||
 |-
 |  8
-|| [[ ]]
+|| [[Riemannian manifolds II]]
-|| 6
+|| 5.6-5.14
 ||
 |-
 |  9
-|| [[ ]]
+|| [[Introduction into Lorentzian manifolds and relativity]]
-|| 6
+|| instructor provided material
 ||
 |-
 |  10
-|| [[ ]]
+|| [[Einstein manifolds]]
-|| 6
+|| instructor provided material
 ||
 |-
 |  11
-|| [[ ]]
+|| [[Global theorems in differential geometry]]
-|| 6
+|| 3.7, 5.4
-||
+|| Hopf-Rinow and Gauss-Bonnet theorem
 |-
 |  12
-|| [[ ]]
+|| [[Differential forms and integral theorems]]
-|| 6
+|| instructor provided material
 ||
 |-
 |  13
-|| [[ ]]
+|| [[Introduction to Lie groups]]
-|| 6
+|| 6.1-6.5
 ||
 |-
 |  14
-|| [[ ]]
+|| [[Auxillary topics]]
 ||
 ||
 |}

@@ Line 3: / Line 3: @@
 '''Sample textbook''': Victor V. Prasolov: Differential Geometry, Springer (2022)
-'''Catalog entry'''
+'''Catalog entry''':
 ''Prerequisite'': A course in Linear Algebra and Calculus sequence
 ''Contents'':
-(1)
+This course offers an in-depth study of differential geometry, focusing on the geometric properties of differentiable manifolds. Topics include smooth manifolds, vector fields, differential forms, tensors, connections, Riemannian metrics, and curvature. Emphasis will be placed on understanding the interplay between geometry and topology, as well as applications in physics, particularly in general relativity.
 ==Topics List==
@@ Line 19: / Line 19: @@
 || Understand how to compute the curvature of a space/planar curve
 |-
-|  2-4
+|  2
-|| [[Quantum computing]]
+|| [[]]
 || 2
 ||
 |-
-|  5-6
+|  3
-|| [[Hash-based Digital Signature Schemes]]
+|| [[ ]]
 || 3
 ||
 |-
-|  7-8
+|  4
-|| [[Code-based cryptography]]
+|| [[ ]]
 || 4
 ||
 |-
-|  9-10
+|  5
-|| [[Lattice-based Cryptography]]
+|| [[ ]]
 || 5
 ||
 |-
-|  11-12
+|  6
-|| [[Multivariate Public Key Cryptography]]
+|| [[ ]]
 || 6
 ||
 |-
-|  13-end
+|  7
-|| [[Homomorphic encryption]]
+|| [[ ]]
 ||
 ||
 |}

@@ Line 12: / Line 12: @@
 ==Topics List==
 {| class="wikitable sortable"
-! Week !! Topic !! Chapter 1 from the Bernstein-Johannes Buchmann-Dahmen book !! MAT1313, or CS2233/2231, or instructor consent.
+! Week !! Topic !! Chapter from textbook !! SLO
 |-
-|  1-3
+|  1
-|| [[Introduction to post-quantum cryptography]]
+|| [[Curves in space and on the plane]]
 || 1
-|| MAT1313 or CS2233/2231, or equivalent.
+|| Understand how to compute the curvature of a space/planar curve
 |-
 |  2-4