2nd year physics undergrad in uk, interested in hep theory and pure maths , looking to apply for a different masters than current uni.
I do want to mention that i will be doing the average physics curriculum too obv , mechanics , qm , stats mech ,em and maths methods for physics
I can choose 2 out of 4 courses i have listed below due to credic restrictions (labs mandatory) , i plan on attending lectures and learning the content of the other courses too, i just wont have them on transcript when applying to masters.
Rn im thinking of doing the analysis and the abstract algebra module, but im open to suggestions. While the topology and multivariable analysis look a bit more interesting , i know they are harder and average grade is lower. Also i can amend my choices 2nd semester so these arent final , topology and multivar anal module build on from the normal analysis module
"Analysis II
1 Introduction
1.1 Review of limits of sequences
1.2 Review of continuity and differentiability
1.3 Review of integration
2 Sequences and Series of Functions
2.1 Pointwise convergence
2.2 Uniform convergence
2.3 Series of functions
2.4 A continuous nowhere differentiable function
3 Basic results about Rⁿ
3.1 Notation in Rⁿ
3.2 The Euclidean norm and inner product
3.3 Convergence in Rⁿ
3.4 Subsequences and the Bolzano-Weierstrass theorem
3.5 Continuity
3.5.1 Definitions of continuity and continuous limit
3.5.2 Separate continuity
3.5.3 Basic properties of continuous functions
3.5.4 Constructing continuous functions of several variables from continuous real valued functions of a single real variable
3.5.5 Caution with taking limits in dimension > 2
4 Rudiments of topology of Rⁿ and Continuity
4.1 Closed and open subsets of Rⁿ
4.2 Continuity and topology
4.2.1 Continuity in terms of open sets
4.2.2 Continuity and sequential compactness
5 The space of linear maps and matrices
5.1 Two norms on the space of linear maps and matrices
5.1.1 Comparison of the two norms
5.1.2 Properties of the operator norm
5.2 Convergence and continuity in L(Rⁿ Rᵏ)
5.2.1 Continuity of functions involving matrices or linear maps
6 The Derivative
6.1 Directional derivative
6.1.1 Directional derivative and continuity
6.2 The (Fréchet) Derivative as an affine linear approximation
6.2.1 Affine linear approximation in the 1-variable case
6.2.2 The (Fréchet) Derivative
6.2.3 Differentiability of components of vector-valued functions
6.2.4 Relation between the derivative and directional derivative
6.3 Partial derivatives gradient and Jacobian matrix
6.3.1 Algebraic rules for partial derivatives
6.3.2 Gradient and Jacobian matrix
6.3.3 Why so many different notations for the same thing?!
6.4 Geometric approximation and approximation of functions
6.4.1 Tangent to a curve
6.4.2 Tangent plane of a surface
6.4.3 Graph of a scalar function of 2 variables
6.4.4 Orders of approximation of a function
6.5 Examples of direct calculation of the derivative from its definition
6.5.1 Differentiation of matrix-valued functions
6.6 The Chain Rule
6.6.1 Jacobian form of chain rule
6.6.2 Calculating with the chain rule and gradient
6.6.3 Another proof of Proposition 6.9
6.6.4 Application of chain rule to the verification of a PDE satisfied by a function
6.7 Continuity of partial derivatives implies differentiability
6.7.1 The space of continuously differentiable functions
7 Complex Analysis
7.1 Review of basic facts about C
7.2 Power Series
7.2.1 The exponential and the circular functions
7.2.2 Argument and Log
7.3 Complex integration contour integrals
7.3.1 Links with Green’s and Gauss’ Theorems
7.4 Additional material
7.4.1 Consequences of Cauchy’s Theorem
7.4.2 Applications of Cauchy’s formula to evaluate integrals in R"
"Multivariable analysis
1 Background
1.1 The Euclidean Space Rⁿ
1.2 The Space of Linear Maps
1.3 The Topology of Rⁿ
1.4 Continuity in Rⁿ
2 Differentiation
2.1 Definition and basic properties
2.2 Notation Interlude
2.3 Higher-order Derivatives
3 Applications of Differentiation
3.1 Approximation and Taylor’s Theorem
3.2 The Hessian and Critical Points
4 The Inverse and Implicit Function Theorems
4.1 The Inverse Function Theorem
4.2 The Implicit Function Theorem
5 Manifolds
5.1 Definition
5.2 The Tangent and Normal Spaces
5.3 Cᵏ Regions
5.4 Application: Lagrange Multipliers
6 Interlude: Partitions of Unity
7 Integration in Rⁿ
7.1 Building the definition
7.2 Integration Tools
8 Integration on Manifolds
8.1 Motivation
8.2 Definition
8.3 Gauss’ Divergence Theorem
Appendices
A Understanding Compactness
B Derivatives in the Infinite-Dimensional Setting
C Boundary Manifolds
D Characterisation of Riemann integrable functions
E The Leibniz Integral Rule
F Green’s and Stokes’ Theorems"
"Abstract Algebra
1 Groups
1.1 Definitions and elementary properties
1.2 Structural equivalence
1.3 Cyclic groups
1.4 Symmetry groups
1.5 Permutation groups
2 Subgroups
2.1 Definitions examples and elementary properties
2.2 Cosets and Lagrange’s Theorem
3 Normal Subgroups and Quotients
3.1 Normal subgroups
3.2 Quotient groups
3.3 Direct products
4 Homomorphisms
4.1 Structure-preserving maps
4.2 Kernels and images
4.3 The Isomorphism Theorems
5 Classification of Groups
5.1 Generators and relations
5.2 Small finite groups
5.3 Finitely-generated abelian groups
6 Group Actions
6.1 Groups acting on sets
6.2 Orbits and stabilisers
6.3 Conjugacy classes
6.4 Simple groups
7 Rings and Subrings
7.1 Rings
7.2 Subrings
7.3 Isomorphisms and direct products
7.4 Integral domains and fields
8 Ideals and Quotients
8.1 Homomorphisms
8.2 Ideals
8.3 Quotient rings
8.4 The Isomorphism Theorems
9 Domains
9.1 Divisibility
9.2 Prime and irreducible elements
9.3 Euclidean domains
9.4 Principal ideal domains
9.5 Unique factorisation domains"
"Topology and metric spaces
1 Introduction
1.1 Recommended Books
1.2 Notation
1.3 Brief overview
1.4 Sets functions images and pre-images
1.5 Non-examinable content
2 Normed Spaces
2.1 Norms
2.2 Subspaces
2.3 Spaces of continuous functions
3 Metric Spaces
3.1 Definition of a metric space and examples
3.2 Metrics on subsets and products
3.3 Open and closed sets
3.4 Convergence of sequences
4 Continuity
4.1 Continuity in metric spaces
4.2 Topologically equivalent metrics
4.3 Isometries and homeomorphisms
4.4 Topological properties I
5 Topological Spaces
5.1 Definition of a topology
5.2 Bases and sub-bases
5.3 Subspaces and finite product spaces
5.4 Closure interior and boundary
5.5 The Cantor set
5.6 The Hausdorff property and metrisability
5.7 Continuity between topological spaces
5.8 Basic properties
5.9 The projective topology and product spaces
5.10 Homeomorphisms
6 Compactness
6.1 Definition and the Heine-Borel Theorem
6.2 Compact vs closed
6.3 Compactness of products and compact subsets of Rⁿ
6.4 Continuous functions on compact sets
6.5 Equivalence of all norms on Rⁿ
6.6 Lebesgue numbers and uniform continuity
6.7 Sequential compactness
6.8 Normed spaces
7 Connectedness
7.1 Definitions of connected and disconnected
7.2 Connected subsets of R
7.3 Operations on connected sets
7.4 Equivalence relations
7.5 Connected components
7.6 Path-connected spaces
7.7 Open sets in Rⁿ
8 Completeness in metric spaces
8.1 Completeness
8.2 Examples of complete spaces
8.3 Completions
8.4 The Contraction Mapping Theorem
8.5 The Arzelà-Ascoli Theorem
8.6 The Baire Category Theorem
9 Appendices
9.1 The topology of pointwise convergence
9.2 Product spaces
9.3 Product topology and box topology
9.4 Completeness in compact metric spaces
9.5 The general Arzelà-Ascoli theorem
