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many_maths_challenges.txt
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many_maths_challenges.txt
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Things which we should get into mathlib! Last updated 30/11/21.
Number fields.
--------------
1) Define the zeta function of a number field.
2) Prove that the zeta function of a number field has a meromorphic extension to the complex numbers.
3) Define the regulator of a number field.
4) State the analytic class number formula.
5) Prove the analytic class number formula.
6) Do everything for global function fields.
Group cohomology.
-----------------
1) Define group cohomology
2) Prove the Hochschild-Serre spectral sequence.
3) Prove the long exact sequence of terms of low degree.
4) Define profinite group cohomology for discrete G-modules.
Galois theory
-------------
1) Define the topology on an infinite Galois group
2) State the fundamental theorem of Galois theory for algebraic normal separable extensions of infinite degree.
3) Prove it. [note that we have everything for finite Galois groups]
Local Fields
------------
1) Set up the basic theory of fields complete with respect to a discrete valuation (integer ring, uniformiser, PID, finite extensions, extension of valuation)
2) Set up the basic theory of Galois groups of finite extensions of p-adic fields (Frobenius elements, inertia groups, higher ramification groups, lower numbering, upper numbering)
Galois cohomology
-----------------
1) State local Tate duality.
2) Prove local Tate duality.
3) State the global duality theorems (Greenberg-Wiles etc).
4) Formalise proofs of all the theorems in Milne's book "arithmetic duality theorems".
Harmonic analysis
-----------------
1) State the Pontrjagin duality theorem.
2) Prove the Pontrjagin duality theorem.
Elliptic curves
---------------
1) Define the group structure on the k-points of an elliptic curve, proving that it is a group.
2) Prove that if k is a number field then the group of k-points of an elliptic curve are finitely-generated.
3) Define the L-function of an elliptic curve over a number field, proving it converges for Re(s) sufficiently large.
4) State the conjecture asserting that the L-function has holomorphic continuation to the complex plane.
5) State the Birch and Swinnerton-Dyer conjecture for elliptic curves over the rational numbers. [note that Jamie Bell has a preliminary
version of a lot of this stuff: https://github.com/jamiebell2805/BSD-conjecture/
Algebra
-------
1) Eigenspaces and generalised eigenspaces.
2) Structure theorem for finitely-generated modules over a PID.
3) Jordan decomposition
Homological Algebra
-------------------
1) Definitions of Ext and Tor
2) Long exact sequences
3) Relation to flatness
Algebraic Topology
------------------
1) Define homology groups H_n(X;R) and cohomology groups H^n(X;R) of a topological space X.
2) Define homotopy groups pi_n(X) of a topological space.
Representation Theory
---------------------
1) Schur's Lemma.
2) Basic representation theory of finite groups (e.g. irreducible characters form a basis of class functions on the group)
3) Extension to compact Lie groups.
Modular forms
-------------
1) Definitions and basic properties and examples.
2) Finite-dimensionality of a space of modular forms of fixed level and weight.
3) Hecke operators and eigenforms.
4) State Deligne's theorem attaching Galois representations to eigenforms.
5) State the modularity theorem (all elliptic curves are modular).