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Subject Panel: Geometry, Number Theory and Topology Suggested Title of Dissertation: Algebraic Fundamental Group Dissertation Su Subject panel: Geometry, Number Theory and Topology Suggested title of dissertation: Algebraic fundamental group Dissertation supervisor: Martin Gallauer Suggested second assessors: Description of proposal: To a topological space X with a base point x 2 X, one associates the fundamental group π1(X; x) which measures to what extent loops in X based at x can be continuously contracted, and therefore yields interesting information about the shape of X around x. At least if X is nice enough, the fundamental group also determines, and is in turn determined by, the covers of X. This latter perspective makes sense in the world of algebraic geometry as well, thus associating to any scheme X with \base point" x 2 X, a group π1(X; x), called the algebraic (or, ´etale)fundamental group. If X is connected, the algebraic fundamental group determines, and is determined by, the finite ´etalecovers of X. For example, if X = Spec(k) is the spectrum of a field, with base point corresponding to a separable algebraic closure k¯ of k, then its algebraic fundamental group is nothing but the absolute Galois group Gal(k=k¯ ) of k. From that point of view, the study of the algebraic fundamental group can be interpreted as a higher-dimensional analogue of Galois theory for fields. The dissertation will center around this important invariant of a scheme. Possible avenues of investigation: The first task should be to digest the notion of finite ´etalecovers and the definition of the algebraic fundamen- tal group with the help of many examples. The latter often have either a topological flavour (relating to the fundamental groups of associated topo- logical spaces), or a number theoretic flavour (reflecting the analogy with Galois theory mentioned above). Depending on the student's background and interests, these analogies can form the basis of a dissertation. Among the further possible avenues of investigation are: • Arithmetic topology: There is a curious analogy between knots (in man- ifolds) and primes (in number theory), and much of this analogy is through the identification of fundamental groups on both sides. • Higher algebraic homotopy groups: As the notation suggests, the fun- 1 damental group is only one of many invariants, called homotopy groups πn(X), one for each integer n ≥ 1. These are in turn derived from a richer invariant, called the ´etalehomotopy type. • Monodromy representations: One way of studying a group is through its representations. It turns out that representations of the algebraic fundamental group π1(X; x) (for X connected) admit a nice geometric description: They are precisely the local systems on X. Pre-requisite knowledge (listed as essential, recommended, useful) • Essential: { B3.1 Galois Theory { C2.6 Introduction to Schemes { C3.7 Category Theory • Recommended: { B3.5 Topology and Groups { B3.4 Algebraic Number Theory • Useful: { C3.1 Algebraic Topology Useful reading • Tam´asSzamuely, Galois Groups and Fundamental Groups, Cambridge Studies in Advanced Mathematics, vol. 117, Cambridge University Press, 2009 • Hendrik Lenstra, Galois theory for schemes, lecture notes available at http://websites.math.leidenuniv.nl/algebra/GSchemes.pdf Further references • Masanori Morishita, Knots and Primes: An Introduction to Arithmetic Topology, Springer, 2012 • Michael Artin; Barry Mazur, Etale homotopy, Springer Lecture Notes in Mathematics 100, 1969 2 • James Milne, Etale´ Cohomology, Princeton Mathematical Series 33, 1980 I can provide further references once the line of investigation has been decided upon. 3.
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