 SYMMETRY IN NEWCASTLE
 Location: (Online Campus)
 Dates: Mon, 23^{rd} Nov 2020  Mon, 23^{rd} Nov 2020

Schedule (Zoom):
18.3019.30: William Hautekiet
19.3020.00: Break
20.0021.00: Florian Breuer
 Speaker: Dr William Hautekiet, Faculté des Sciences, Université libre de Bruxelles
 Title: Automorphism groups of transcendental field extensions
 Abstract for Automorphism groups of transcendental field extensions:
It is wellknown that the Galois group of an (infinite) algebraic field extension is a profinite group. When the extension is transcendental, the automorphism group is no longer compact, but has a totally disconnected locally compact structure (TDLC for short). The study of TDLC groups was initiated by van Dantzig in 1936 and then restarted by Willis in 1994. In this talk some of Willis' concepts, such as tidy subgroups, the scale function, flat subgroups and directions are introduced and applied to examples of automorphism groups of transcendental field extensions. It remains unknown whether there exist conditions that a TDLC group must satisfy to be a Galois group. A suggestion of such a condition is made.
 Speaker: Prof. Florian Breuer, School of Mathematical and Physical Sciences, The University of Newcastle
 Title: Realising general linear groups as Galois groups
 Abstract for Realising general linear groups as Galois groups:
I will show how to construct field extensions with Galois groups isomorphic to general linear groups (with entries in various rings and fields) from the torsion of elliptic curves and Drinfeld modules. No prior knowledge of these structures is assumed.
 Abstract for Automorphism groups of transcendental field extensions:
It is wellknown that the Galois group of an (infinite) algebraic field extension is a profinite group. When the extension is transcendental, the automorphism group is no longer compact, but has a totally disconnected locally compact structure (TDLC for short). The study of TDLC groups was initiated by van Dantzig in 1936 and then restarted by Willis in 1994. In this talk some of Willis' concepts, such as tidy subgroups, the scale function, flat subgroups and directions are introduced and applied to examples of automorphism groups of transcendental field extensions. It remains unknown whether there exist conditions that a TDLC group must satisfy to be a Galois group. A suggestion of such a condition is made.
 Abstract for Realising general linear groups as Galois groups:
I will show how to construct field extensions with Galois groups isomorphic to general linear groups (with entries in various rings and fields) from the torsion of elliptic curves and Drinfeld modules. No prior knowledge of these structures is assumed.
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 CARMA COLLOQUIUM
 Speaker: Prof. Florian Breuer, School of Mathematical and Physical Sciences, The University of Newcastle
 Title: The Parallel Worlds of Number Theory
 Location: Room V205, Mathematics Building (Callaghan Campus) The University of Newcastle
 Time and Date: 4:00 pm, Thu, 10^{th} May 2018
 Abstract:
There is an intriguing analogy between number fields and function fields. If we view classical Number Theory as the study of the ring of integers and its extensions, then function field arithmetic is the study of the ring of polynomials over a finite field and its extensions. According to this analogy, most constructions and phenomena in classical Number Theory, ranging from the elementary theorems of Euler, Fermat and Wilson, to the Riemann Hypothesis, Elliptic curves, class field theory and modular forms all have their function field analogues. I will give a panoramic tour of some of these constructions and highlight their similarities and differences to their classical counterparts.
This lecture should be accessible to advanced undergraduate students.
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