Abstract. I will explain what is known about several different classifications of 4-manifolds (up to diffeomorphism, homeomorphism, homotopy equivalence...) and the relations between the different classifications.
The talks are all happening at University of Southampton (in person), on Highfield Campus (SO17 1BJ). For the building and the room, see the individual postings. Email the organiser ( b (dot) koeck (at) soton.ac.uk ) if you have questions.
More Pure Maths talks at Southampton: we run Pure Maths Colloquia, Topology Seminar, and Pure Postgraduate Seminar (link soon).
Abstract. I will explain what is known about several different classifications of 4-manifolds (up to diffeomorphism, homeomorphism, homotopy equivalence...) and the relations between the different classifications.
Abstract.
What are the transitive permutation groups of prime (or prime power) degree?
Are there infinitely many simple groups of order a product of six primes?
Are there infinitely many counterexamples to a theorem of Cauchy on permutation groups?
Solving these and various other problems depends on certain conjectures in Number Theory regarding prime values of polynomials, namely Schinzel's Hypothesis and its quantified form, the Bateman--Horn Conjecture. Proving these is a very difficult open problem, but joint work with Alexander Zvonkin (LaBRI, Bordeaux) gives strong computational evidence that in the above contexts (and many others) they are true. No background in Number Theory is required for this talk, beyond knowing the definition of a prime number (and a few examples).