The Topology Seminar will meet on Fridays, usually alternating with the AGC seminar, 3:00 p.m. – 4:00 p.m. in room MB 124 (Mathematics Building). Below you find the information for the topology talks. Everybody who is interested is invited to join. Please send email to firstname.lastname@example.org for more information. For completeness we also have included the talks in the AGC-Seminar.
January 12: All participants
January 19: Jens Harlander
Title: Who cares about finite topological spaces? Part I
Abstract: There are three things that I will try to explain:
1) The category of finite spaces is the same as the category of pre-ordered set.
2) The homotopy classification of finite spaces is done.
3) A finite CW-complex has the weak homotopy type of a finite topological space.
This has been known since the mid 1960’s. Recently, paying attention to 3), Barmack and Minian have translated famous conjecture from low dimensional topology, such as the Andrews-Curtis and Whitehead’s asphericity conjecture, into the language of finite spaces. I will focus on the Whitehead conjecture in a second talk. Most of what I have to say is suitable for undergraduate students. So don’t worry and come.
January 26: Jens Harlander
Title: Who cares about finite topological spaces? Part II
Abstract: I will review the homotopy classification for finite topological spaces and then move from homotopy types to weak homotopy types. The main result here is that every finite CW-complex has the weak homotopy type of a finite topological space.
February 2: Jens Harlander
Title: Who cares about finite topological spaces? Part III
Abstract: I will talk about translations of famous open conjectures into the world of finite topological spaces. Versions of the conjectures can be solved in the finite world. However, the methods fall short of resolving the original conjectures.
March 16: Uwe Kaiser
Title: Categorification in Algebra and Topology
Abstract: The idea of categorification has been introduced by theoretical physicist Louis Crane in 1994. In 1999 Mikhail Khovanov constructed a categorification of the Jones polynomial, which has been quickly followed by categorifications of more general quantum invariants. Since then categorification has become a major topic in topology, algebra and representation theory. We will discuss a few of the basic original ideas, in particular the motivation from Topological Quantum Field Theory and the intended goal of the so called the Crane-Frenkel program.
March 23: no seminar
April 6: Jens Harlander
Title: Finite or Infinite?
Abstract: The fundamental group of a space is a surprisingly strong invariant of a space. Topologists encounter this group in terms of a presentation, a (finite) set of data consisting of a list of generators and a list of relations that hold among them. It is notoriously difficult to decipher properties of the group from a presentation. The most obvious property being whether the group is finite or infinite. We will look at examples and techniques to tackle such questions.
April 20: Kayla Neal
Title: Pentagonal Tilings
Abstract: I will talk about the history of pentagonal tilings and the mathematics of the discoveries. There will also be discussion of the restrictions for monohedral pentagonal tilings to have a convex, equilateral prototile. Then I will discuss research of mosaic knots on squares and hexagons and conclude with research questions on knot mosaics on equilateral, convex pentagonal tilings.