# Topology Seminar Archives

The following are selected archives of the Topology Seminar at Boise State University.

## 2018–2019

**Date:** September 7 2018

**Speaker:** Jens Harlander

**Title:** Some conjectures in 2-dimensional combinatorial topology

**Abstract:** In the early part of the 20th century a variety of homotopy notions emerged, such as simple homotopy, combinatorial in nature, or the homotopy theory of chain complexes, algebraic and part of homological algebra. It is well understood how the different notions relate to each other, and what obstructions exist for direct comparisons. However, fundamental open problems remain, particularly in dimension 2, where questions concerning group presentations enter the discussion. I will talk about three conjectures in dimension 2.

**Date:** September 28 2018

**Speaker:** Jens Harlander

**Title:** What is homological group theory?

**Date:** October 26 2018

**Speaker:** Kennedy Courtney

**Title:** Topological Invariants of Food Webs

**Abstract:** A food web is an interconnected network of food chains in an ecosystem. Food webs are easily modeled by directed graphs and have been well-studied from the graph theoretic perspective. However, viewing food webs as graphs does not seem to easily reveal qualities that are important in ecology. We seek to address this problem by analyzing graphs of food webs through a more sophisticated topological approach, namely through the directed forest complex.

**Date:** November 30 2018

**Speaker:** Uwe Kaiser

**Title:** Topological complexity and motion planning

**Abstract:** I will give several definitions of the topological complexity of a configuration space due to Michael Farber. These are related to motion planning in that space. I will discuss how they compare with each other, homotopy invariance, upper and lower bounds, and show several examples.

**Date:** January 25 2019

**Speaker:** Jens Harlander

**Title:** The Corson/Trace characterization of diagrammatic reducibility

**Abstract:** It is difficult to tell from a given presentation P if the group presented G(P) is finite or infinite. If the associated 2-complex K(P) exhibits a strong asphericity condition, diagrammatic reducibility, then G(P) is trivial or infinite. How do we decide G(P)=1 in the presence of DR? In 2000 Corson and Trace gave a surprisingly simple and checkable answer which I will share in my talk.

**Date:** February 1 2019

**Speaker:** Jens Harlander

**Title:** The Corson/Trace characterization of diagrammatic reducibility, 2

**Date:** March 8 2019

**Speaker:** Stephan Rosebrock, PH Karlsruhe, Germany

**Title:** On the asphericity of labeled oriented trees

**Abstract:** The Whitehead conjecture asks whether a subcomplex of an aspherical 2-complex is alwaysaspherical. This question is open since 1941. Howie has shown that the existence of a finite counterexample implies (up to the Andrews-Curtis conjecture) the existence of a counterexample within the class of labelled oriented trees. Labelled oriented trees are algebraic generalisations of Wirtinger presentations of knot groups. In this talk we start with an introduction into the field. Then we present several possibilities to show asphericity in the class of labelled oriented trees. There are many known classes of aspherical LOTs given by the weight test of Gersten, the I-test of Barmak/Minian, LOTs of Diameter 3 (Howie), LOTs of complexity two (Rosebrock) and several more.We introduce a new notion of relative asphericity and proove with this notion the asphericity of injective labelled oriented trees.

**Date:** April 5, 2019

**Speaker:** Uwe Kaiser

**Title:** Obstructions to ribbon concordance

**Abstract:** I will discuss the concept of ribbon concordance and how it relates to concordance and to the infamous slice-ribbon conjecture. Then I will state some classical results by Cameron Gordon, and survey a recent paper by Ian Zemke, which contains several interesting corollaries concerning crossing numbers.

## 2017–2018

**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.