## Thesis Research

**Advisor:** Julie Bergner

**Area of research:** Algebraic topology, more specifically homotopy theory

Throughout mathematics we talk about various types of invertible maps. In set theory, we have bijections,
in group theory we have isomorphisms, and in topology we have homeomorphisms. These three types of maps
give us a notion of equality. However in homotopy theory, sometimes we want to say two spaces are weakly
equivalent even if there is not a homeomorphism between the spaces. For example, we say a map between
spaces is a weak homotopy equivalence if it induces isomorphisms on all homotopy groups. In my thesis, I
investigate various settings that have notions of weak equivalences by implementing a machine called the
*classification diagram*. In particular, I provide a complete description of the classification
diagram of finite sets and show that it is weakly equivalent to the classification diagram of graphs. I
also determine that if we are working in a situation where every map is a weak equivalence, then the
classification diagram has a beautiful description in terms of its classifying space.

## Women in Topology (WIT) Research

**Team Leaders:**

- Kristine Bauer, University of Calgary
- Brenda Johnson, Union College

**Team Members:**

- Christina Osborne, University of Virginia
- Emily Riehl, Johns Hopkins University
- Amelia Tebbe, University of Illinois Urbana-Champaign

**Papers:**

- [BJORT]
__Directional derivatives and higher order chain rules for abelian functor calculus__, to appear in WIT II proceedings,*Topology Appl.* - [OT]
__A first step toward higher order chain rules in abelian functor calculus__, submitted.

## Talks Given

Jan. 2018 | The classifying diagram of the category of finite sets
The Joint Mathematics Meetings: AMS Contributed Paper Session on Knots & Diagram Categories, San Diego, CA |

Nov. 2017 | Old and new chain rules
–
(undergraduate audience)
The MAA MD-DC-VA Section, Christopher Newport University |

Nov. 2017 | Investigations of the classifying and classification diagrams
Lloyd Roeling Topology Conference, University of Louisiana at Lafayette |

Sep. 2017 | Old and new chain rules
–
(undergraduate audience)
Mathematics Seminar, Randolph-Macon College |

Apr. 2017 | A higher order chain rule for abelian functor calculus
–
slides
AWM Research Symposium: Topics in Homotopy Theory Session, University of California, Los Angeles |

Apr. 2017 | Directional derivatives and higher order chain rules for abelian functor calculus –
slides
AMS Spring Central Sectional Meeting: Homotopy Theory Session, Indiana University, Bloomington |

Jan. 2017 | The first step towards higher order chain rules for abelian functor calculus
Graduate Student Seminar, University of California, Riverside |

Dec. 2016 | The first step towards higher order chain rules for abelian functor calculus
–
slides
Union College Mathematics Conference, Union College |

Nov. 2016 | The first step towards higher order chain rules for abelian functor calculus
Topology Seminar, University of Virginia |

Apr. 2016 | Introduction to calculus on functors: Goodwillie and beyond
Graduate Student Seminar, University of California, Riverside |

Jan. 2016 | Replacing a map by a cofibration
Topology Seminar, University of California, Riverside |

Oct. 2015 | Group actions and fiber bundles
Topology Seminar, University of California, Riverside |

Aug. 2015 | Model structures and simplicial presheaves
USC K-Theory Summer School, University of Southern California |

Apr. 2015 | Introduction to model categories
Graduate Student Seminar, University of California, Riverside |

## Conferences Attended

Jan. 2018 | Joint Mathematics Meetings
San Diego, Texas |

Nov. 2017 | The Mathematical Association of America MD-DC-VA Section
Christopher Newport University |

Nov. 2017 | Lloyd Roeling Topology Conference
University of Louisiana at Lafayette |

Jul. 2017 | Homotopy Theory: Tools and Applications
University of Illinois at Urbana-Champaign |

Apr. 2017 | AWM Research Symposium
University of California, Los Angeles |

Apr. 2017 | AMS Spring Central Sectional Meeting
Indiana University, Bloomington |

Dec. 2016 | Union College Mathematics Conference
Union College |

Nov. 2016 | VA Topology Conference: Mapping Class Groups and Low Dimensional Topology
University of Virginia |

Apr. 2016 | Graduate Student Topology and Geometry Conference
Indiana University, Bloomington |

Nov. 2015 | Southern California Women in Mathematics Symposium
Pomona College |

Feb. 2015 | West Coast Lie Theory Workshop
University of California, Riverside |

Jan. 2015 | Joint Mathematics Meetings
San Antonio, Texas |

Oct. 2013 | Southern California Women in Mathematics Symposium
University of California, Riverside |

## Workshops and Schools Attended

Apr. 2016 | Women in Topology Workshop
Banff International Research Station |

Aug. 2015 | West Coast Algebraic Topology Summer School
University of Oregon |

Aug. 2015 | USC K-Theory Summer School
University of Southern California |