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:


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