Papers

Elementary Higher Toposes:

• A Theory of Elementary Higher Toposes:
  We define an elementary higher topos and show it generalizes elemenary toposes and higher toposes.
• Complete Segal Objects:
  We define an internal version of a higher category and show it has a similar the same characteristics of a higher categories (such as objects, morphisms, composition, …). Then we use it to define univalence.

Foundations of Higher Categories:

• A Model for the Higher Category of Higher Categories:
  We construct complete Segal spaces which model simplicial spaces, Segal spaces, complete Segal spaces and spaces along with their universal fibrations.
• Cartesian Fibrations and Representability:
  We introduce a general method to construct fibrations that model functors in reflective subcategories of simplicial spaces. We also show how this comes with a model structure that we can understand very well. Using this new method we can define representable Cartesian fibrations.
• Yoneda Lemma for Simplicial Spaces:
  We study the theory of left fibrations over simplicial spaces, by showing that left fibrations are fibrant objects in a model structure. We use that to prove the Yoneda lemma for simplicial spaces.
• An Introduction to Complete Segal Spaces:
  This is a very intuitive introduction to higher category theory via complete Segal spaces. It discusses subjects such as composition, functoriality, adjunctions and colimits.

Applications of Algebraic Topology:

• RGB image-based data analysis via discrete Morse theory and persistent homology with Ruth Davidson, Chuan Du, Rosemary Guzman, Adarsh Manawa and Christopher Szul:
  We use a code developed at Australian National University that can detect fundamental topological features of a grayscale image and enhance it so that it can also analyze RGB images. As a result we can perform data analysis directly on RGB images representing water scarcity variability as well as crime variability.

Notes & Slides:

• Anakyzing RGB Images using Topology with Ruth Davidson, Chuan Du, Rosemary Guzman, Adarsh Manawa and Christopher Szul: In this talk we discuss how to use a code developed at Australian National University to do image analysis with discrete Morse theory. We show how to use the code in two different scenarios: water scarcity and crime data.
• Functoriality in Higher Categories: Cartesian Fibrations: In this talk I show the need to use fibrations when dealing with functors in higher categories. Then I introduce the two most important classes of fibrations, right fibrations and Cartesian fibrations.
• A Theory of Elementary Higher Toposes: In this talk I give a definition of an elementary higher topos and show how it generalizes existing definitions while retaining desired features.
• An Introduction to TFTs: This is a talk I gave in the graduate student homotopy seminar. I introduce the basic notions of topological field theories and show that even simple computations necessitate using higher categorical tools.
• Representable Cartesian Fibrations: Notes from my talk in the conference Homotopy theory: tools and applications at University of Illinois at Urbana Champaign on July 19th, 2017. It focuses on a new way to define Cartesian fibrations and how it enables us to do a lot of new cool things.
• Representable Cartesian Fibrations: These are notes for a talk I gave in the topology seminar at University of Illinois at Urbana Champaign on April 18th, 2017.
  It gives general picture of how the study of representable Cartesian fibrations can help us do topos theory.
• Composition Fibrations: These are notes for a talk I gave in the AMS Sectional Meeting (Special Session on Homotopy Theory) on April 2nd, 2017 in Indiana University.
  I define a class of maps and prove that it preserves categorical equivalences under base change.
• I have been running a higher category theory seminar in Spring 2017 and collected some notes from that seminar.
  Special thanks to William Balderrama, Martino Fassina, Jesse Huang, Aristotelis Panagiotopoulos, Matej Penciak, Joseph Rennie, Brian Shin and Josh Wen for their great talks and careful notes.
• Complete Segal Objects & Univalent Maps: These are notes from a talk I gave in the Workshop on Homotopy Type Theory and Univalent Foundations of Mathematics in the Fields Institute on May 17th, 2016. In this talk I define and discuss internal higher categorical objects. There is also a recording of the talk.
• A New Approach to Straightening: These are my slides for the talk I gave in GSTGC (Graduate Student Geometry Topology Conference) 2016 on April 2nd, 2016
  . I show a method to introduce the unstraightening construction to a larger mathematical audience.
• I took my preliminary exam March 3rd, 2015. Here is my prelim syllabus and the slides of my prelim talk.