Nima Rasekh's Academic Home Page

I am a postdoctoral researcher at the Max Planck Institute for Mathematics. I mostly think about homotopy theory, particularly if it includes some category theory. Concretely I am interested in foundations of higher category theory and higher topos theory. Before that I was a PhD student at the University of Illinois at Urbana-Champaign, where I worked with my advisor Charles Rezk.

You can find out more about my research in my research statement. Here is my current CV.

If you would like to know further about my work then feel free to email me under rasekh [at] mpim-bonn.mpg.de .

Research Interests

Some buzz words: Higher Category Theory, Higher Topos Theory, Elementary Topos Theory, Homotopy Type Theory
Concretely I developed a theory of elementary higher toposes and am in process of studying topological and categorical properties.

Publications & Preprints & Notes

Elementary Higher Toposes:

• Truncations and Blakers-Massey in an Elementary Higher Topos - Talk Notes - Slides - Other Slides - Talk Recording:
  We prove that every elementary higher topos has a universal truncation functor using the join construction. Moreover, we show it satisfies the Blakers-Massey theorem.
• Yoneda Lemma for Elementary Higher Toposes:
  We prove that every object in an elementary higher topos embeds into its ``object of maps''.
  This corresponds to the Yoneda lemma for spaces.
• Every Elementary Higher Topos has a Natural Number Object - Talk Notes:
  We compute the loop space of the circle and use that to show that every elementary higher topos as a natural number object and countable limits and colimits.
• A Theory of Elementary Higher Toposes - Talk Notes:
  We define an elementary higher topos and show it generalizes elemenary toposes and higher toposes.
• Complete Segal Objects - Talk Notes - Talk Recording (from Workshop in the Fields Institute):
  We define an internal version of a higher category and show it has the same characteristics as a higher category (such as objects, morphisms, composition, ...). Then we use it to define univalence.
  

Theory 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.
• 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.
• Cartesian Fibrations and Representability - Talk Notes 1 - Talk Notes 2 - Talk Notes 3:
  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.

Other Work:

• 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.
• Analyzing 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.
• 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 ANU 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.
• 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.
• 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.