Nima Rasekh's Academic Home Page
I am currently a postdoctoral researcher at the Max Planck Institute for Mathematics. I am currently on the academic job market, seeking tenure track positions.
For more information about me check out:
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.
Here is an excellent article by Emily Riehl discussing the rise of higher category theory. Here is another general science article talking about some of the things I like to think about.
I also taught a course on higher categories. You can find the lecture notes here.
Before coming to Bonn I spent three years as a collaborateur scientifique (postdoctoral researcher) at the École Polytechnique Fédérale de Lausanne working in the research group of Kathryn Hess. I was a PhD student at the University of Illinois at Urbana-Champaign, where I worked with my advisor Charles Rezk.
Email: rasekh [at] mpim-bonn.mpg.de
Address:Vivatsgasse 7, 53111 Bonn
Some buzz words: higher category theory, higher topos theory, elementary topos theory, homotopy type theory, topological Hochschild homology, cotangent complex, shadows.
A chronological list of my papers can be found on my ArXiv page or my Google Scholar page. Thematically, my work breaks down into three broad themes:
Foundations of Homotopy Theory:
- Constructing Coproducts in locally Cartesian closed ∞-Categories - Talk Slides - Talk Recording,
Published in Homology, Homotopy and Applications, doi:
joint with Jonas Frey:
We show that locally Cartesian closed ∞-categories with subobject classifier have an initial object and coproducts generalizing a well-known result in elementary topos theory.
- Univalence in Higher Category Theory:
We study univalence in locally Cartesian closed ∞-categories using internal ∞-categories (as complete Segal objects), generalizing a definition by Gepner and Kock in the presentable setting.
- Filter Quotients and Non-Presentable (∞,1)-Toposes - Talk Notes - Slides 1 - Slides 2,
Published in Journal of Pure and Applied Algebra, doi:
We generalize the filter quotient construction from elementary toposes to (∞,1)-categories to construct new non-presentable (∞,1)-toposes.
- Truncations and Blakers-Massey in an Elementary Higher Topos - Talk Notes - Slides 1 - Slides 2 - Slides 3 - 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 - Slides,
Published in Theory and Applications of Categories:
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 elementary 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 (∞,n)-Categories:
- A homotopy coherent nerve for (∞,n)-categories
joint with Lyne Moser and Martina Rovelli:
We construct a homotopy coherent nerve from enriched categories to Segal categories and prove it gives us a Quillen equivalence.
- Yoneda Lemma for D-Simplicial Spaces-Talk Recordings-Talk Slides:
We define Cartesian fibrations for several models of (∞, n)-categories, such as n-fold complete Segal spaces. We study various properties of these fibrations and in particular prove a Grothendieck construction.
- Quasi-Categories vs. Segal Spaces: Cartesian Edition,
Published in Journal of Homotopy and Related Structures, doi:
We generalize the equivalence between quasi-categories and complete Segal spaces due to Joyal and Tierney to an equivalence between the Cartesian fibrations.
- Cartesian Fibrations of Complete Segal Spaces,
To appear in Higher Structures:
We define a notion of Cartesian fibrations for complete Segal spaces using the complete Segal object approach to right fibrations.
- 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,
Published in Homology, Homotopy and Applications, doi:
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.
Homotopy Coherent Hochschild Homology:
- Shadows are Bicategorical Traces-Talk Recordings-Other Talk Recordings-Talk Slides-Other Talk Slides
joint with Kathryn Hess:
We characterize shadows, originally due to Ponto, as enriched THH and use that to give an alternative proof for Morita equivalence.
- The cotangent complex and Thom spectra,
Published in Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, doi
joint with Bruno Stonek:
We use the results in the previous paper to compute the cotangent complex of Thom spectra, generalizing a result by Basterra and Mandell. As part of the work we also compare various definitions of cotangent complexes that can be found in the literature using methods from Goodwillie calculus.
- Thom spectra, higher THH and tensors in ∞-categories-Talk Notes-Slides,
Published in Algebraic & Geometric Topology, doi
joint with Bruno Stonek and Gabriel Valenzuela:
We compute tensor and in particular THH of Thom spectra using tensors of presentable ∞-categories, building on work of Gepner, Groth and Nikolaus and generalizing and streamlining a result by Schlichtkrull.
- 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.