Nima Rasekh's Home Page
I am a graduate student in my last year at the
University of Illinois at UrbanaChampaign.
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,
which I have been working on with my advisor Charles Rezk.
You can find out more about my research in my research statement.
Here is my current CV.
You can also find out more about me in the recent issue of the
Math Times.
Before I came to Illinois, I was an undergrad at
Shiraz University in the city of
Shiraz, Iran. I also was a
master student at the University of Western Ontario in Canada.
Information about the 20172018 TA award can be found here.
If you would like to know further about my work then feel free to email me under rasekh2 [at] illinois.edu .
Overview
Email: rasekh2 [at] illinois.edu
Office: 113 Altgeld Hall
Office Phone: (217) 3009542
Some buzz words: Higher Category Theory, Higher Topos Theory,
Elementary Topos Theory, Homotopy Type Theory, Simplicial Homotopy Theory
More concretely I have thinking about developing a theory of a elementary higher topos.
As is customary in higher category theory, in order to be able to formulate my ideas I first ventured
into the realm of foundations.
Collectively, I ended up working on the following projects in order to get to a theory of
elementary higher topoi:
 Defining an elementary higher topos
by generalizing the notion of an object classifier
using representable Cartesian fibrations.
 A theory of complete Segal objects
and how they generalize the theory of complete Segal spaces. In particular, how we can use
representable Cartesian fibrations to talk about categorical constructions for complete Segal
objects.
 A new characterization of Cartesian fibrations
with the end goal of defining and studying
representable Cartesian Fibrations.
 How complete Segal spaces model higher categories and how some definitions and
theorems look like if you look at them from the
perspective of complete Segal spaces.

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.

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.

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.
 RGB imagebased 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.
 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.

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.

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.
I have been a Teaching Assisstant in UIUC for the following courses:
Course website for Math 414: Mathematical Logic (Spring 2017)
Here are links to seminars I have organized.
Here are links to the homepage of some people whose work I know or like:
Matt Ando,
Nerses Aramian,
David Gepner,
Zhen Huan,
Chris Kapulkin,
Joachim Kock,
Jacob Lurie,
Aaron MazelGee,
Randy McCarthy,
Eric Peterson,
Charles Rezk,
Emily Riehl,
Mike Shulman,
Sarah Yeakel
Here are some pages about topology:
List of topologists (by Andrew Baker)
Some links for different conferences and seminars related to homotopy theory:
A list of topology seminars (by Niles Johnson)
Another list of topology seminars (by Sarah Whitehouse)
The Midwest Topology Seminar
The Talbot Workshop
Graduate Student Topology and Geometry Conference
The UIUC Topology Seminar
The UIUC Graduate Student Topology and Geometry Seminar
The UIUC Graduate Student Homotopy Seminar
Last modified 5 January 2018 by
Nima Rasekh.
Email:
rasekh2 [at] illinois.edu