Peter Haine

Seminars

Below are some seminars that I have helped organize:

1. Juvitop seminar on Differential Cohomology, Fall 2019
• with Araminta Amabel
2. Miniature seminar on Factorization Homology, Spring 2019
• with Araminta Amabel
3. Juvitop seminar on Ambidexterity, Fall 2018
• with Hood Chatham & Andy Senger
4. MIT Student Colloquium for Undergraduates in Mathematics, 2015– 2016
• with Ahaan Rungta, Sandeep Silwal, Soohyun Park, & Victor Wang

Service

I am involved in a number of service activities and am interested in problems around diversity in mathematics and higher education in general.

1. Since Fall 2018, I have been the coordinator for PRIMES Circle at MIT
2. January 2017 & January 2018, I was a mentor for the MIT Math Department's Directed Reading Program
3. Fall 2017, I was the organizer for the MIT Topology Seminar
4. Summer 2017, I was a mentor for MIT's √Mathroots program
5. Since Spring 2016, I have been a member of the MIT Math Department's Diversity and Community Building Committee

Expository Writing

Books

1. Differential forms, with Victor Guillemin
• World Scientific Publishing Company, March 2019, ISBN: 9789813272774. DOI: 10.1142/11058.

Expository Papers & Notes

1. The lci locus of the Hilbert scheme of points & the cotangent complex (last updated September 2019)
• Notes for a talk at the Fall 2019 Thursday Seminar.
2. On coherent topoi & coherent 1-localic ∞-topoi (last updated July 2019)
• arXiv:1904.01877
• Fills a small gap in the literature by proving that the ∞-categories of coherent ordinary topoi (in the sense of SGA4) and coherent 1-localic ∞-topoi (in Lurie's sense) are equivalent. We also collect some examples of coherent geometric morphisms coming from algebraic geometry.
3. Nonabelian Poincaré Duality (last updated March 2019)
• Some informal notes sketching Ayala and Francis' proof of Nonabelian Poincaré Duality.
4. Ambidexterity §4 (last updated November 2018)
• Notes on §4 of Hopkins and Lurie's paper Ambidexterity in K(n)-local stable homotopy theory phrased for those not familiar with the fibrational perspective.
5. Notes on Étale Cohomology (last updated October 2017)
• Notes on the fundamental theorems of étale cohomology following Chapter VI of Milne's Étale Cohomology.
6. An Introduction to Goodwillie Calculus (last updated September 2017)
• Notes on the basic setup of Goodwillie calculus in the ∞-categorical setting.
7. An Overview of Motivic Cohomology (last updated September 2017)
• Notes for a talk at the MIT graduate student lunch seminar giving a very broad overview of motivic cohomology.
8. On the K-theory of Finite Fields (last updated September 2016)
• Notes for a talk given at the 2016 Kan Seminar.
9. Lifting Enhanced Factorization Systems to Functor 2-categories (last updated July 2016)
• arXiv:1604.06812
• A note on a technical result about defining enhanced factorization systems on functor 2-categories.

Research

1. Pyknotic objects, I. Basic Notions, with Clark Barwick (last updated April 2019)
• arXiv:1904.09966
2. Exodromy for stacks, with Clark Barwick (last updated January 2019)
• arXiv:1901.09414
3. Extended étale homotopy groups from profinite Galois categories (last updated January 2019)
• arXiv:1812.11637
4. On the homotopy theory of stratified spaces (last updated September 2019)
• arXiv:1811.01119
5. Exodromy, with Clark Barwick & Saul Glasman (last updated July 2019)
• arXiv:1807.03281
6. Stability analysis of non-Newtonian rimming flow, with Vladimir Chugunov, Sergei Fomin, & Ravi Shankar (February 2016)
• Applied Mathematical Modelling, Volume 40, Issue 4, 15 February 2016, 2999–3010.
• arXiv:1509.05102

About

I was an undergraduate at MIT and now am a third year PhD candidate in mathematics at MIT working with Clark Barwick. I plan to graduate in June 2021. My mathematical interests center around homotopy theory, algebraic K-theory, and algebraic geometry. Recently I've been applying tools from homotopy theory to solve problems in algebraic and arithmetic geometry.

Upcoming Travel

1. December 2, 2019 I will speak at the Johns Hopkins Topology Seminar.
2. Spring 2020 I will be a Program Associate for the MSRI program on Higher Categories and Categorification.

Elements

i = 0;

while (!deck.isInOrder()) {
    print 'Iteration ' + i;
    deck.shuffle();
    i++;
}

print 'It took ' + i + ' iterations to sort the deck.';