{"id":208,"date":"2023-09-30T16:17:51","date_gmt":"2023-09-30T23:17:51","guid":{"rendered":"https:\/\/live-usc-dornsife.pantheonsite.io\/zhanhu\/?page_id=208"},"modified":"2026-09-03T13:45:22","modified_gmt":"2026-09-03T20:45:22","slug":"research","status":"publish","type":"page","link":"https:\/\/dornsife.usc.edu\/ivan\/research\/","title":{"rendered":"Research"},"content":{"rendered":"\n\n \n \n\n\n\n\n\n\n\n
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\u201cLike any science or art, mathematics is not about how much you know about existing knowledge discovered by other mathematicians; it\u2019s about how much you can contribute to that field. And the only way to contribute to that field is by boldly breaking existing rules that other people take for granted. As long as you don’t give up, there is hope, so you must believe in yourself. Always be ambitious, and think outside the box. That\u2019s what made you, you; and that\u2019s what\u2019s going to keep you, you.\u201d<\/p>\n

\u2014 Ivan Z. Feng<\/span><\/p>\n<\/div>\n

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Publication
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My paper The Dual Burnside Process<\/em> is now published open access in the Journal of Theoretical Probability<\/em>.<\/p>\n

Published:

Read the article \u2192
\n<\/a><\/p>\n<\/div>\n<\/div>\n

My primary research areas are probability (Markov chain theory) and algebraic combinatorics. Currently, I’m working on Burnside processes, mixing times, and card shuffling with my advisor, Prof. Jason Fulman<\/a>. I am also collaborating with J. E. Paguyo<\/a> on a twisted Bose-Einstein Markov chain and its applications to sampling Catalan structures.<\/p>\n

I also like analytic number theory and algebraic geometry. In particular, I’m interested in the Riemann zeta function<\/a> and its related topics, which basically include contour integration and infinite products in complex analysis, L-functions and the prime number theorem in analytic number theory, the Lichtenbaum conjectures in algebraic K-theory, and the Weil conjectures in \u00e9tale cohomology.<\/p>\n

Selected papers<\/strong>:<\/p>\n

The Dual Burnside Process<\/a> [Journal of Theoretical Probability<\/em> 39<\/strong>, article 78 (2026); published 8\/27\/2026; arXiv<\/a>]<\/p>\n

Analysis of a twisted Bose-Einstein Markov chain with applications to sampling Catalan structures<\/a> (joint work with J. E. Paguyo; 5\/15\/2026, 8\/22\/2026)<\/p>\n

Explicit cutoff profiles for colored top-m-to-random shuffles<\/a> (4\/10\/2026, 5\/27\/2026; companion notes and special examples<\/a>)<\/p>\n

Cutoff profiles for colored top-m-to-random shuffles with growing block size<\/a> (6\/28\/2026)<\/p>\n

The Math Teaching Atlas: Trails, Anchors, and Compass in Action<\/a> [Published in the Journal of Humanistic Mathematics<\/em>, Volume 16, Issue 1, 344\u2013364 (January 2026); expanded arXiv version<\/a>]<\/p>\n

Periodicity Uncovered: A Deep Dive into Bott\u2019s Theorems in K-Theory and Fiber Bundles<\/a> (5\/10\/2023)<\/p>\n

Zeta Functions: From Prime Numbers to K-Theory<\/a> (Master’s thesis, 5\/2022)<\/p>\n

Values of Zeta Functions at Integers<\/a> (1\/21\/2022)<\/span><\/p>\n

A Coherent Introduction to Algebraic K-Theory<\/a> (11\/20\/2021)<\/span><\/p>\n

Ivan’s preprints on arXiv<\/a><\/p>\n

More papers coming soon… \ud83d\ude3a<\/span><\/p>\n

Selected talks and presentations<\/strong>:<\/p>\n

Colored top-m-to-random shuffles (2026): Lightning talk at the 4th Simons Math Summer Workshop: Algebraic Methods in Probability<\/a> (July 2026)<\/p>\n

The Magic of Colored Top-m-to-Random Shuffles (2026), USC Graduate Student Colloquium: Presentation Recording<\/a>, Companion Notes<\/a><\/p>\n

The Math Teaching Atlas (2026): USC Math Education Seminar, Great Teaching Leaves Trails<\/em> (March 23), and MAA MathFest poster, Trails, Anchors, and Compass in Action<\/em> (August 7): Presentation Recording<\/a>, Paper<\/a>, Poster<\/a>, Flyer<\/a>, Handout<\/a>, Notes<\/a><\/p>\n

Zeta: Primes, Zeros, and Beyond<\/a> (March 25, 2025): Presentation Recording<\/a>, Flyer<\/a>, Handout<\/a>, Notes<\/a><\/p>\n

Exploring Leray\u2019s Formulation of the Navier-Stokes Equations (2024): Presentation Recording<\/a>, Notes<\/a><\/p>\n

Weak convergence theorems in the Theory of Probability (2022): Presentation Recording<\/a>, Notes<\/a><\/p>\n

Selected coursework notes<\/strong>:<\/p>\n

Math 500: Ivan’s Colloquia Notes and Reports<\/a><\/p>\n

Math 520: Ivan’s Solutions to Some Exercises from Ahlfors’s Complex Analysis<\/em><\/a><\/p>\n

Math 532: Ivan’s Solutions to Some Exercises from Aigner’s A Course in Enumeration<\/em><\/a><\/p>\n

Math 541: Ivan’s Solutions to Exercises in Heilman’s Lecture Notes<\/a><\/p>\n

Math 595: Ivan’s Thoughts on Teaching<\/a><\/p>\n

Math 610: Ivan’s Selected Homework Solutions<\/a><\/p>\n

Math 641: Ivan’s Solutions to Selected Homework Problems<\/a><\/p>\n

Graduate math courses<\/a> taken at USC<\/strong>:<\/p>\n

Math 500 (Graduate Colloquium) [Spring 2023];
\n505a (Applied Probability) [Fall 2024];
\n507a, b (Theory of Probability) [Fall 2022, Spring 2024];
\n510a, b (Algebra) [Fall 2020, Spring 2021];
\n514 (Algebraic Geometry) [Spring 2025];
\n520 (Complex Analysis) [Spring 2021];
\n525a, b (Real Analysis) [Fall 2020, Spring 2021];
\n532 (Combinatorial Analysis) [Fall 2022];
\n535a (Differential Geometry) [Spring 2021];
\n540 (Topology) [Fall 2020];
\n541a (Introduction to Mathematical Statistics) [Spring 2023];
\n555a, b (Partial Differential Equations) [Fall 2021, Spring 2022];
\n565a (Ordinary Differential Equations) [audited, Spring 2022];
\n580 (Introduction to Functional Analysis) [Fall 2023];
\n590 (Directed Research) [Summer 2024], [Fall 2024, with Prof. Evgeni Dimitrov<\/a> and Prof. Julian Chaidez<\/a>], [Summer 2025];
\n594a, b, z (Master\u2019s Thesis) [Fall 2021, Spring 2022];
\n595 (Practicum in Teaching the Liberal Arts: Mathematics) [Fall 2022];
\n599 (Special Topics) [Fall 2025];
\n605 (Topics in Probability) [Fall 2023];
\n610 (Topics in Algebra) [Summer 2023];
\n633 (Topics in Combinatorics; topic: Combinatorics and Markov chains) [Fall 2026, in progress];
\n641 (Topics in Topology) [Spring 2023];
\n655 (Topics in Differential Equations) [Spring 2024];
\n790 (Research) [Spring 2025, with Prof. Aravind Asok<\/a>];
\n794a, b, c (Doctoral Dissertation) [Fall 2025], [Spring 2026], [Fall 2026, in progress].<\/p>\n

More courses coming soon… \ud83d\ude05<\/p>\n

Other fun work<\/strong>:<\/p>\n

Two Methods (Topology & Geometry) to Prove the Hairy Ball Theorem<\/a><\/p>\n

An Easy Introduction to the Motivic Zeta Function for A^l and P^l<\/a><\/p>\n

An interesting extension of a basic calculus problem<\/a><\/p>\n

DRP Notes<\/a> (Mathematical Statistics with Applications<\/em>, 7th edition)<\/p>\n

\"A<\/p>\n

Note: This picture was taken along the Los Angeles coast in August 2022. I’ve placed it here because the view is incredibly beautiful, with a hint of mysterious and surreal atmosphere. During that trip, I was so captivated by the seaside scenery that I ended up writing two poems titled “Seashore Duet” (which can be found on the 2022 poetry page<\/a>) to immortalize my memories. In August 2023, exactly one year later and just before the semester started, I revisited the same place since I was feeling nostalgic. I left another poem there, titled “Coastal Symphony” (which can be found on the 2023 poetry page<\/a>).<\/em><\/p>\n\n\n\n<\/div>\n\n\n <\/div><\/div>\n\n\n\n\n \n \n\n\n\n\n\n\n\n

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Last update: Sep 3, 2026<\/em><\/p>\n

Back to my homepage<\/em><\/a><\/p>\n\n\n\n<\/div>\n\n\n <\/div><\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":390,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"footnotes":""},"class_list":["post-208","page","type-page","status-publish","hentry"],"acf":[],"yoast_head":"\nResearch - Ivan Z. Feng<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/dornsife.usc.edu\/ivan\/research\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Research - Ivan Z. Feng\" \/>\n<meta property=\"og:url\" content=\"https:\/\/dornsife.usc.edu\/ivan\/research\/\" \/>\n<meta property=\"og:site_name\" content=\"Ivan Z. 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