{"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":"2026-04-05T11:33:08","modified_gmt":"2026-04-05T18:33:08","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<\/p>\n<\/div>\n

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Here\u2019s the latest version of my paper The Math Teaching Atlas: Trails, Anchors, and Compass in Action<\/em>.<\/p>\n

Last update: Mar

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Here\u2019s the latest version of my preprint The Dual Burnside Process<\/em>.<\/p>\n

Last update: Nov

View PDF \u2192
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My primary research areas are probability (Markov chain theory) and algebraic combinatorics. Currently, I’m working on the Burnside process and card shuffling with my advisor Prof. Jason Fulman<\/a> and J. E. Paguyo<\/a>.<\/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 the 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

My doctoral advisor is Prof. Jason Fulman<\/a>. In my first 2.75 Ph.D. years, my temporary advisor was Prof. Kyler Siegel<\/a>, and my MGSA<\/a> mentor was Wes Wise<\/a>. My master’s thesis is on zeta functions analyzed mainly from analytic number theory and algebraic K-theory. And my thesis supervisor is\u00a0Prof. Eric M. Friedlander<\/a>, a world-famous K-theorist.\u00a0The thesis is now available for\u00a0view and download\u00a0in the USC Digital Library<\/a>. My thesis committee members were\u00a0Prof. Eric\u00a0<\/span>Friedlander (Chair), Prof. Susan Montgomery<\/a>, and Prof. Sheldon Kamienny<\/a>. <\/span>My undergraduate thesis is on the basic properties of the Riemann zeta function as well as the origin and history of the Riemann Hypothesis, under the supervision of Dr. Yan Zhao.<\/p>\n

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

The Dual Burnside Process<\/a> (10\/29\/2025)<\/p>\n

The Math Teaching Atlas: Trails, Anchors, and Compass in Action<\/a> (04\/06\/2025)
\n[Published in the Journal of Humanistic Mathematics<\/em><\/a> (Volume 16, Issue 1, January 2026)]<\/p>\n

Periodicity Uncovered: A Deep Dive into Bott\u2019s Theorems in K-Theory and Fiber Bundles<\/a> (05\/10\/2023)<\/p>\n

Zeta functions: From Prime Numbers to K-theory<\/a>\u00a0(04\/16\/2022)<\/p>\n

Values of Zeta Functions at Integers<\/a>\u00a0(01\/21\/2022)<\/span><\/p>\n

A Coherent Introduction to Algebraic K-theory<\/a>\u00a0(11\/20\/2021)<\/span><\/p>\n

More papers coming soon…\u00a0\ud83d\ude3a<\/span><\/p>\n

Selected\u00a0<\/strong>coursework notes<\/strong>:<\/p>\n

Math 500: Ivan’s Colloquia\u00a0Notes and Reports<\/a><\/p>\n

Math 520:\u00a0Ivan’s Solutions to Some\u00a0Exercises from\u00a0Ahlfors’s\u00a0Complex Analysis<\/em><\/a><\/p>\n

Math 532: Ivan’s Solutions to Some\u00a0Exercises from\u00a0Heidelberg’s\u00a0A Course in Enumeration<\/em><\/a><\/p>\n

Math 541: Ivan’s Solutions to\u00a0Exercises 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\u00a0math 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)\u00a0[Spring 2021];
\n525a, b (Real Analysis)\u00a0[Fall 2020, Spring 2021];
\n532 (Combinatorial Analysis) [Fall 2022];
\n535a (Differential Geometry) [Spring 2021];
\n540 (Topology) [Fall 2020];
\n541a (Introduction to Mathematical Statistics)\u00a0[Spring 2023];
\n555a, b (Partial Differential Equations) [Fall 2021, Spring 2022];
\n565a (Ordinary Differential Equations) [“Audit”, Spring 2022];
\n580 (Introduction to Functional Analysis) [Fall 2023];
\n590 (Directed Research) [Summer 2024], [Fall 2024, with Prof. Evgeni Dimitrov<\/a>\u00a0and Prof. Julian Chaidez<\/a>], [Summer 2025];
\n594a, b, z (Master\u2019s Thesis)\u00a0[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];
\n655 (Topics in Differential Equations) [Spring 2024];
\n641 (Topics in Topology) [Spring 2023];
\n790 (Research) [Spring 2025, with Prof. Aravind Asok<\/a>];
\n794a,b (Doctoral Dissertation) [Fall 2025], [Spring 2026] (In progress).<\/p>\n

More courses coming soon…\u00a0\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

Weak convergence theorems in the Theory of Probability (2022): Presentation\u00a0Recording<\/a>, Notes<\/a><\/p>\n

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

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

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

\"\"<\/p>\n

Note: This picture was taken at Manhattan Beach, Playa del Rey 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: Feb 17, 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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