# Mathematics PhD dissertations that opened a new field of research

I propose this as a companion wiki page to the one about PhD dissertations which contain a solution to an open problem in the style of big-list questions, thinking in terms of the well-known paradigm that splits mathematical research into problem solving and theory building. Theories are at times developed to solve famous open problems, but sometimes the concrete problems they solve are quickly dwarfed by the possibilities that a new theory opens.

Can you name modern mathematicians who already in their PhD theses (or earlier in their career) developed a substantial new theory or laid the foundations of a new field of research?

• The list will be too long. – Alexandre Eremenko Apr 24 '18 at 12:11
• en.wikipedia.org/wiki/Tate%27s_thesis – Steve Huntsman Apr 24 '18 at 12:14
• en.wikipedia.org/wiki/… – Steve Huntsman Apr 24 '18 at 12:15
• Perhaps Scholze's Perfectoid spaces? – TKe Apr 24 '18 at 14:25
• @Steve Huntsman: Shannon's thesis does not qualify: it was a master thesis:-) – Alexandre Eremenko Apr 25 '18 at 2:52

John Forbes Nash Jr. got a Nobel Prize for his.

Nash earned a Ph.D. degree in 1950 with a 28-page dissertation on non-cooperative games.

The thesis, written under the supervision of doctoral advisor Albert W. Tucker, contained the definition and properties of the Nash equilibrium, a crucial concept in non-cooperative games. It won Nash the Nobel Memorial Prize in Economic Sciences in 1994.

There are many examples. Here are a few that come to mind:

Simon Donaldson's thesis The Yang-Mills equations on Kahler manifolds contains the first major steps in his work on the differential topology of four manifolds. The following paraphrases its abstract. He gave a new proof of a theorem of Narasimhan and Seshadri characterizing those holomorphic bundles over a projective curve that admit a flat connection and used it to prove the simplest interesting case of the conjecture of Hitchin and Kobayashi. He studied the moduli space of self-dual connections on a simply-connected four manifold and used it to deduce obstructions to the realization of a matrix as the intersection pairing on the second cohomology of such a manifold.

John Tate's thesis is another well known example, although I'm not competent even to summarize it. It has its own wikipedia page.

Mikio Sato's doctoral thesis (based on some already published work) introduced the theory of hyperfunctions as boundary values of holomorphic functions. See this survey by P. Schapira and this interview with Sato. (Nothing about Sato's education is standard.)

• I read that article my Mikio Sato. I thought it very interesting. Too bad most famous mathematicians seem to be as unforthcoming as Sato in describing their formative influences. – Mozibur Ullah May 3 '18 at 4:51

I’ll pick Philippe Delsarte’s 1973 thesis “An algebraic approach to the association schemes of coding theory” which basically expressed classical extremal problems in designs and codes as algebraic questions involving eigenspaces of related association schemes.

Here is a link to a talk on what is now known as “Delsarte Theory”.

Maybe not up to the Nash standard, but pretty good for a PhD!

Thomas G. Kurtz' PhD dissertation at Stanford University (1967) was titled Convergence of operator semigroups with applications to Markov processes. He went on to write with his PhD student Stewart N. Ethier the book Markov Processes: Characterization and Convergence (John Wiley & Sons Inc., 1986), which is "the standard reference for the advanced theory of Markov processes". Its first chapter is Operator semigroups. He made a stellar research career on these foundations. Much of the modern theory of stochastic processes is variations of what he pioneered: "establishing the convergence of Markov processes and characterising the limiting process" (quotes from the Wikipedia page). I do not claim that he single-handedly created the area, but his contribution is immense.

There has been a similar question here: https://www.quora.com/Which-are-the-best-PhD-theses-ever-in-pure-mathematics

I would suggest Kurt Godel. His doctorate thesis proved the completeness theorem, and a year later he published his incompleteness theorems.