Daniel Kriz

MIT

I am an NSF postdoctoral fellow at MIT. I graduated in 2018 from Princeton University.

My advisers were Christopher Skinner and Shouwu Zhang. Previously, I was also an undergraduate at Princeton, and graduated in 2014.

In the Fall of 2016, I visited the I.H.E.S..

Mathematical Interests

I am interested in algebraic number theory, particularly in arithmetic geometry and Iwasawa theory. My work concerns mainly the study of p-adic objects such as Shimura varieties over p-adic rings, p-adic modular forms, p-adic L-functions and Iwasawa theory. Recently, I have developed the Iwasawa-theoretic aspects of the superinsgular p-adic L-functions that I previously constructed by formulating and proving Iwasawa main conjectures for them. The formulation and proofs of these main conjectures rely on techniques exploiting new interplays between Iwasawa theory and p-adic Hodge theory. In particular these main conjectures have applications to the arithmetic of elliptic curves of classical interest, including the cubic twists and congruent number families. As a corollary, I prove the 1879 conjecture of Sylvester, proving that for any prime p = 4,7,8 mod 9, x^3 + y^3 = p has a solution with x,y rational numbers. Moreover, combined with previous Selmer distribution results, I show that 100% of square free d = 5,6,7 mod 8 are congruent numbers, and hence establish Goldfeld's conjecture for the congruent number family y^2 = x^3 -d^2x and solve the congruent number problem in 100% of cases.

Previously, I have constructed supersingular Rankin-Selberg p-adic L-functions for imaginary quadratic fields in the style of Katz, Bertolini-Darmon-Prasanna and Liu-Zhang-Zhang, resolving questions about the existence of such p-adic L-functions dating back to the 70s. I also established special value formulas for these p-adic L-functions and explored their arithmetic applications. Before that, I did work with Chao Li on establishing the rank and p-parts of the Birch and Swinnerton-Dyer conjecture for quadratic twist families of elliptic curves over Q, in particular showing that a positive proportion of quadratic twists satisfy rank BSD whenever the curve has a rational 3-isogeny (thus verifying a weak version of Goldfeld's conjecture for such curves). We also establish similar results for the Mordell sextic twists family y^2 = x^3 + k, showing a positive proportion have rank 0 (resp. 1).

In the past, I also did work in geometric topology regarding the Heegaard-Floer and Khovanov homologies of knot theory.

Research

Here are some of my recent papers.

Supersingular main conjectures, Sylvester's conjecture and Goldfeld's conjecture.

A New p-adic Maass-Shimura operator and supersingular Rankin-Selberg p-adic L-functions.

Prime twists of elliptic curves (with Chao Li), Mathematical Research Letters, vol. 26 no. 4.

Goldfeld's conjecture and congruences between Heegner points (with Chao Li), Forum of Mathematics, Sigma, Volume 7, 2019, e15.

A Galois cohomological proof of Gross's factorization theorem, submitted.

Generalized Heegner cycles at Eisenstein primes and the Katz p-adic L-function, Algebra and Number Theory vol. 10, no 2, 2016, pp. 309-374. Arxiv (minor differences from the published version)

On a conjecture concerning the maximal cross number of unique factorization indexed sequences, appeared in J. Number Theory, vol 133, 9 (September 2013), 3033-3056. This paper is a result of the research I did as a participant in the Duluth REU, run by Professor Joe Gallian at the University of Minnesota Duluth.

A spanning tree cohomology theory for links (with Igor Kriz), appeared in Advances in Mathematics, vol 255, 1 (April 2014), 414-454. Arxiv

Field theories, stable homotopy theory, and Khovanov homology (with Po Hu, Igor Kriz), Topology Proceedings, vol 48, 2016, pp. 327-360. Arxiv

Expository

Here is an introductory article to Étale cohomology, written for the final project of my algebraic geometry class with Nick Katz in Spring 2012.

An Excursion into Étale Cohomology.

Seminars

In Spring 2020, I am organizing a seminar on Recent Advances in Iwasawa Theory and the Arithmetic of Elliptic Curves, which will go over my recent work on supersingular main conjectures for imaginary quadratic fields, and the proofs of Sylvester's conjecture and Goldfeld's conjecture for the congruent number family.

In Spring 2020, I am also co-organizing the MIT STAGE seminar. The topic of this semester will be p-adic modular forms.

Music

I have played piano since I was 5 years old and violin since I was 10. I also compose music, usually for solo piano, and sometimes for ensembles. I find great inspiration in playing, performing and composing. Below you will find pdfs (and perhaps recordings) of some of my compositions.

Snowfall, for solo piano.

Pan's March, for piano and voice (and chorus).

The Grind, for voice and rock band.

Lullaby, a simple piece for solo piano.

Email

dkriz@mit.edu

This page last modified on January 29, 2020.