Daniel Kriz

Princeton University

Daniel Kriz

I graduated in 2018 from Princeton University.

My advisers were Christopher Skinner and Shouwu Zhang. I have now moved to MIT. 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. Most recently 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 have also established special value formulas for these p-adic L-functions and explored their arithmetic applications. Previously, 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). I have also done work in geometric topology regarding the Heegaard-Floer and Khovanov homologies of knot theory.

Research

Here are some of my recent papers.

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

Prime twists of elliptic curves (with Chao Li), to appear in Mathematical Research Letters.

Goldfeld's conjecture and congruences between Heegner points (with Chao Li), accepted to Forum of Mathematics, Sigma.

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.

Email

dkriz@mit.edu

This page last modified on January 10, 2019.