**Luciena Xiao (肖霄)
**

MC 253-27

1200 E California Blvd

Pasadena, CA 91106

Office: 302 Linde Hall

I am a fourth-year Ph.D. student in Mathematics at Caltech. My advisor is Elena Mantovan.

My research interests include Shimura varieties, the Langlands program, arithmetic geometry, and post-quantum cryptography.

Here is my CV (updated September 2019).

Abstract: The Hecke orbit conjecture plays an important role in understanding the geometry of Shimura varieties. It predicts that prime-to-p Hecke correspondences on mod p reductions of Shimura varieties characterize the foliation formed by central leaves. In other words, every prime-to-p Hecke orbit is Zariski dense in the central leaf containing it. The full conjecture is known for for Hilbert modular varieties (C.-F. Yu, 2004) and for Siegel modular varieties (Chai and Oort, 2005). In my dissertation, I generalize the method used by Chai and Oort to Newton strata on Shimura varieties of PEL type which contain hypersymmetric points, and prove that every prime-to-p Hecke orbit in the irreducible component of any such Newton stratum is dense in the central leaf containing that orbit.

Number Theory Seminar, UCLA, November 2019

Number Theory Seminar, Caltech, October 2019

Geometry & Arithmetic of Surfaces Workshop, UW-Madison, WI, USA, 2019

*
Algebraic Groups and Geometrization of the Langlands Program,
*ENS Lyon, France, 2018

The 11th annual WiMSoCal Symposium, Pepperdine University, CA, USA, 2018

Arizona Winter School 2020: Nonabelian Chabauty, March 7-11, 2020 at the University of Arizona, AZ, USA

INdAM special research activity in Padova on "p-adic Langlands program", May 12 - July 12, 2019, University of Padua, Italy

AMS Spring Eastern Sectional Meeting, April 13-14, 2019, University of Connecticut Hartford, Hartford, CT, USA

Geometry & Arithmetic of Surfaces Workshop, February 9-10, 2019, UW-Madison, WI, USA

Conference on open questions in cryptography and number theory, Sept 17-21, 2018, UC Irvine, CA, USA

Algebraic Groups and Geometrization of the Langlands Program, April 23 - June 29, 2018, Univ Lyon 1 and ENS Lyon, France

Sino-French conference in Algebraic and Complex Geometry, April 23-27, 2018, Université Lyon 1, France

Anachronism(s) in the History of Mathematics, April 13-14, 2018, Caltech, CA, USA

The 11th annual WiMSoCal Symposium, March 24, 2018, Pepperdine University, CA, USA

Arizona Winter School 2018: Iwasawa Theory, March 3-7, 2018, University of Arizona, AZ, USA

UNCG Summer School: Modular Forms, May 22-26, 2017, UNC at Greensboro, NC, USA

Arizona Winter School 2017: Perfectoid Spaces, March 11-15, 2017, University of Arizona, AZ, USA

Women and Math: Aspects of Algebraic Geometry, May 11-22, 2015, Institute for Advanced Study, NJ, USA

I am a participant of the Certificate of Practice in University Teaching program at Caltech.

Ma 1d Sequence and Series (Instructor): Winter 2019, Winter 2018. Lecture Notes.

Ma 1a Calculus of One and Several Variables and Linear Algebra (Recitation Instructor, Grader): Fall 2017, Fall 2018, Fall 2019

Ma 5/105b: Introduction to Abstract Algebra (Teaching Assistant): Winter 2017

Ma 5/105a: Introduction to Abstract Algebra (Teaching Assistant): Fall 2016

Ma 1c: Calculus of Several Variables (Recitation instructor, Grader): Spring 2016

Ma 1b: Linear Algebra (Recitation instructor, Grader): Winter 2015

Ma 2/102a: Differential Equations (Recitation instructor, Grader): Fall 2015

Introduction to SIKE. This my final project report for CS152 Introduction to Cryptography in Fall 2018. It gives a brief introduction to the Supersingular Isogeny Key Encapsulation suite submitted to the NIST standardization process on post-quantum cryptography by Jao et al.

I'm a certified Mental Health First Aider by Mental Health First Aid USA. Feel free to reach out to me if you are experiencing a mental health or substance use challenge. I'm not a therapist, but I can provide initial help until appropriate professional resources are available.

During the last two years of my undergraduate studies, I translated
Brian P. Copenhaver's English translation of the *Hermetica* into
Chinese.
My translation was published in July 2019 as a volume in the "经典与解释"(Hermes:
Classici et commentarii) book series.

Here's a commentary (in Chinese) I wrote for the first ever Chinese translation of the prose Edda, where I give some background on the Eddas and explain Old Norse poetic forms via two examples.