Luciena Xiao (肖霄)

Department of Mathematics
MC 253-27
1200 E California Blvd
Pasadena, CA 91106
Office: 302 Linde Hall

E-mail:  xxiao at caltech dot edu

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).


Papers

On the Hecke Orbit Conjecture for PEL type Shimura varieties. In preparation. Draft available by request.

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.

 

Conferences

Invited Talks:

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

I will be at the following conferences:

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

Conferences I attended:

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


Teaching

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


Miscellaneous

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.