Jen Berg

About

I am currently in my last year as a RTG Lovett Instructor (postdoc) at Rice University. I received my PhD in 2016 at the University of Texas at Austin and my B.S. at UIUC in 2010. I grew up in Chicago, IL. These days, my main interests outside of math include cooking/baking, painting, attending concerts, reading, playing board games, various additional arts & crafts projects, and bouldering.

Research

My research focuses on problems in arithmetic geometry and algebraic number theory. In particular, I often work on questions related to failures of the Hasse principle. Many of my current and recent projects concern obstructions to integral and rational points on surfaces (e.g. K3 surfaces) and higher dimensional varieties. My thesis advisor was Felipe Voloch. My postdoc mentor is Tony Várilly-Alvarado.

Upcoming and Recent Travel

(* denotes invited speaker)
  • April 6-7, 2019. AWM Research Symposium: WIN Special Session Rice University, Houston, TX.*
  • March 22-24, 2019 AMS Special Session: Arithmetic Geometry and Its Connections University of Hawaii, Honolulu, HI.*
  • January 9, 2019. AMS Special Session: Number Theory, Arithmetic Geometry, and Computation JMM, Baltimore, MD.*
  • November 2-4, 2018. Pop-up Conference in Number Theory UIC, Chicago, IL*
  • October 23, 2018. Algebra Seminar University of Oregon, Eugene, OR.*
  • September 11, 2018. IDA CCR-Princeton Colloquium Princeton, NJ.*
  • August 20-24, 2018. Arithmetic Geometry, Number Theory, and Computation MIT, Cambridge, MA.
  • July 2-6, 2018. Rational Points on Schiermonnikoog Schiermonnikoog, Netherlands. (group photo)
  • June 18-22, 2018. The 13th Brauer group conference Pingree Park, Colorado.* (group photo)
  • May 27 - June 1, 2018. Rational and Integral Points Via Analytic and Geometric Methods CMO, Oaxaca, Mexico.* (talk video) (group photo)
  • May 14 - 18, 2018. Birational Geometry and Arithmetic ICERM, Providence, RI.
  • April 27, 2018. Geometry Seminar Texas A&M, College Station, TX.*
  • April 17, 2018. Algebraic Geometry and Number Theory Seminar Rice University, Houston, TX.*
  • April 6-8, 2018. TAGS Texas A&M, College Station, TX.
  • April 3, 2018. Algebra and Number Theory Seminar Emory University, Atlanta, GA.*

Preprints

  1. Odd order obstructions to the Hasse principle on general K3 surfaces (with Tony Várilly-Alvarado). arXiv: 1808.00879
  2. Abstract: We show that odd order transcendental elements of the Brauer group of a K3 surface can obstruct the Hasse principle. We exhibit a general K3 surface Y of degree 2 over the rationals together with a three torsion Brauer class A that is unramified at all primes except for 3, but ramifies at all 3-adic points of Y . Motivated by Hodge theory, the pair (Y, A) is constructed from a cubic fourfold X of discriminant 18 birational to a fibration into sextic del Pezzo surfaces over the projective plane. Notably, our construction does not rely on the presence of a central simple algebra representative for A. Instead, we prove that a sufficient condition for such a Brauer class to obstruct the Hasse principle is 3-adic insolubility of the fourfold X (and hence the fibers) and local solubility at all other primes.

  3. Obstructions to integral points on affine Chatelet surfaces arXiv: 1710.07969
  4. Abstract: We consider the Brauer-Manin obstruction to the existence of integral points on affine surfaces defined by x^2−ay^2=P(t) over a number field. We enumerate the possibilities for the Brauer groups of certain families of such surfaces, and show that in contrast to their smooth compactifications, the Brauer groups of these affine varieties need not be generated by cyclic (e.g. quaternion) algebras. Concrete examples are given of such surfaces over the rationals which have solutions in the p-adic integers for all p and solutions in the rationals, but for which the failure of the integral Hasse principle cannot be explained by a Brauer-Manin obstruction. The methods of this paper build on the ideas of several recent papers in the literature: we study the Brauer groups (modulo constant algebras) of affine Chatelet surfaces with a focus on explicitly representing, by central simple algebras over the function field of X, a finite set of classes of the Brauer group which generate the quotient. Notably, we develop techniques for constructing explicit representatives of non-cyclic Brauer classes on affine surfaces, as well as provide an effective algorithm for the computation of local invariants of these Brauer classes via effective lifting of cocycles in Galois cohomology.

Publications

  1. Insufficiency of the Brauer-Manin Obstruction for Rational Points on Enriques Surfaces, with F. Balestrieri, M. Manes, J. Park, and B. Viray. Directions in Number Theory, September, 2016.
  2. Congruences for Ramanujan's f and Omega Functions Via Generalized Borcherds Products,with A. Castillo, R. Grizzard, V. Kala, R. Moy, C. Wang. The Ramanujan Journal, August 2013.
  3. p-groups Have Unbounded Realization Multiplicity, with Andrew Schultz. Proceedings of the American Mathematical Society, August 2012.

In Progress

  1. Conic Bundles over Elliptic Curves (joint with Masahiro Nakahara)

Conferences Organized

  • RTG Lectures in Arithmetic Geometry at Rice (with Anastassia Etropolski and Anthony Várilly-Alvarado)

Contact

  • Office: Herman Brown Hall (HBH) 408
  • Office Hours: W 4-5:30, Th 3-4, or by appointment
  • Email: jb93 [at] rice [dot] edu

Teaching

Upcoming Semester

  • Math 354 (Honors Linear Algebra)
  • Math 499

At Rice

  • Spring 2018: Math 464/564 (Abstract Algebra III), Math 499
  • Fall 2017: Math 111, Math 499
  • Spring 2017: Math 306, Math 499
  • Fall 2016: Math 101, Math 499
P.M.&Y. students learning RSA encryption.
Group photo from Patterns, Math, & You

In the summer of 2018 I co-led a 2 week intensive program for middle school students entitled Patterns, Math, and You at Rice University.

In the summer of 2012 I worked as a teaching assistant at the Summer Program for Women in Math (SPWM) at George Washington University.

At UT Austin

  • Fall 2015: Learning Assistant and Calc Lab coordinator.
  • Spring 2014: Won Natural Sciences Council TA Award!
  • Spring 2014: Graded Prelim Algebra M380D for Felipe Voloch.
  • Fall 2013: SI for M408N for Prof. Anna Spice.
  • Spring 2013: M408C for Prof. John Dollard
  • Fall 2012: M305G for Dr. Amanda Hager, Graded for M373K for Sean Keel
  • Spring 2012: M408D for Prof. Ray Heitmann
  • Fall 2011: M408C for Prof. Kathy Davis
  • Spring 2011: On departmental fellowship.
  • Fall 2010: M408K for Dr. Elif Seckin

Photos

Below are some photos from my two main hobbies: baking and art.