Reading and Research

My research interests include analytic number theory, sieve methods and their applications, and number theory in function fields. The title of my thesis is "Distribution of Frobenius traces for Drinfeld modules using analytic methods". My thesis advisor is Prof. A. C. Cojocaru.

Papers

• A. Castillo, Distribution of the trace of Frobenius on average for rank 2 Drinfeld modules, submitted (2015), 21 pages.
• A. Castillo, The normal order of the divisor-counting function for rank 2 Drinfeld modules with nontrivial endomorphism ring, submitted (2015), 18 pages.
• A. Castillo and R. Dietmann, On Hilbert’s Irreducibility Theorem, submitted (2015), 11 pages.
• A. Castillo, C. Hall, R. J. Lemke Oliver, P. Pollack, and L. Thompson, Bounded gaps between primes in number fields and function fields, Proc. Amer. Math. Soc. 143 (2015), no. 7, 2841–2856.
• J. Berg, A. Castillo, B. Grizzard, V. Kala, R. Moy, and C. Wang, Congruences for Ramanujan's f and ω functions via generalized Borcherds products, The Ramanujan Journal, November 2014, Volume 35, Issue 2, 327-338.

Seminar and Conference Talks

• "Distribution of the trace of Frobenius for Drinfeld modules", Illinois Number Theory Conference, University of Illinois at Urbana-Champaign, August 2015
• "Distribution of the trace of Frobenius for Drinfeld modules", Elementary, analytic, and algorithmic number theory: Conference in honor of Carl Pomerance’s 70th birthday, University of Georgia in Athens, June 2015
• "Bounded gaps between primes in number fields and function fields", Upstate Number Theory Conference, Cornell University, April 2015
• "Hilbert's Irreducibility Theorem and Galois resolvents", AMS Spring Southeastern Sectional Metting, University of Alabama in Huntsville, March 2015
• "Hilbert's Irreducibility Theorem and Galois resolvents", Automorphic Forms Workshop, University of Michigan, March 2015
• "Hilbert's Irreducibility Theorem and Galois resolvents", UIC-Northwestern Graduate Student Number Theory Day, Northwestern University September 2014
• "The normal order method and invariants of Drinfeld modules over finite fields", UIC Number Theory Seminar, October 2014
• "Distribution of the trace of Frobenius for Drinfeld modules", ENFANT, Hausdorff Center for Mathematics, July 2014
• "The normal order method and invariants of Drinfeld modules over finite fields", Midwest Number Theory Conference for Graduate Students and Recent PhDs, University of Illinois at Urbana-Champaign, June 2014

In the academic years 2012-2013 and 2013-2014, I co-organized the Graduate Number Theory Seminar at UIC with Cara Mullen.

Recent Travel

• Elementary, analytic, and algorithmic number theory: Research inspired by the mathematics of Carl Pomerance (In honor of Carl Pomerance’s 70th birthday), University of Georgia in Athens, June 8-11, 2015
• Upstate Number Theory Conference, Cornell University, April 10-12, 2015
• AMS Spring Southeastern Sectional Metting, University of Alabama, Huntsville, AL, March 27-29, 2015.
• Automorphic Forms Workshop, Ann Arbor, MI, March 2-5, 2015.
• Joint Math Meetings, San Antonio, TX, January 10-13, 2015.
• AIM Workshop: Bounded gaps between primes, American Institute of Mathematics, Palo Alto, CA, November 17-21, 2014.
• LMS-CMI Research School: Bounded gaps between primes, Clay Mathematics Institute, Oxford, UK, September 22-26, 2014.
• Workshop: Analytic Number Theory - ENFANT and ELEFANT, Hausdorff Center for Mathematics, Bonn, Germany, July 2014.
• Seminaire de Mathematiques Superieures 2014: Counting Arithmetic Objects, MSRI & CRM, Montreal, Canada, July 2014.
• Number Theory at Illinois: A Conference in Memory of the Batemans and Heini Halberstam, University of Illinois at Urbana-Champaign, June 2014.
• Midwest Number Theory Conference for Graduate Students and Recent PhDs, University of Illinois at Urbana-Champaign, June 2014.

Other lectures

• "Brun's Theorem towards twin primes and improvements: a survey", UIC Graduate Number Theory Seminar, January 2015
• "A survey of conjectures about the distribution of Frobenius for elliptic modules", UIC Graduate Number Theory Seminar, October 2014
• "Bounded gaps between primes and applications of the Maynard-Tao method", UIC Graduate Number Theory Seminar, September 2014
• "An introduction to function field arithmetic and Drinfeld modules", learning seminar lectures, Spring 2014, UIC
• "Effective versions of Hilbert's Irreducibility Theorem", Number Theory Working Seminar lectures, Spring 2013, UIC
• "Introduction to modular forms", learning seminar lectures, Winter 2012, UIC
• "Probabilistic Galois theory", Number Theory Working Seminar lectures, Fall 2012, UIC
• "Number fields", learning seminar lectures, Summer 2012, UIC
• "Local and global fields", learning seminar lectures, Spring 2012, UIC

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