NATO grant awarded on Isogeny based post-quantum cryptography, with Shaul Zemel of Hebrew University. Contact us if you are interested in the topic.
Artificial Intelligence, Machine Learning
Machine learning for moduli space of genus two curves and an application to post-quantum cryptography, 2024
Hybrid AI-Math, Feb. 27-29, 2024,
Institute of Mathematics and Statistics,
State University of Rio de Janeiro, RJ, BrazilAn introduction to Equivariant Neural Networks
A machine learning approach to Julia reduction
Neural networks and weighted moduli spaces
Recent advances in mathematics and artificial intelligence, Cont. Math., 2024, (in progress)
Artificial Intelligence in Mathematics (Special Session, AMS Meeting, Milwaukee), April 2024.
Weighted heights and WGCD's
Deep learning and Vojta's conjecture on weighted varieties (in progress)
Vojta's conjecture on weighted projective varieties and an application on greatest common divisors
Local and global heights on weighted projective varieties, Houston J. Math. (2024)
Weighted greatest common divisors and weighted heights, J. Number Theory 213 319--346 (2020)
Computing heights on weighted projective spaces, Cont. Math. 724 149--160 (2019)
Cryptography
Grant on: Isogeny based post-quantum cryptography,
with Shaul ZemelFrey, Gerhard; Shaska, Tony Curves, Jacobians, and cryptography. Algebraic curves and their applications, 279–344,Contemp. Math., 724,2019.
Abelian varieties and number theory. Edited by Moshe Jarden and Tony Shaska. Contemporary Mathematics, 767. 2021.
Advances on superelliptic curves and their applications. NATO Advanced Study Institute (ASI) on Hyperelliptic Curve Cryptography, Ohrid 2014. NATO Science for Peace and Security Series D: Information and Communication Security, 41. IOS Press 2015.
Algebraic aspects of digital communications. New Challenges in Digital Communications, Vlora 2008. NATO Science for Peace and Security Series D: Information and Communication Security, 24. IOS Press 2009.
Advances in coding theory and cryptography. Edited by T. Shaska, W. C. Huffman, D. Joyner and V. Ustimenko. Series on Coding Theory and Cryptology, 3. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007.
Isogenies, genus 2 Jacobians
On Isogenies Among Certain Abelian Surfaces. Michigan Math. J. 71 (2022), no. 2, 227–269.
Isogenous components of Jacobian surfaces , Eur. J. Math. 6 (2020), no. 4, 1276–1302.
Genus two curves with many elliptic subcovers, Comm. Algebra 44 (2016), no. 10, 4450–4466.
Genus 2 curves with degree 5 elliptic subcovers, Forum Math. 21 (2009), no. 3, 547–566.
Genus 2 fields with degree 3 elliptic subfields, Forum Math. 16 (2004), no. 2, 263 -- 280.
Genus 2 curves with (3,3)-split Jacobian and large automorphism group. ANTS 2002, Lecture Notes in Comput. Sci., 2369, 2002
Curves of genus 2 with (n,n)-decomposable Jacobians, J. Symbolic Comput. 31 (2001), no. 5,603–617.
Curves of genus two covering elliptic curves. Thesis (Ph.D.)–University of Florida. 2001. 72 pp. ISBN: 978-0493-20012-5, ProQuest LLC
Groups and algebraic curves
Automorphism groups of rational maps (in preparation)
Broughton, A.; Shaska, T.; Wootton, A. On automorphisms of algebraic curves. Algebraic curves and their applications, 175–212, Contemp. Math., 724, 2019.
Malmendier, A.; Shaska, T. From hyperelliptic to superelliptic curves. Albanian J. Math. 13 (2019), no. 1, 107–200.
Galois group of prime degree polynomials with non-real roots , Lect. Notes in Computing, 13, 2005, 243--255.
Determining the automorphism group of a hyperelliptic curve , ISSAC 05, 248--254, ACM, New York, 2003.
The locus of curves with prescribed automorphism group, Communications in arithmetic fundamental, Sūrikaisekikenkyūsho Kōkyūroku, No. 1267 (2002), 112–141.
Elliptic subfields and automorphisms of genus 2 function fields , Algebra, arithmetic and geometry with applications,703--723, Springer, 2004.
Coding theory
On the automorphism groups of some AG-codes based on Ca,b curves.
Codes over Fp2 and Fp×Fp, lattices, and theta functions.
Codes over rings of size four, Hermitian lattices, and corresponding theta functions
On some applications of graphs to cryptography and turbocoding.
On the homogeneous algebraic graphs of large girth and their applications.
Codes over rings of size p2 and lattices over imaginary quadratic fields
Quantum codes from superelliptic curves
Theta functions and symmetric weight enumerators for codes over imaginary quadratic fields. Des. Codes Cryptogr. 76 (2015), no. 2, 217–235.
Weight distributions, zeta functions and Riemann hypothesis for linear and algebraic geometry codes.
Self-inversive polynomials, curves, and codes
Words of wisdom
"I can't believe they destroyed a perfectly good old farm to build this damn university" (From an old colleague)
“… Maybe you’ll find yourself in a mediocre department, where your work will be intentionally undervalued, people with much lesser research record will be promoted before you, and the hypocrites and frauds will run wild. Don’t get discouraged, don’t give up! Remember why you got into math?! It wasn’t for the money, recognition, or fame, it was for that special feeling that you get when you find the perfect solution or understand a beautiful argument. That hasn’t changed, son! If you still got that magical feeling, then you are doing fine.” (From Dad)
David Hilbert’s radio address - English translation.
“… There is a secret to mathematics. Do what you can, not what you dreamed of doing! And try to learn from any paper that you write.” (John Thompson after my dissertation defense)
“Certainly the best times were when I was alone with mathematics, free of ambition and pretense, and indifferent to the world.” Langlands, in Mathematicians: An Outer View of the Inner World, p142. (From James Milne website)