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The Langlands Program: A grand unified theory

The Langlands program connects fields that do not appear to be related, such as algebra, geometry, analysis, and physics. Due to its wide scope and deep results, the Langlands program is considered a grand unified theory. In this talk we will give a general overview of the Langlands program and discuss work in progress on the exceptional group G_2 which is [Read More...]

Presenter: Melissa Emory, University of Toronto
Authors: Melissa Emory, Maria Fox, Julee Kim
Symposium Year: 2022
Session: New Directions in Number Theory
Presentation Time: June 17, 2022; 3:30 pm

Bounding Lifts of Markoff Triples mod p*

In 2016, Bourgain, Gamburd, and Sarnak proved that Strong Approximation holds for the Markoff surface in most cases. That is, the modulo p solutions to the equation $x^2+y^2+z^2=3xyz$ are covered by the integer points for most primes p. In this talk, we will discuss how the algorithm given in the paper of Bourgain, Gamburd, and Sarnak can be used to obtain [Read More...]

Presenter: Elisa Bellah, University of Oregon
Authors: Elena Fuchs and Lynnelle Ye
Symposium Year: 2022
Session: New Directions in Number Theory
Presentation Time: June 17, 2022; 3:55 pm

Prime components in Apollonian packings

An Apollonian circle packing is a fractal arrangement formed by repeatedly inscribing circles into the interstices in a Descartes configuration of four mutually tangent circles. The curvatures of the circles in such a packing are often integers, and so it is natural to ask questions about their arithmetic properties. For example, it is known by work of [Read More...]

Presenter: Catherine Hsu, Swarthmore College
Authors: Holley Friedlander, Elena Fuchs, Piper H, Catherine Hsu, Katherine Sanden, Damaris Schindler, and Katherine Strange
Symposium Year: 2022
Session: New Directions in Number Theory
Presentation Time: June 17, 2022; 4:20 pm

Newton polygon stratification of the Torelli locus in PEL-type Shimura varieties

We study the intersection of the Torelli locus with the Newton polygon stratification of the modulo $p$ reduction of certain PEL-type Shimura varieties. Using a clutching method, we produce infinitely many new examples of Newton polygons that occur for smooth curves that are cyclic covers of the projective line. Most of these examples demonstrate unlikely [Read More...]

Presenter: Wanlin Li, Centre de Recherches Mathématiques
Authors: Wanlin Li, Elena Mantovan, Rachel Pries, Yunqing Tang
Symposium Year: 2022
Session: New Directions in Number Theory
Presentation Time: June 17, 2022; 4:45 pm

Orienteering with one endomorphism

In supersingular isogeny-based cryptography, the path-finding problem reduces to the endomorphism ring problem. Can path-finding be reduced to knowing just one endomorphism? Our Women in Numbers 5 (WIN5) working group investigated these questions in our study of oriented supersingular elliptic curves. An endomorphism gives an explicit orientation of a [Read More...]

Presenter: Sarah Arpin, University of Colorado Boulder
Authors: Sarah Arpin, Mingjie Chen, Kristin E. Lauter, Renate Scheidler, Katherine E. Stange, Ha T. N. Tran
Symposium Year: 2022
Session: New Directions in Number Theory
Presentation Time: June 18, 2022; 10:15 am

Two-dimensional polynomials with dynamical Mahler measure 0

The Mahler measure of an integer polynomial in one or more variables is an important quantity that shows up many places in number theory and mathematics more generally. The integer polynomials with Mahler measure 0 have been classified -- in the one-variable case, these are products of cyclotomic polynomials. We introduce a dynamical generalization of [Read More...]

Presenter: Alison Beth Miller, AMS Mathematical Reviews
Authors: Annie Carter, Matilde Lalín, Michelle Manes, Alison Beth Miller, and Lucia Mocz.
Symposium Year: 2022
Session: New Directions in Number Theory
Presentation Time: June 18, 2022; 10:40 am

Extremal Primes for elliptic curves*

An extremal prime $p$ for an elliptic curve $E$ is one for which the trace of the Frobenius at $p$ is maximal or minimal in view of the Hasse bound. In this talk, assuming GRH, we present a joint distribution result involving the Chebotarev Density Theorem. As a consequence, we obtain an upper bound for the number of primes satisfying the maximality [Read More...]

Presenter: Neha Prabhu, Savitribai Phule Pune University, India
Symposium Year: 2022
Session: New Directions in Number Theory
Presentation Time: June 18, 2022; 11:05 am

Arboreal representations for rational maps with few critical points

An arboreal Galois representation is the dynamical analog of the Galois representations attached to elliptic curves. A dynamical analog to Serre's open image theorem, conjectured by Jones, remains largely open. We prove a version of Jones' Conjecture for quadratic and cubic polynomials assuming the abc-Conjecture and Vojta's Conjecture. We also exhibit a [Read More...]

Presenter: Michelle Manes, University of Hawaii at Manoa
Authors: Jamie Juul, Holly Krieger, Nicole Looper, Michelle Manes, Bianca Thompson, Laura Walton
Symposium Year: 2022
Session: New Directions in Number Theory
Presentation Time: June 18, 2022; 11:30 am

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