Session Information (Location: IPAM Lecture Hall 1200)

Algebra and Number Theory

Session Organizers: Dagan Karp (Harvey Mudd) and Adriana Salerno (Bates College)
Session Speakers: Rosemary Guzman (University of Chicago), Pamela Harris (Williams College), Pablo Solis (Stanford University), Bianca Thompson (Harvey Mudd College), and Anthony Varilly-Alvarado (Rice University).

LOCATION: IPAM Lecture Hall 1200

2:00 - 2:25 Rosemary Guzman (University of Chicago)
2:30 - 2:55 Pablo Solis (Stanford University)
3:00 - 3:30 Coffee Break
3:30 - 3:55 Bianca Thompson (Harvey Mudd College)
4:00 - 4:25 Anthony Varilly-Alvarado (Rice University)
4:30 - 4:55 Pamela Harris (Williams College)

Rosemary Guzman (University of Chicago)
Title: The quantitative geometry of k-free hyperbolic 3-manifold groups
Abstract: Consider a discrete, torsion-free subgroup $\Gamma$ of the orientation-preserving isometry group of hyperbolic 3-space. We will examine the case where $\Gamma$ is cocompact and has the "k-free property," which is to say that every subgroup of rank less than or equal to k is free. The quotient M of hyperbolic 3-space modulo $\Gamma$ defines a closed, orientable hyperbolic 3-manifold, whose fundamental group is isomorphic to $\Gamma$ via the canonical isomorphism. A celebrated theorem of Mostow states that if M, N are two closed, connected, orientable, hyperbolic n-manifolds which are homotopy equivalent in dimensions n = 3, then M, N are equivalent up to isometry.

This unique geometric-topological relationship has been the framework for many important results, including notable results providing lower bounds on the volume of M, and results relating volume to homology. We will discuss a new quantitative-Mostow-Rigidity-type result relating to k = 5-free hyperbolic 3-manifold groups and their associated quotient manifold as mentioned above.

Pamela Harris (Williams College)
Title: Invisible Lattice Points
Abstract: This talk is about the invisibility of points on the integer lattice \( \mathbb{Z} \times \mathbb{Z} \), where we think of these points as (infinitely thin) trees. Standing at the origin one may notice that the tree at the integer lattice point \( (1, 1) \)blocks from view the trees at \( (2, 2), (3, 3) \), and, more generally, at \( (n,n) \) for any \( n \in \mathbb{Z}_{\geq 0} \). In fact any tree at \( (l,m) \) will be invisible from the origin whenever l and \( m \) share any divisor \(d \), since the tree at \( (\ell/D, m/D) \), where \( D = \text{gcd} (\ell, m) \) blocks \( (l, m) \) from view. With this fact at hand, we will investigate the following questions. If the lines of sight are straight lines through the origin, then what is the probability that the tree at \( (\ell, m) \) is visible? Meaning, that the tree \( (\ell, m) \) is not blocked from view by a tree in front of it. Is possible for us to find forests of trees (rectangular regions of adjacent lattice points) in which all trees are invisible? If it is possible to find such forests, how large can those forests be? What happens if the lines of sight are no longer straight lines through the origin, i.e. functions of the form \( f(x) = ax \) with \( a \in \mathbb{Q} \), but instead are functions of the form \( f(x) = ax^b \) with \( b \) a positive integer and \( a \in \mathbb{Q} \)? Along this mathematical journey, I will also discuss invisibility as it deals with the underrepresentation of women and minorities in the mathematical sciences and I will share the work I have done to help bring more visibility to the mathematical contributions of Latinx and Hispanic Mathematicians.

Math work is joint with Bethany Kubik, Edray Goins, and Aba Mbirika. Diversity work with Alexander Diaz-Lopez, Alicia Prieto Langarica, and Gabriel Sosa.

Pablo Solis (Stanford University)
Title: Hunting Vector Bundles on P1 x P1
Abstract: Motivated by ideas in commutative algebra, Eisenbud and Schreyer conjectured there should be vector bundles on P1 x P1 with natural cohomology and prescribed Euler characteristic. I'll give some background on vector bundles and explain what natural cohomology means and prove the conjecture in "most" cases.

Bianca Thompson (Harvey Mudd College)
Title: An Accessible guide to Uniform Bounds of Twists
Abstract: The study of discrete dynamical systems boomed in the age of computing. We could suddenly compute high iterates of functions and look at their behavior over time. We could create the beautiful fractal, the Mandelbrot via iterating 0 in the function z^2+c and allowing c to vary. This gives mathematicians a wealth of questions to explore. One question tied to the Mandelbrot set is how many rational points points have iterates that end in a cycle as we allow c to vary? Is this number of rational points uniformly bounded as c varies? It turns out this is a hard question to answer. Instead we will explore places where this question can be answered; special families of rational functions, twists.

Anthony Varilly-Alvarado (Rice University)
Title: Diophantine Equations and Algebraic Geometry
Abstract: Diophantine equations are often profitably studied through the lens of Algebraic Geometry. Geometric insights can shed light on the structure of solutions to these equations (or the lack thereof). Using examples such as Fermat's Last Theorem, I will explain how a geometric perspective leads to some deep conjectures on the number theory of spaces defined by polynomial equations.

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