April 29, 2021
Speaker: Vlad Sworski
Title: An Introduction to Genus 2 Isogeny Graphs
Abstract: With the advent of quantum computers, the field of post-quantum cryptography has emerged to develop quantum-resistant cryptosystems. One potential contender is that of SIDH (Supersingular Isogeny Diffie-Hellman). This method relies heavily on isogeny graphs between elliptic curves. Takashima proposed concerns with the current models, and as a result, study of genus 2 isogeny graphs has begun as a potentially more secure alternative. We will begin by briefly reminding the viewer of the genus 1 construction before discussing the nature of the genus 2 graph and its isogenies. We will not go into great detail on the cryptography itself.
April 22, 2021
Speaker: Jae Hwang Lee
Title: Classical McKay Correspondence
Abstract: When we have a finite subgroup of SL_2(C), it is possible to derive two graphs: i) resolution graphs, and ii) McKay graphs. Once we see those, we can recognize how they related to each other. I would like to present this correspondence, the so-called classical McKay correspondence.
April 1, 2021
Speaker: Kirk Bonney
Title: The Genus of a Modular Curve
Abstract: Modular curves are important for their relationship to elliptic curves and modular forms. Like elliptic curves, they are Riemann surfaces. Unlike elliptic curves, the size of the genus varies widely over the family of modular curves. This talk will introduce and dissect the genus formula for a modular curve and (hopefully) work through an example.
March 25, 2021
Speaker: Seth Ireland
Title: Grassmannians and their tangents
Abstract: Grassmannians are geometric objects which parameterize linear subspaces of a vector space. In this talk, I’ll introduce Grassmannians, show how we can think about them as projective varieties, and examine their tangent spaces.
March 11, 2021
Speaker: Amie Bray
Abstract: In this talk, we discuss the construction of the l-modular polynomial for elliptic curves, describe the l-isogeny graph of an elliptic curve, and consider its applications in cryptography. We then look at the same story for Drinfeld modules and explain why rank two Drinfeld modules are a bad choice for supersingular isogeny cryptography.
February 18, 2021
Speaker: Jae Hwang Lee
Title : Yoneda / ne = Yoda
Abstract: We will try to understand Yoneda lemma by exploring examples.
February 10, 2021
Speaker: Kelly Emmrich
Abstract: There is an interesting interaction between topology and algebra via the fundamental group. After a quick review of covering spaces, we will consider covering transformations-especially those contained in the Deck group-and develop the idea of Galois covers. To end the talk, we will uncover (pun definitely intended) the Galois correspondence for covering spaces.
February 3, 2021
Speaker: Michael Moy
Title: An overview of persistent homology and work towards persistence stability of metric thickenings
Abstract: I'll give an introduction to some theoretical work on persistent homology and metric thickenings, intended as an accessible overview rather than a detailed description. Given a sequence of evolving topological spaces, often simplicial complexes built on datasets, persistent homology returns a reductive description of how long certain features last, called a persistence diagram. An important result in the area is the stability of persistent homology, which says that for certain common methods of constructing these simplicial complexes, if two initial vertex sets are close then the resulting persistence diagrams are close. Recent work to better understand simplicial complexes and persistent homology has led to the definition of metric thickenings, which agree with the usual notion of simplicial complexes in the case of finite vertex sets but can have different topologies for infinite vertex sets. Common methods of constructing simplicial complexes then have analogous methods for constructing metric thickenings. I'll describe some of my recent work towards proving stability for these metric thickenings.
January 28, 2021
Speaker: Jae Hwang Lee
Title : Dimension of a Variety
Keyword : Krull dimension, transcendence degree, algebraic independence, dimension at a point, tangent space, Hilbert polynomial
Dimension of a variety is usually taught at the end of the basic algebraic geometry course. For a variety, it is clearly complicated. The most familiar one is Krull dimension. The second one is from algebraic independence. The third one is the most geometric one using tangent spaces at a point. The last one would be the least familiar one using Hilbert polynomials. We explore these definitions for dimension of a variety going through specific examples.