Fordham Algebra Seminar
This is an in house algebra seminar for department of mathematics at Fordham University. We will meet at 1 pm on Thursday, at Lincoln Center campuse. The seminar will be broadcasted via zoom to the other campus. The organizer is Han-Bom Moon. If you want to receive a reminder email, please let the organizer know. All is welcome to participate! For the videos, you need Fordham credential to access.
Spring 2024
Date: Feb. 1.
Speaker: Patrick McFaddin
Place: LL 526 (LC)
Title: Analogues of simplicial/singular homology for schemes
Abstract: This talk will be an overview of the theory of higher Chow groups (a la Bloch) and Chow groups with coefficients (a la Rost). Both of these homology theories emulate topological constructions in an algebro-geometric framework. This is accomplished by enriching Chow groups with additional arithmetic data.
Date: Feb. 8.
Speaker: Sebastian Bozlee
Place: JMH 405 (RH)
Title: Spaces of generators of algebras
Abstract: A classical problem in number theory is the following: Given a finite field extension K of the rational numbers, can the ring of integers of K be obtained by adjoining just one element theta of K to the integers? If so, the number field is said to be monogenic and theta is said to be a monogenerator. I will present a geometric interpretation of the problem and use this to construct a space of monogenerators for any finite extension of commutative rings.
Date: Feb. 15.
Speaker: Han-Bom Moon
Place: LL 526 (LC)
Title: Algebra of conformal blocks
Abstract: The algebra of conformal blocks is an algebraic creature constructed via physics, in particular conformal field theory. I will explain how this object is related to moduli theory in algebraic geometry, representation theory, and if time permits, low dimensional topology.
Date: Feb. 22.
Speaker: Cris Poor
Place: LL 526 (LC)
Title: The Perfect Cone Decomposition
Abstract: We will give an introduction to lattice packings of spheres by proving that the hexagonal lattice packing is optimal in the plane. We will discuss Voronoi's theorem that a lattice gives a local maximum for the packing density if and only if the lattice is both perfect and eutactic. We will describe the covering of the cone of definite n-by-n real matrices by perfect cones.
Date: Feb. 29.
Speaker: A. Raghuram
Place: LL 526 (LC)
Title: Cohomology of locally symmetric spaces
Abstract: I will introduce the notion of a locally symmetric space, its cohomology, and draw attention to some interesting research problems that drive the subject. Time permitting, I will discuss some recent results of mine obtained in collaboration with my student Nasit Darshan on the problem of nonvanishing cuspidal cohomology.
Date: March 7.
Speaker: Lisa Marquand (NYU)
Place: LL 526 (LC)
Title: Finite groups of symplectic birational transformations of manifolds of OG10 type
Abstract: Compact Hyperkähler manifolds are one of the building blocks of Kähler manifolds with trivial first chern class. Although occurring in high dimensions, their geometry is governed by the existence of a non-degenerate bilinear form on the second cohomology, mimicking the intersection product on a K3 surface. One can study geometric problems by studying this associated indefinite lattice; in particular studying isometries. In this talk, we will use this strategy to classify finite groups of birational symplectic automorphisms for a specific deformation type of examples: OG10 type manifolds. In particular, there are 211 possible finite groups that occur, with 66 of those having unique action on the associated lattice. This talk is based on joint work with Stevell Muller.
Date: March 14.
Speaker: Giancarlo Castellano
Place: LL 526 (LC)
Title: Rational structures and periods of group representations over arbitrary ground fields
Abstract: In the field of special values of $L$-functions, \emph{periods} are nonzero complex numbers obtained by comparing rational structures on isomorphic complex representations of some (fixed) group $G$. A necessary condition for a complex representation $\pi$ to have an $F$-rational structure ($F \subset \mathbb{C}$ a subfield) is that $F$ contains the \emph{rationality field} of $\pi$, a subfield $\mathbb Q(\pi)$ defined in terms of \emph{twists} of $\pi$ by field automorphisms of $\mathbb C$. As part of my PhD project, I investigated the existence and (essential) uniqueness of $F$-rational structures over ground fields $K$ other than $\mathbb C$ or $overline{\mathbb Q}$, and found that the familiar proofs from the complex case remain valid under suitable assumptions on $\pi$ and on the extension $K/F$. In this talk, I will relate these results in a mostly self-contained way. If time permits, I will discuss rationality fields of irreducible rational representations of $\mathrm{GL}_n(k)$, $k$ an algebraic number field, as a concrete example.
Date: April 4.
Speaker: Qiyao Vivian Yu (Columbia)
Place: LL 526 (LC)
Title: On counting totally imaginary number fields
Abstract: A number field is said to be a CM-number field if it is a totally imaginary quadratic extension of a totally real number field. We define a totally imaginary number field to be of CM-type if it contains a CM-subfield, and of TR-type if it does not contain a CM-subfield. For quartic totally imaginary number fields when ordered by discriminant, we show that about 69.95% are of TR-type and about 33.05% are of CM-type. For a sextic totally imaginary number field we classify its type in terms of its Galois group and possibly some additional information about the location of complex conjugation in the Galois group.
Date: April 11.
Speaker: David Swinarski
Place: LL 526 (LC)
Title: A series of graph curves with automorphisms
Abstract: This is practice for a talk at an upcoming AMS Sectional Meeting. I will begin the Fordham Algebra Seminar presentation by giving the 20 minute talk, then use the remaining time to answer questions from the audience and elaborate on these topics. Here is the abstract for the AMS talk: Trivalent graph curves (in the sense of Bayer and Eisenbud) are maximally degenerate nodal curves of arithmetic genus g; the graph records how the irreducible components meet. We study a particular series of odd genus g=2k+1 graph curves with an action of the dihedral group of order 6k. Their automorphism groups satisfy the hypotheses needed to apply Kempf-Morrison-Swinarski’s approach to establish geometric invariant theory semistability via state polytope calculations.
Date: April 25.
Speaker: Candace Bethea (Duke)
Place: LL 526 (LC)
Title: Equivariant curve counting and local enrichments
Abstract: Enumerative geometry seeks integral solutions to geometric questions, such as how many rational degree d curves with n marked points lie on a given surface. Motivic and equivariant homotopy have been used in recent years to produce enrichments of classical enumerative results over non-closed fields and under the presence of a group action respectively. In this talk, we will discuss connections between equivariant homotopy theory and equivariant curve counting, specifically presenting a count of nodal orbits in an invariant pencil of plane conics enriched in the Burnside Ring of a finite group. We will also discuss joint work in progress with Kirsten Wickelgren on defining a local degree using stable equivariant homotopy theory to produce further equivariantly enriched enumerative results.