Date: Tuesday, April 29
Time: 2:00-3:00 PM
Location: LL 1019A
Speaker: Sebastian Bozlee (Fordham)
Title: Moduli of curves via moduli of subrings
Abstract: For the past few years at Fordham I have been studying "territories:" moduli spaces of subalgebras of finite dimensional algebras. I will present my favorite theorems on territories and explain how they help us to better understand the moduli space of all reduced algebraic curves.
Date: Tuesday, March 25
Time: 2:00-3:00 PM
Location: LL 1019A
Speaker: Dave Swinarski (Fordham)
Title: Invariant polynomials and Mukai's models of moduli spaces of curves and K3 surfaces
Abstract: Beginning in the 1980s, Mukai introduced birational models of some moduli spaces of curves and some moduli spaces of K3 surfaces. They are defined as geometric invariant theory quotients. Little is known about the boundaries of these spaces. We describe an approach to efficiently evaluate certain invariant polynomials associated to these GIT problems. This allows us to show that several singular curves and surfaces are GIT semistable in Mukai's models.
Date: Tuesday, February 4
Time: 3:00-4:00 PM
Speaker: Kyoung-Seog Lee (Postech)
Title: Ulrich bundles on the intersection of two quadrics
Abstract: Ulrich bundles are special vector bundles on algebraic varieties drawing lots of attention since they were defined and studied by Eisenbud and Schreyer. Especially, there have been lots of studies about Ulrich bundles on Fano manifolds these days. In the first part of this talk, I will briefly review some basic definitions and results about Ulrich bundles. Then I will discuss Ulrich bundles on the intersection of two quadrics. The last part of this talk is based on a joint work in progress with Jiwan Jung and Han-Bom Moon.
Date: September 12, 1 pm.
Venue: LL 701 (LC)
Speaker: Zeyu Shen (Rutgers)
Title: G_0 of affine, simplicial toric varieties
Abstract: Let X be a Noetherian scheme. G_0(X) is the Grothendieck group of coherent sheaves on X. In other words, it is the quotient of the free abelian group on the isomorphism classes of coherent sheaves on X by the subgroup generated by the relations [G]=[F]+[H] for every short exact sequence of coherent sheaves on X: 0->F->G->H->0. As a basic example, I will discuss how to compute G_0(Spec A) for any principal ideal domain A. So far, there is no result on the G-theory of simplicial toric varieties. I seek to address this issue by computing the G_0 of affine, simplicial toric varieties first. I will describe how I computed G_0(X) for any affine, simplical toric variety X in terms of the filtration on G_0(X) by codimension of support. I will also mention how to use the Brown-Gersten-Quillen spectral sequence to compute the G_0 of any affine toric surface and G_0 of any affine, simplicial toric 3-fold X as a group extension of the Chow group A^1(X) by A^2(X). The order of the Chow group A^1(X) can be computed for any affine, simplicial toric variety X.
Date: September 19, 1 pm.
Venue: LL 701 (LC)
Speaker: Leo Herr (Virginia Tech)
Title: The log Grothendieck group of varieties
Abstract: The ordinary Grothendieck group of varieties is a group with simple generators and relations built out of varieties analogous to the construction of the integers from the natural numbers. Called the ``poor man's motives,'' this group recovers Hodge numbers, even for open, singular varieties, and point counts over finite fields.
Log schemes are morally pairs of a scheme with a divisor (X, D). Used for compactifications and degenerations of varieties, they admit a log Hodge theory due to Kato-Usui. This theory is not yet understood for open or singular varieties, so we define a log Grothendieck group to get log Hodge numbers. Unfortunately, this does not work! We modify log Hodge numbers to get weaker well-defined invariants from our theory, leaving lots of open questions.
Date: September 26, 1 pm.
Venue: LL 701 (LC)
Speaker: Yoon-Joo Kim (Columbia)
Title: Isotrivial fibrations of hyper-Kähler manifolds
Abstract: A hyper-Kähler manifold is a higher dimensional generalization of a K3 surface. It often admits a morphism called a Lagrangian fibration, which is again a generalization of an elliptic fibration for K3 surfaces. A Lagrangian fibration is called isotrivial if all its smooth fibers are isomorphic. Isotrivial elliptic fibrations of K3 surfaces are well-studied and essentially classified in the literature. In this talk, I will present joint work with Radu Laza and Olivier Martin about the structure of isotrivial Lagrangian fibrations of higher-dimensional hyper-Kähler manifolds. We first divide them into two types A and B depending on their birational behavior. After that, we give a complete classification of type A isotrivial Lagrangian fibrations.
Date: October 17, 1 pm.
Venue: LL 701 (LC)
Speaker: David Swinarski (Fordham)
Title: Geometric invariant theory and Kempf's worst 1-parameter subgroup
Abstract: In this two-part minicourse, we will discuss four topics.
1. The ABCs of GIT
2. The worst 1-parameter subgroup associated to a nonsemistable point
3. Hyeon, Lee, and Park's Generic Semistability Theorem
4. The Kempf-Morrison Lemma
Date: October 24, 1 pm.
Venue: LL 701 (LC)
Speaker: David Swinarski (Fordham)
Title: Geometric invariant theory and Kempf's worst 1-parameter subgroup part II
Abstract: In this two-part minicourse, we will discuss four topics.
1. The ABCs of GIT
2. The worst 1-parameter subgroup associated to a nonsemistable point
3. Hyeon, Lee, and Park's Generic Semistability Theorem
4. The Kempf-Morrison Lemma
Date: November 7, 1 pm.
Venue: LL 701 (LC)
Speaker: Patrick McFaddin (Fordham)
Title: Categorical approaches to interleaving distance
Abstract: In the field of topological data analysis, and more specifically persistent homology, interleaving distance is a measure of similarity between persistence modules (or "barcodes"), obtained as the minimal shift necessary for alignment. In many cases of interest, such persistence modules may be viewed as functors from the poset category of the real line to the category of vector spaces. In this talk, we will introduce interleaving distance in the context of persistent homology, and then discuss a slew of categorical generalizations. This is joint work with Tom Needham (Florida State University).
Date: November 14, 1 pm.
Venue: LL 701 (LC)
Speaker: Tamar Blanks (Fordham)
Title: Invariants valued in cycle modules
Abstract: In the theory of invariants introduced by J.-P. Serre, one classically assigns some value—such as a quadratic form or a Galois cohomology class—to each $G$-torsor over each field, for an algebraic group $G$. On the other hand Serre's definition is quite general: an invariant is a natural transformation between a functor $H : \text{Fields}_k \to \text{Sets}$ and a functor $M : \text{Fields}_k \to \text{Ab}$. Somewhere in the middle is the idea of taking $M$ to be a \emph{cycle module}, which generalizes Milnor K-theory and satisfies some desirable algebraic and geometric properties. This talk will explore what is known about invariants valued in cycle modules, focusing on new developments in the past ten years or so, and discuss some interesting ways one might generalize these new results.
Date: November 21, 1 pm.
Venue: LL 701 (LC)
Speaker: David Swinarski (Fordham)
Title: Geometric invariant theory and Kempf's worst 1-parameter subgroup part III
Abstract: In this minicourse, we will discuss four topics.
1. The ABCs of GIT
2. The worst 1-parameter subgroup associated to a nonsemistable point
3. Hyeon, Lee, and Park's Generic Semistability Theorem
4. The Kempf-Morrison Lemma
Date: December 5, 1 pm.
Venue: LL 701 (LC)
Speaker: David Swinarski (Fordham)
Title: The worst 1-parameter subgroup for toric curves with one unibranch singular point
Abstract: Kempf proved that when a point is unstable in the sense of Geometric Invariant Theory, there is a "worst'' destabilizing 1-parameter subgroup $\lambda$. It is natural to ask: what are the worst 1-parameter subgroups for the unstable points in the GIT problems used to construct the moduli space of curves $\overline{M}_g$? Here we consider Chow points of toric curves with one unibranch singular point. We translate the problem as an explicit problem in convex geometry (finding the closest point on a polyhedral cone to a point outside it). We prove that the worst 1-PS has a combinatorial description that persists once the embedding dimension is sufficiently large, and present some examples. This is joint work with Joshua Jackson (Cambridge).
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.
Date: Sep. 21.
Place: LL 902 (LC)
Speaker: Patrick McFaddin
Title: Arithmetic tori, toric varieties, and K-theory
Abstract: Complex algebraic tori (and their compactifications) play a tremendously important role in algebra and algebraic geometry. One reason for their ubiquity is that compactifications of tori can be encoded using combinatorial data in the associated lattices of characters and cocharacters. Over base fields which are not algebraically closed, the study of (compactifications of) tori becomes much more subtle, and Galois actions on these associated lattices must be considered. In this talk, I will discuss tori and toric varieties defined over an arbitrary base field, together with a few examples. I will then give an overview of work of Merkurjev and Panin who studied these objects using a certain non-commutative category of K-motives. Lastly, I will discuss partial progress toward determining the Galois-lattice structure of the Grothendieck group of a toric variety.
Date: Oct. 5.
Place: LL 902 (LC)
Speaker: David Swinarski
Title: The worst destabilizing 1-parameter subgroup for cuspidal curves
Abstract: Kempf proved that when a point is unstable in the sense of geometric invariant theory, there is a "worst" destabilizing 1-parameter subgroup $\lambda$. It is natural to ask: what are the worst destabilizing 1-ps for the unstable points in familiar GIT problems, such as those used to construct the moduli space of curves $M_g$? We first consider Chow points of rational curves with one unibranch singular point. Then the problem can be restated as an explicit problem in convex geometry (finding the proximum of a polyhedral cone to a point outside it). We present some observations based on computer experiments, and prove the answer for cusps $y^2 = x^{2r+1}$ of order $r$. This is joint work with Joshua Jackson (Sheffield).
Date: Oct 12.
Place: LL 902 (LC)
Speaker: Sebastian Bozlee
Title: A stratification of moduli of arbitrarily singular curves
Abstract: What does the space of all algebraic curves look like? One way to answer the question is to break up this gigantic, unruly space into nice pieces, called strata, and then to describe those.
For nodal (i.e. not particularly singular) curves such strata have long been constructed and are used throughout the theory. In this talk, I will describe how to construct a similar stratification of moduli of arbitrarily singular curves, indexed by combinatorial data generalizing dual graphs. The key idea is to combine the geometry of moduli of smooth curves with moduli of subalgebras.
Date: Oct 19.
Place: LL 902 (LC)
Speaker: Cris Poor
Title: Formal series of Jacobi forms
Abstract: Do all formal series of Jacobi forms invariant under a certain involution converge? We discuss recent progress on this open problem. The Fourier-Jacobi expansions of paramodular forms, necessarily convergent, are examples of series of Jacobi forms invariant under an involution. The open problem is the converse: do any such formal series converge and thereby define paramodular forms? Evidence for this converse is given by the success of computer programs that rigorously determine specific vector spaces of paramodular forms. We reduce the open problem to the behavior of a vector space under the action of a Hecke algebra.
Date: Nov. 9.
Place: LL 902 (LC)
Speaker: Giancarlo Castellano
Title: Special values of Rankin–Selberg $L$-functions for $\GL_n \times \GL_m$ over totally real fields
Abstract: In recent years, considerable advancements have been made in the arithmetic of automorphic $L$-functions. Ever since the seminal work of Raghuram–Shahidi (2008), progress has been especially fruitful in relating special values of Rankin–Selberg automorphic $L$-functions for $\GL_n \times \GL_{n-1}$ to representation-theoretic invariants of cohomological automorphic representations known as \emph{Whittaker–Betti periods}. The case where both representations are cuspidal was settled in Raghuram (2016) for arbitrary number fields; work of Grobner–Harris (2016), Grobner (2018), Grobner–Lin (2021) and Li–Liu–Sun (2021) progressively extended those results to the case where the representation of $\GL_{n-1}$ is allowed to be isobaric.
In contrast, a lot less seems to be known about special values of Rankin–Selberg automorphic $L$-functions for $\GL_n \times \GL_m$ with $1 \le m < n - 1$. Among the most prominent results we have, on the one hand, a theorem of Grobner–Sachdeva over CM-fields (with a parity restriction on $m$ and $n$), and on the other, results of Harder–Raghuram and Raghuram about \emph{quotients} of special $L$-values (the base field being assumed totally real and totally imaginary, respectively) where such quotients are brought into connection with quantities known as \emph{relative periods}.
In this talk, I will relate the main outcomes of my PhD work. On the one hand, I obtained a result on special values of Rankin–Selberg $L$-functions for $\GL_n \times \GL_m$ over a totally real field, with $n$ odd and $m < n$ even, by a method similar to the one used in the work of Grobner–Sachdeva. As a consequence, one obtains a corollary on quotients of such special values, which, when compared and combined with Harder–Raghuram's results, yields a relation between Whittaker periods, on the one hand, and Harder–Raghuram relative periods on the other. In the special case $m = n - 1$, the latter result is to be compared with earlier work of Raghuram about relative periods being equal to quotients of Whittaker periods of opposite signatures, up to an explicit power of the imaginary unit $\mathrm{i}$.
Date: Nov. 16.
Place: LL 902 (LC)
Speaker: Han-Bom Moon
Title: Derived category of moduli space of vector bundles
Abstract: The derived category of moduli spaces of vector bundles on a curve is expected to be decomposed into the derived categories of symmetric products of the base curve. I will briefly explain the statement and how representation theory is involved in the proof of this statement.
Date: Nov. 30.
Place: LL 902 (LC)
Speaker: Tamar Blanks
Title: Witt invariants of Weyl groups
Abstract: The trace form is a simple and powerful tool in the study of fields and field extensions. For a finite-dimensional extension $K/F$, it is the symmetric bilinear form over $F$ sending $(x,y) \in K \times K$ to the trace $Tr_{K/F}(xy)$. In generalizing this notion, one can ask: given a particular type of algebraic structure (such as field extensions), what kind of quadratic form data is available? What invariants exist, if we want the invariant to be valued in the Witt ring of quadratic forms?
More precisely, for a smooth linear algebraic group $G$ over a field $k$, a Witt invariant of $G$ is a natural transformation from $H^1(-,G)$ to $W(-)$, that is, a rule that assigns an element of the Witt ring to each $G$-torsor in a way that is compatible with extension of scalars. In 2003, J.-P. Serre showed that when $G=S_n$, all Witt invariants come from the trace form. I will discuss recent work extending Serre's result from $S_n$ to other Weyl groups, and place this work in the broader context of cohomological invariants. I will then outline some related challenges and open questions in this area.