Poster-abstracts

Ricardo Jaime Acuna (Washington University in St. Louis)

Title: A geometric interpretation of folding

Abstract: There is a procedure called folding that can produce a smaller dimensional cluster algebra, from the data of a valued quiver. I give a geometric interpretation of this process.

Rahul Ajit (University of Utah)

Title: Canonical modules and F-Rationality of (extended) Rees Algebra

Abstract: We show that canonical module of extended Rees algebra matches with canonical module of Rees algebra in positive degrees in any charecteristics. Using this, we describe the test modules of these algebras and answer a conjecture of Hara-Watanabe-Yoshida in full generality. This is a joint work with Hunter Simper. Our result gives a way to compute test ideals for non-principal ideals as well in Macaulay2. This is a joint work with Karl Schwede and Hunter Simper. 

Devin Akman (Washington University in St. Louis)

Title: A-Polynomials are Rare

Abstract: We can assign an algebro-geometric invariant called the A-polynomial to a certain class of 3-manifolds. There is a criterion in algebraic K-theory characterizing irreducible polynomials which are factors of some A-polynomial. Our recent results on zero loci of regulator maps imply that a family of curves obtained by varying coefficients of a polynomial contains only finitely many smooth A-factors.

Kamyar Amini (Virginia Tech University)

Title: I-functions of flag manifolds and of their cotangent bundles

Abstract: The K-theoretic J-function in Gromov-Witten theory is a generating function for one-point genus-zero Gromov-Witten invariants. Ciocan-Fontanine, Kim, and Maulik defined the moduli space of quasimaps and introduced the I-function, which is analogous to the J-function in Gromov-Witten theory, focusing on maps to GIT quotients. Okounkov utilized this moduli space to define a vertex function that acts as a generating function counting quasimaps to Nakajima varieties, such as the cotangent bundle of flag manifolds. By considering the hyper-quot compactification of the moduli space of maps to flag manifolds, we introduce a new class using the I-function and a generating function on flag manifolds, through which we recover Okounkov's vertex function for the cotangent bundles.

Matthew Bertucci (University of Utah)

Title: Taylor Conditions On Varieties over Finite Fields

Abstract: We extend Poonen's Bertini theorem over finite fields to more general conditions on the Taylor coefficients of allowable hypersurfaces at each point. Namely, we consider Taylor conditions arising from subsheaves of the sheaf of principal parts. This is suggested by an analogous result of Margaret Bilu and Sean Howe in the motivic setting.

Haohua Deng (Duke University)

Title: Hodge-theoretic completion of period maps

Abstract: Half a century ago, Griffiths proposed the question on finding completion of general period maps with significant geometric and Hodge-theoretic meanings. I will show some recent breakthroughs on this project, as well as some of its applications.

Fernando Figueroa (Northwestern University)

Title: Algebraic Tori in the complement of quartic surfaces

Abstract: Log Calabi-Yau pairs can be thought of as generalizations of Calabi-Yau varieties. Previously Ducat showed that all coregularity 0 log Calabi-Yau pairs $(\mathbb{P}^3,S)$ are crepant birational to a toric model. A stronger property to ask for is for the complement of $S$ to contain a dense algebraic torus. In this work we start the classification of coregularity zero, slc quartic surface for which their complements contain a dense algebraic torus.

We fully classify the case of reducible surfaces. In the case of irreducible surfaces we are able to classify the cases where the singular locus is non planar.

This is joint work with Eduardo Alves da Silva and Joaquín Moraga.

Nathaniel Gallup (University of California Davis)

Title: Well-Ordered Flag Spaces as Functors of Points

Abstract: Using Grothendieck's ``functor of points'' approach to algebraic geometry, we define a new infinite-dimensional algebro-geometric flag space as a $k$-functor (for $k$ a ring) which maps a $k$-algebra $R$ to the set of certain well-ordered chains of submodules of an infinite rank free $R$-module. This generalizes the well known construction of a $k$-functor that is represented by the classical (i.e. finite-dimensional) full flag scheme. We prove that as in the finite-dimensional case, there is an action of a general linear group on our flag space, that the stabilizer of the standard flag is the subgroup $B$ of upper triangular matrices, and that the Bruhat decomposition holds, meaning that our space is covered by the disjoint Schubert cells $\sh(B \sigma B) / B$ indexed by permutations $\sigma$ of an infinite set. Finally, in the case of flags indexed by the ordinal $\omega + 1$, we define an analog of the Bruhat order on this infinite permutation group and prove that when $k$ is a domain, Ehresmann's closure relations still hold, i.e. that the closure $\overline{\sh(B \sigma B) / B}$ is covered by the Schubert cells indexed by permutations smaller than $\sigma$ in the infinite Bruhat order.

Jack J Garzella (University of California San Diego)

Title: A gpu-accelerated algorithm for the quasi-F-split Fedder type criterion

Abstract: In this work in progress, we recast the Fedder's criterion of Kawakami, Takamatsu, and Yoshikawa in terms of matrix multiplication, allowing for a faster algorithm. We implement a gpu-accelerated polynomial multiplication and matrix multiplication to implement the algorithm. We use this to find examples of K3 surfaces with any quasi-F-split height over \mathbb{F}_5.

Javier González Anaya  (Harvey Mudd College)

Title: Moduli spaces of points in flags of affine spaces and polymatroids

Abstract: We introduce two novel moduli spaces of labeled points in flags of affine spaces. The first moduli space parametrizes distinct weighted points, with configurations defined up to translation and scaling. The second moduli space allows points to collide freely, without any notion of equivalence between configurations. We demonstrate that the first moduli space admits a toric compactification which coincides with the polypermutohedral variety of Crowly-Huh-Larson-Simpson-Wang. Conversely, the second moduli space is toric and coincides with the polystellahedral variety of Eur-Larson. We also study the relation between our moduli spaces and Hassett's compactifications of M_{0,n} and Chen-Gibney-Krashens moduli space of points in affine space. This is joint work with P. Gallardo and J.L. Gonzalez.

Aaron Goodwin (University of California Riverside)

Title: Compactifications of Moduli Spaces of Plane Curves with Marked Points

Abstract: We study compactifications of the moduli space of a degree d plane curve marked by n labeled points up to projective equivalence via Geometric Invariant Theory (GIT). For cubic curves we provide a complete description of the GIT walls and show that the moduli-theoretic wall-crossing can be understood through analysis of the singularities of the plane curves and the position of the points.

Zengrui Han (Rutgers University)

Title: Stringy Hodge numbers of Pfaffian double mirrors and Homological Projective Duality

Abstract: The Pfaffian double mirrors provide historically the first example of derived equivalent but not birationally equivalent Calabi-Yau manifolds. This construction fits into a conceptual framework of Kuznetsov called Homological Projective Duality. In this project we study the relationship between the Hodge-theoretic aspects and homological aspects of Pfaffian double mirrors. More precisely, we obtain results on the stringy Hodge numbers of Pfaffian double mirrors, and use them to make numerical prediction on the Lefschetz decomposition of categorical crepant resolution of Pfaffian varieties. 

Daigo Ito (University of California Berkeley)

Title: A new proof of the Bondal-Orlov reconstruction and its implications

Abstract: Bondal-Orlov and Ballard showed that a Gorenstein (anti-)Fano variety X can be reconstructed from the triangulated category structure of its perfect derived category. On the other hand, Balmer showed that for any variety X, it is possible to reconstruct X by considering the tensor triangulated structure on its derived category using the derived tensor product. In this poster, we will observe how much of Balmer’s reconstruction can be understood without the monoidal structure, using the recently introduced Matsui spectrum of a triangulated category. From this perspective, we provide a new proof of the reconstruction theorems of Bondal-Orlov and Ballard as well as related results by Favero. We will also discuss a natural framework that arises from those observations.

Zhiyuan Jiang (University of California San Diego)

Title: Kaehler Abundance via Algebraic Reduction

Abstract: There has been a lot of recent exciting activity aimed at establishing the MMP for Kaehler varieties.  By results of Campana, Das, Hacon, H\”oring, Paun, Peternell, and Ou, we know abundance and the MMP for threefolds and we have partial results for the MMP for fourfolds. We introduce a new approach to establish abundance for Kaehler varieties.  The goal is to reduce abundance for Kaehler varieties to abundance for projective varieties, using the algebraic reduction map.

Alexandros Kafkas (Purdue University)

Title: Residues of Logarithmic Connections and the Equivariant Riemann-Roch Formula

Abstract: Given a finite group action on a nonsingular projective surface $X$, a line bundle $L$ on $X$ is called equivariant if the action of $G$ extends to $L$ in a compatible way. This induces a $G$-action on the cohomology groups $H^*(X, L)$. The virtual Euler characteristic of $L$ is the character of the virtual representation $H^0(X, L) - H^1(X, L) + H^2(X, L)$ of $G$. We will give an approach to calculating the virtual Euler characteristic by using logarithmic connections and a residue theorem.

Irit Huq-Kuruvilla (Virginia Tech University)

Title: Quantum K Rings of Partial Flags, Coulomb Branches, and the Bethe Ansatz

Abstract: There are two major predictions for the quantum K ring of a partial flag: One comes from quantum field theory, where the ring is the OPE ring associated to a 3D GLSM. The other comes from work identifying the quasimap version of the ring with the Bethe algebra associated to a quantum integrable system. These predictions have a mysterious set of equations in common, which can be regarded as a special case of Nekrasov-Shatashvili's gauge/Bethe correspondence. We give a purely geometric explanation for this coincidence in terms of the abelian/non-abelian correspondence and use it to prove both sets of predictions.

Ruoxi Li (University of Pittsburgh)

Title: Motivic classes in finite characteristic with applications to Higgs bundles and bundles with connections

Abstract: We will first discuss the motivations of motivic classes from point counting. Then we will give the definitions of the motivic classes of schemes, in particular we mention the condition in finite characteristic. We will introduce symmetric powers and motivic zeta functions, and the measure of the function will be a local zeta function for finite fields. For the second part, we will focus on the motivic classes of stacks and the relations between two motivic classes. In particular, we will give the formulae for the volumes of moduli of bundles with connections.

Haggai Liu (Simon Fraser University)

Title: Moduli Spaces of Weighted Stable Curves and their Fundamental Groups

Abstract: The Deligne-Mumford compactification, $\overline{M_{0,n}}$, of the moduli space of $n$ distinct ordered points on $\mathbb{P}^1$, has many well understood geometric and topological properties. For example, it is a smooth projective variety over its base field. Many interesting properties are known for the manifold $\overline{M_{0,n}}(\mathbb{R})$ of real points of this variety. In particular, its fundamental group, $\pi_1(\overline{M_{0,n}}(\mathbb{R}))$, is related, via a short exact sequence, to another group known as the cactus group. Henriques and Kamnitzer gave an elegant combinatorial presentation of this cactus group. 

In 2003, Hassett constructed a weighted variant of $\overline{M_{0,n}}(\mathbb{R})$: For each of the $n$ labels, we assign a weight between 0 and 1; points can coincide if the sum of their weights does not exceed one. We seek combinatorial presentations for the fundamental groups of Hassett spaces with certain restrictions on the weights. 

In particular, we express the Hassett space as a blow-down of $\overline{M_{0,n}}$ and modify the cactus group to produce an analogous short exact sequence. The relations of this modified cactus group involves extensions to the braid relations in $S_n$. To establish the sufficiency of such relations, we consider a certain cell decomposition of these Hassett spaces, which are indexed by ordered planar trees.

Arjun Nigam (Duke University)

Title: Lines on the intersection of degree $n-2$ hyperplanes in $\mathbb{P}^n_k$

Abstract: Let $V$ be a rank $n$ vector bundle over a smooth $S$-scheme $X$ of dimension $n$, $s$ a section of $V$ with zero locus $Z$, and $E$ a motivic ring spectrum over $S$. One can define the Euler class $e(V,s,E)$ as a morphism in the category $SH(X)$ of $\mathbb{A}^1$-motivic spectra over $X$. An analogous definition can be made when $E$ is a $\mathbb{P}^1$-motivic ring spectra. Under nice conditions, we can push forward the Euler class to the category of $\mathbb{A}^1$-motivic spectra/$\mathbb{P}^1$-motivic spectra over $S$. This pushforward is the Euler number, and when $X$ is a moduli space, the Euler numbers of certain vector bundles encode interesting enumerative results. We consider a specific moduli problem and explicitly compute the associated Euler number. 

Yu Shen (Michigan State University)

Title: Morita theory on root gerbes

Abstract: We study Morita theory of Azumaya algebras on root gerbes X . There, we find explicit equivalent conditions for Morita equivalence. During this study, we find examples of a decomposable category become indecomposable after a Brauer twist.

Benjamin Tighe (University of Oregon)

Title: A curious family of Calabi-Yau 3-Folds

Abstract: In 1994, Aspinwall and Morrison introduced a family of Calabi-Yau 3-folds arising as an intermediary between the Dwork pencil and its mirror quintic family.  The Hodge theory of this family was subsequently studied by Szendroi.  He showed that the rational period map has degree at least 5 and later conjectured the members of a general fiber are not birational nor derived equivalent.  In joint work with Nicolas Addington, we confirm this conjecture by showing the integral period map is generically injective.  I will outline this result and discuss why it is surprising. 

Sridhar Venkatesh (University of Michigan)

Title: The intersection cohomology Hodge module on toric varieties

Abstract: The intersection cohomology complex IC_X on a toric variety X has been well studied starting with the works of Stanley and Fieseler, and more recently, the works of de Cataldo-Migliorini-Mustata and Saito. However, it has a richer structure as a Hodge module (denoted IC^H_X) in the sense of Saito’s theory, and so we have the graded de Rham complexes gr_k(DR(IC^H_X)), which are complexes of coherent sheaves carrying significant information about X. We will describe the generating function of the cohomology sheaves of gr_k(DR(IC^H_X)) and give a precise formula relating it with the stalks of the perverse sheaf IC_X (in particular, this implies that the generating function depends only on the combinatorial data of the toric variety). This is joint work with Hyunsuk Kim.

Xiaohan Yan (IMJ-PRG, Sorbonne Université)

Title: Serre Duality in Quantum K-Theory 

Abstract: In Gromov-Witten (GW) theory, a "quantum" version of Serre duality connects the GW invariants of a subvariety with those of the total space of its dual bundle (usually a simpler target), and has applications in mirror symmetry and GLSMs. We extend such duality to K-theoretic settings and prove it for primitive bundles in genus-zero K-theoretic GW theory, where the enumerative invariants are defined as holomorphic Euler characteristics on the moduli stacks of stable maps. Moreover, the assumption on primitivity can be removed for GIT quotients like flag varieties through consideration of torus-equivariant theory. This poster is based upon work in arXiv:2401.03054.