RJ Acuna (Washington University)

Title: The points where the real regulator is trivial are isolated in toric families of elliptic curves.

Abstract: To a lattice polygon D in R2 with 0 as the unique interior lattice point we can assign a 1 parameter family of toric curves {a*phi(x,y) -1=0} inside C*2. Let pi:C->A1 be the family obtained by compactifying it. The toric symbol {-x,-y} lifts to a motivic class xi in H2(C,Z(2)) which restricts to xi_a  on the fibers of pi. To it we can assign its regulator class R_a in H1(C_a, C/Z(2)). Applying the Gauss-Manin connection in the direction a(d/da) gives a basis omega of the relative canonical sheaf. We can pair omega with a basis of H_1(C_a,Z) = Z<alpha,beta> and write int_alpha omega = r_alpha, r_beta = int_beta omega. Then nu(a) = r_beta(a) alpha +r_alpha(a) beta is an admissible normal function. The real regulator is the vector r(a) in R2 given by the imaginary parts of r_beta(a), and r_alpha(a). We’ve determined that the points a in A1 where the real regulator  r(a) = <0,0> are isolated. Moreover, for the analogous problem for genus g, we can show that the real regulator is nontrivial for smooth members of the family.

Haohua Deng (Duke)

Title: Hodge structures near infinity

Abstract: In the poster I will show by examples how the period maps degenerate near the boundary of moduli spaces and its geometric consequences. Some recent results on compactifying the period image will also be introduced.

Anne Fayolle (Utah)

Title: Tame ramification and centers of $F$-purity

Abstract: I introduce a notion of tame ramification for general finite covers. When specialized to the separable case, it extends to higher dimensions the classical notion of tame ramification for Dedekind domains and curves and sits nicely in between other notions of tame ramification in arithmetic geometry. However, when applied to the Frobenius map, it naturally yields the notion of centers of $F$-purity, which lets us describe how these behave under finite covers —it all comes down to a transitivity property for tame ramification in towers. This is joint work with Javier Carvajal-Rojas

Fernando Figueroa (Princeton)

Title: Bounds for many T-singularities in stable surfaces.

Abstract: T-singularities are the two-dimensional singularities that admit a one-parameter Q-Gorensetein smoothing. These are precisely the singularities showing up in a normal degeneration of canonical surfaces in the KSBA moduli space. If we fix K^2 and Chi, then there is a finite list of the possible T-singularities in the surfaces parametrized by the KSBA moduli space. Explicitly writing down this list is a complicated question. Previously Rana and Urzúa have classified the case of surfaces with one T-singularity. 

In the poster we present a bound for the case of multiple T-singularities, based on the configurations appearing in the exceptional divisor for the minimal resolution.

This is based on joint work with J. Rana and G. Urzúa.

Alex Galarraga and Alex Wang (U. Washington)

Title: Classification of Degree Sets of Superelliptic Curves over Henselian Fields

Abstract: Given a variety V over a field K without a K-rational point, a natural question to ask is if we can find points of V over some extension L/K.  The degree set D(V) over K is defined to be the set of all degrees of extensions L such that V has a point over L. When V is a superelliptic curve and K is the field of fractions of a Henselian ring R, we give a method for computing the degree set, and as a result show that these degree sets are either a union of positive integer multiples or cofinite in the naturals. Further, we explicitly compute all possible degree sets of a hyperelliptic curve over a strictly Henselian field with genus g < 13.

Qi Ge (U. Alberta)

Title: Formalization of aspects of arithmetic geometry

Abstract: We present the formalization of certain results related to Galois group of local fields and their representations in Lean 4.

Javier Gonzalez Anaya (Harvey Mudd College)

Title: Higher-dimensional Losev-Manin spaces

Abstract: The classical Losev-Manin space can be interpreted as a toric compactification of the moduli space of n points in the affine line modulo translation and scaling. Motivated by this, we study its higher-dimensional toric counterparts, which compactify the moduli space of n distinct labeled points in affine space modulo translation and scaling. We show that these moduli spaces are a fibration over a product of projective spaces, that they are isomorphic to the normalization of a Chow quotient, and prove that a related generalization of the moduli space of pointed rational curves proposed by Chen, Gibney, and Krashen is not a Mori dream space for n>>1. This is joint work with J.L. Gonzalez, P. Gallardo and E. Routis.

Sean Griffin (UC Davis)

Title: Partial resolutions of the nilpotent cone, and the Delta Conjecture

Abstract: The Delta Conjecture gives two combinatorial formulas for the bigraded Hilbert series (and more generally the bigraded Frobenius series) of the global sections of a certain vector bundle on the Hilbert scheme of n points in the plane. One of these formulas, the "rise formula," has been proven independently by Blasiak, Haiman, Morse, Pun, and Seelinger, and by D'Adderio and Mellit. In this poster, I will explain how the theory of partial resolutions of the nilpotent cone of Borho and MacPherson can be used to give new formulas for the t=0 part of the Delta Conjecture.

Based on joint work with Maria Gillespie.

Aaron Goodwin (UC Riverside)

Title: Compactifications of the Moduli of Pointed Plane Curves

Abstract: We study compactifications of moduli spaces of smooth curves in the plane with marked points. In particular, we give results for GIT compactifications of pointed plane cubics.

Matthew Manchung Hase-Liu (Columbia U)

Title: The mapping space of a smooth projective curve to a smooth hypersurface of low degree

Abstract: Browning and Vishe analyzed the space of rational curves on smooth hypersurfaces of low degree by a clever use of spreading out and an application of the Hardy-Littlewood circle method. We reinterpret the circle method geometrically, allowing us to prove a generalization for a fixed smooth projective curve.

Farid Hosseinijafari (Purdue)

Title: Eisenstein Cohomology and Special Values of L-functions

Abstract :In this poster, I present the construction of rank-one Eisenstein cohomology for the split group of type $G_2$ over a totally imaginary field. Building upon the foundational work established by Harder and Raghuram, I showcase my findings concerning the rationality of the ratio of special values of automorphic L-functions associated with $G_2$ using the Langlands-Shahidi method, within the framework of rank-one Eisenstein cohomology theory.

Casey Hill  (U. Kentucky)

Title: Computing the Algebra of Conformal Blocks for SL_4

Abstract: Conformal blocks are finite-dimensional vector spaces that arise from the WZNW model of conformal field theory. These have applications in algebraic geometry, particularly in describing the moduli of principal bundles and the moduli of curves. We will discuss recent progress on computing a presentation of the algebra of conformal blocks for SL_4. We also describe equations, the tropical variety, and a large family of toric degenerations for the case of a cone with genus 0 and 3 marked points.

Jae Hwang Lee (Colorado State)

Title: A Quantum H*(G)-module via Quasimap Invariants

Abstract: For X a smooth variety or Deligne-Mumford stack, the quantum cohomology ring QH*(X) is a deformation of the usual cohomology ring H*(X), where the product structure is modified to incorporate quantum corrections. These correction terms are defined using Gromov-Witten invariants. For a GIT quotient V//G, the cohomology ring H*(V//G) also has the structure of a H*(G)-module. In this work, we use quasimap invariants with light points and a modified version of the WDVV equation to define a quantum deformation of this H*(G)-module structure. Using localization, we explicitly compute this structure for the Hirzebruch surface of type 2, which is a semipositive variety. We conjecture that this brand-new quasimap quantum module structure is isomorphic to the Batyrev ring when the target is a semipositive toric variety. This would be an interesting geometric interpretation of the Batyrev ring for the semipositive case.

Haggai Liu (SFU) 

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.

We study a weighted variant of $\overline{M_{0,n}}(\mathbb{R})$ known as a Hassett space: 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. Our goal is to find combinatorial presentations for the fundamental groups of Hassett spaces with certain restrictions on the weights. To proceed with our goal, we use two main approaches: The first approach is to recursively compute them using blowups, Seifert Van-Kampen, and knowledge for smaller n. The second approach is to express the Hassett space as a blow-down of $\overline{M_{0,n}}$ and  modify the cactus group directly.

Peter Mcdonald (Utah)

Title: Multiplier ideals and klt singularities via (derived) splittings

Abstract: This project gives a new characterization of the multiplier of a variety X by looking at images of maps R\pi_*\omega_Y=\pi_*\omega_Y\to O_X where \pi:Y\to X ranges over all regular alterations. As a corollary to this result, we give a derived splinter characterization of klt singularities akin to the characterization of rational singularities given by Bhatt and Kovács.

Lucas Mioranci (UIC)

Title: Algebraic hyperbolicity of very general hypersurfaces in homogeneous varieties

Abstract: A complex projective variety X is (Brody) hyperbolic when it admits no nonconstant holomorphic map \mathbb{C}\to X, that is, when it contains no entire curves. In dimension one, hyperbolic curves are those with genus greater or equal to 2. In higher dimensions, it is a difficult and important problem to characterize hyperbolic varieties. It motivates celebrated conjectures such as the Lang Conjectures, Green-Griffiths Conjecture, and Manin’s Conjecture.

Algebraic hyperbolicity has been introduced as an algebraic analogue for hyperbolicity: we say X is algebraically hyperbolic if there exists an ample divisor H and a real number \epsilon > 0 such that the geometric genus g(C) and the degree of any integral curve C\subset X satisfy the inequality

 2g(C) - 2\ge \epsilon \deg_H (C). 

In particular, algebraically hyperbolic varieties do not contain any rational or elliptic curves. Every hyperbolic variety is algebraically hyperbolic, and Demailly conjectured that the converse holds. 

The algebraic hyperbolicity of very general hypersurfaces in projective space is almost completely classified by the results of Clemens, Ein, Voisin, Pacienza, Coskun and Riedl, and Yeong. By building on their techniques, I extended the classification to the much more general case of homogeneous varieties, thus obtaining explicit bounds for the hyperbolicity in plenty of open cases, including Grassmannians, flag varieties, and their products.

Kelly O'Connor (Colorado State)

Title: Relative Oriented Class Groups of Quadratic Extensions

Abstract: In 2019 Zemková defined relative oriented class groups associated to quadratic extensions of number fields L/K, extending work of Bhargava concerning composition laws for binary quadratic forms over number fields of higher degree. Indeed, this work generalized the classical correspondence between the ideal class group of a quadratic number field and classes of binary quadratic forms to any base number field of narrow class number one. Zemková explicitly computed these relative oriented class groups for totally real quadratic extensions of the rationals. We extend Zemková's result for quadratic extensions L/K of a totally real field of narrow class number 1, under certain conditions on the absolute Galois group of L/Q.

Swaraj Sridhar Pande (U. Michigan)

Title: A Frobenius version of Tian's alpha-invariant.

Abstract: Tian introduced the analytic definition of the alpha-invariant of a Fano manifold to study K-stability. This was later reinterpreted by Demailly in terms of log canonical thresholds, a singularity invariant of divisors on the Fano variety. We will present a positive characteristic version of the alpha-invariant, which relies on replacing the log canonical threshold with the F-pure threshold.

Sun Woo Park (U. Wisconsin)

Title: A probabilistic and a geometric approach to prime Selmer groups of cyclic prime twist families of elliptic curves over global function fields

Abstract: Let $K = \mathbb{F}_q(t)$ be the global function field over the finite field $\mathbb{F}_q$ of characteristic coprime to $2$ and $3$. Let $E$ be a non-isotrivial elliptic curve over $K$. Fix a prime number $l$ such that $\mu_l \subset \mathbb{F}_q$. Assuming some mild conditions, we present two distinct methodologies - a probabilistic approach (utilizing Markov chains) and a geometric approach (utilizing Grothendieck-Lefschetz trace formula) - to verify the Bhargava-Lane-Lenstra-Poonen-Rains heuristics on the moments of prime Selmer groups of $(l-1)$ dimensional abelian varieties obtained from the kernel of the norm map from the Weil restriction of $E$ to $E$ with respect to cyclic order $l$ Galois extensions over $K$. 

Peikai Qi (Michigan State)

Title: Iwasawa lambda invariant and Massey products

Abstract: How does the class group of the number field change in field extensions? This question is too wild to have a uniform answer, but there are some situations where partial answers are known. I will compare two such situations. First, in Iwasawa theory, instead of considering a single field extension, one considers a tower of fields and estimates the size of the class groups in the tower in terms of some invariants called $\lambda$ and $\mu$. Second, in a paper of Lam-Liu-Sharifi-Wake-Wang, they relate the relative size of Iwasawa modules to values of a "generalized Bockstein map", and further relate these values to Massey products in Galois cohomology in some situations. I will compare these two approaches to give a description of the cyclotomic Iwasawa \lambda-invariant of some imaginary quadratic fields and other fields in terms of Massey products.

Debaditya Raychaudhury (U. Arizona)

Title: On the singularities of secant varieties

Abstract: In this work, we study the singularities of secant varieties of smooth projective varieties when the embedding line bundle is sufficiently positive. We give a necessary and sufficient condition for these to have p-Du Bois singularities. In addition, we show that the singularities of these varieties are never higher rational. From our results, we deduce several consequences, including a Kodaira-Akizuki-Nakano type vanishing result for the reflexive differential forms of the secant varieties. Work in collaboration with S. Olano and L. Song.

Sharon Robins (SFU)

Title: Deformations of Smooth Complete Toric Varieties

Abstract: We can study how a given scheme X fits into a family using the tools from the deformation theory. One begins by using infinitesimal methods, studying possible obstructions, and attempting to construct a family called a versal deformation, which collects all possible deformations. If X is a smooth complete toric variety, combinatorial descriptions of the space of first-order deformations and the obstruction to second-order deformation given by the cup product have been studied. We describe a combinatorial iterative procedure for constructing the versal deformation of smooth complete toric varieties. This is joint work with Nathan Ilten.

Shahryar Roshan Zamir (U. Nebraska-Lincoln)

Title: Interpolation in the Weighted Projective Space

Abstract: Given a finite set of points X in the projective space over a field k one can ask for the k-vector space dimension of all degree d polynomials that vanish to order two on X. (These are polynomials whose first derivative vanishes on X.) The Alexander-Hirschowitz theorem (A-H) computes this dimension in terms of the multiplicity of the points and the k-vector space dimension of degree d monomials, with finitely many exceptions. We look at this question in the weighted projective line and space, P(s,t) and P(a,b,c). We define a notion of multiplicity for weighted spaces, give an example of P(a,b,c) where A-H holds with no exceptions and an infinite family where A-H fails for even one point.

Nolan Schock (UIC)

Title: The W(E_6)-invariant birational geometry of the moduli space of marked cubic surfaces

Abstract: The moduli space of marked cubic surfaces is one of the most classical moduli spaces in algebraic geometry, essentially dating back to the nineteenth century work of Cayley and Salmon studying the 27 lines on a cubic surface. Building on works of Naruki, Hacking-Keel-Tevelev, Naruki, and Gallardo-Kerr-Schaffler, we explicitly describe the compactifications of the moduli space of marked cubic surfaces by KSBA weighted stable pairs. We then study the W(E_6)-invariant birational geometry of these compactifications. Namely, we give generators for the cones of W(E_6)-invariant effective curves and divisors, a complete stable base locus decomposition of the W(E_6)-invariant effective cone (for small weights) and a complete description of the log minimal model program with respect to the sum of the boundary and Eckardt divisors.

Mahrud Sayrafi (U. Minnesota)

Title: Splitting of vector bundles on toric varieties

Abstract: In 1964, Horrocks proved that a vector bundle on a projective space splits as a sum of line bundles if and only if it has no intermediate cohomology. Generalizations of this criterion, under additional hypotheses, have been proven for other toric varieties, for instance by Eisenbud-Erman-Schreyer for products of projective spaces. This talk is about a splitting criterion for arbitrary smooth projective toric varieties.

Wanchun (Rosie) Shen, (Harvard)

Title: Higher rational and higher Du Bois singularities

 Abstract: Higher rational and higher Du Bois singularities have recently been introduced as natural refinements of rational and Du Bois singularities, and they are studied extensively in the local complete intersection (lci) case. We explain why their naïve extensions to the general case are not good notions to consider. Instead, we propose new definitions of k-rational and k-Du Bois singularities that agree with the existing definitions in the lci case, but are more flexible in general. We will establish some basic properties of k-rational and k-Du Bois singularities in this new sense, and in particular show that k-rational singularities are k-Du Bois. This is joint work with Sridhar Venkatesh and Anh Duc Vo.

Yu Shen (Michigan State)

Title: Derived categories of maximal Del Pezzo orders.

Abstract:  We consider the moduli functor of simple $A$-modules (with appropriate boundness conditions) for torsion-free simple algebra $A$. The coarse moduli space always exists for such a functor. For an exotic del Pezzo maximal order $A$ , we construct a fully faithful Fouerier- Mukai functor from the derived category of the moduli space into the derived category of $A$-modules. This is a new direction in studying derived categories of maximal orders.

Talon Stark (UCLA) 

Title: The cone conjecture in relative dimension 2

Abstract: We prove that, for a klt CY pair (X/S,\Delta) of relative dimension two, there are finitely many PsAut(X/S,\Delta)-orbits of Mori chambers and of faces of Mori chambers, where PsAut(X/S,\Delta) are the birational automorphisms of X over S that are isomorphisms in codimension 1 and preserve \Delta. This proves the geometric statement that there are finitely many isomorphism classes of small Q-factorial modifications (equivalently, for X terminal, minimal models) and rational contractions of X over S. Joint with Joaquín Moraga.

Arun Suresh (U. Missouri)

Title: Second moment of dihedral actions, incidence varieties and ensuring signal recovery.

Abstract: In this project we apply techniques from algebraic geometry and representation theory to solve a problem in signal processing that we call the generic crystallographic phase retrieval problem. Given a generic basis for R^N, the goal of the project is to recover an unknown M-sparse signal x in R^N (expressed with respect to this basis) from its periodic autocorrelation. We show that the autocorrelation manifests itself as the second moment of a specific dihedral action on R^N. We analyze the problem over irreducible sub-representations of this group action and reinterpret the problem as bounding the dimension of a certain incidence variety, which in turn allows us to determine the sparsity level at which recovery (both full and generic) is guaranteed.

This is based on joint work with Dr. Dan Edidin at the University of Missouri. 

Andrew Tawfeek (U. Washington)

Title: A Tropical Framework for Using Porteous' Formula

Abstract: Given a tropical cycle X, one can talk about a notion of "tropical" vector bundles on X having tropical fibers. By restricting our attention to bounded rational sections of these bundles, one can develop a good notion of characteristic classes that behave as expected classically. We present further results on these characteristic classes and use these properties to prove a Porteous' formula for these bundles, which gives a determinantal expression of the fundamental class of degeneracy loci in terms of tropical Chern classes. 

Benjamin Tighe (University of Oregon)

Title: Applications of the du Bois Complex to Symplectic Singularities

Abstract: The du Bois complex detects both local and global properties with respect to a singular algebraic variety X.  On the one hand, if X is proper, then the du Bois complex generates the Hodge filtration on the cohomology groups; on the other hand, recent work on hypersurfaces and complete intersections (MOPW, JKSY) has shown that the du Bois complex can detect the presence of singularities arising in the minimal model program.

I will outline how certain symmetries arise in the du Bois complex of a singular symplectic variety X and how these symmetries detect certain desirable properties of X.  Applications include the Hodge theory of (singular) irreducible holomorphic symplectic 4-folds and the (non-) existence of symplectic resolutions.

Joshua Turner (UC Davis)

Title: Homology of Affine Springer Fibers

Abstract:  Affine Springer fibers are geometric spaces often studied in geometric representation theory. They have strong connections to the Hilbert scheme of singular curves, as well as knot homology. In this poster, we will show some examples of what these affine Springer fibers look like and how to calculate their equivariant Borel-Moore homology.

Sridhar Venkatesh (U Michigan) 

Title: Higher rational and higher Du Bois singularities for cones and toric varieties.

Abstract: Higher rational and higher Du Bois singularities have recently been introduced as natural refinements of rational and Du Bois singularities, and they are studied extensively in the local complete intersection (lci) case. Inspired by these notions, we investigate the vanishing of the higher cohomologies of (the graded pieces of) the Du Bois complex, and the higher derived pushforwards $R^i f_*\Omega_{\tilde{X}}^p(\log E)$ in two typically non-lci settings: toric varieties and cones over smooth projective varieties. Here, $f:\tilde{X} \to X$ is a log resolution of singularities with exceptional divisor $E$ such that $f$ is an isomorphism over the smooth locus of $X$. This is part of joint work with Wanchun Shen and Anh Duc Vo.

Kaelyn Willingham (U Minnesota)

Title: Neural Network Theory through the lens of Tropical Geometry

Abstract: The field of Deep Learning has taken off in popularity in recent years, both in the academic & industrial sectors, and this is largely due to the sheer effectiveness of neural network modeling. However, a theoretical understanding of why neural networks are so effective remains elusive. Recent work has shown the potential for tropical geometry to be useful in building a rigorous theory of neural networks, and this poster will highlight some of that work. In particular, we'll see that ReLU neural networks can be built algebraically using tropical rational functions and geometrically using Newton polygons & zonotopes. We'll also see how these tropical constructions can allow us to study the nature of fitting for neural network models, as well as their geometric complexity. 

Yilong Zhang (Purdue)

Title: A log resolution of the dual variety of cubic threefold

Abstract: The dual variety of a smooth projective variety parameterizes hyperplane sections that are singular. When the variety is a hypersurface in projective space, the dual variety is also a hypersurface but highly singular. In joint work with Lisa Marquand, we proposed an explicit log resolution of the dual variety of a very general cubic threefold X. Our motivation is to study the rational map from (P^4)* to the moduli of cubic surfaces by sending a hyperplane H to X∩H.  This rational map extends to a morphism on the resolution we proposed.