Date: February 27.
Speaker: Will Traves (U.S. Naval Academy)
Title: Interpolation and Incidence
Abstract: I will give an overview of several geometric problems and how their solutions depend on incidence results involving linear spaces. Extending results of Pappus and Pascal on conic curves, we'll look at an incidence construction (developed jointly with David Wehlau) that determines when 10 points lie on a cubic curve. Our construction uses an important tool from algebraic geometry: the Cayley-Bacharach Theorem. In the 18th century Hermann Grassmann came up with his own solution to this problem that used only elementary methods. I'll explain Grassmann's approach and show how to extend his results. In particular, we will see that a simple extension of his ideas allows us to locate the base-point of a 1-parameter family of cubic curves with just a straightedge: if two cubic curves intersect in 8 known points and we know one additional point on each curve, we can locate their ninth point of intersection with an unmarked ruler. We will also consider a three-dimensional geometry problem originally posed at l'Academie de Bruxelles in 1825. This problem asks for an incidence construction that determines whether 10 points lie on a quadric surface. I'll give an overview of my recent solution to the Bruxelles problem and point to many other open problems in this area.
Date: March 13. (Postponed)
Speaker: Jessie Loucks-Tavitas (California State University Sacramento)
Title: Algebra and geometry of camera resectioning
Abstract: Algebraic vision, lying in the intersection of computer vision and projective geometry, is the study of three-dimensional objects being photographed by multiple pinhole cameras. Three natural questions arise:
Triangulation: Given multiple images as well as (relative) camera locations, can we reconstruct the scene or object being photographed?
Resectioning: Given a 3-D object or scene and multiple images of it, can we determine the (relative) positions of the cameras in the world?
Structure-from-motion: Given only 2D images, can we recover both the camera positions and the object being imaged?
We will discuss and characterize certain algebraic varieties associated with the camera resectioning problem. As an application, we will derive and re-interpret celebrated results in computer vision due to Carlsson, Weinshall, and others related to camera-point duality. This is joint work with Erin Connelly and Timothy Duff.
Date: March 27.
Speaker: Kelly O'Connor (Rose-Hulman Institute of Technology)
Title: Shapes of Rank 2 Unit Lattices
Abstract: The unit group of the ring of integers of a number field K is a finitely generated abelian group which, modulo its torsion subgroup, can be endowed with the structure of a lattice. The rank of this lattice is determined by the number of real and complex embeddings of K, and its shape is determined by the lattice up to scaling, rotation, and reflection. After discussing the various objects at play in the study of rank 2 unit lattices, I will give an overview of recent work on the lattices arising from the unit groups of D_4-quartic number fields with signature (2,1) and the fantastic experience of working on this project with my collaborators-many of whom are faculty at PUIs!
Date: April 3.
Speaker: Fei Ye (CUNY Queensborough Community College)
Title: Decomposition of higher Jacobian ideals of hypersurfaces
Abstract: Recently, higher Nash blowups and higher Jacobian ideas have gained a lot of attention. Like Jacobian ideals, higher Jacobian ideals can be defined as the Fitting ideals of higher order differentials. For hypersurfaces, they can be explicitly described as the ideals of maximal minors of higher Jacobian matrices. In this talk, we will focus on the structure of the second-order Jacobian ideal of a hypersurface and show that it has a power of the first Jacobian ideal as a factor. Using this structure, one can show that the second Nash blowup algebras are contact invariant.
Date: April 24.
Speaker: Reginald Anderson (Claremont McKenna College)
Title: Enumerative Invariants from Derived Categories
Abstract: The study of enumerative invariants dates back at least as far as Euclid’s work circa 300 BC, who observed that through two distinct points in the plane there is a unique line. In 1849, Cayley-Salmon found that there are 27 lines on a nonsingular cubic surface. In 1879, Schubert found that there are 2875 lines on a generic non-singular quintic threefold; Katz correctly counted 609250 conics in a generic nonsingular quintic threefold in 1986. In 1991, physicists Candelas-de la Ossa-Green-Parkes gave a generating function for genus 0 Gromov-Witten invariants of a generic non-singular quintic threefold by studying the mirror space. This observation represented a change in our approach to enumerative problems by counting rational degree d curves inside of the quintic threefold “all at once;” other landmark achievements in modern enumerative geometry include Kontsevich-Manin’s recursive formula for the number of rational plane curves. From the perspective of homological mirror symmetry, enumerative invariants come from the Hochschild cohomology of the Fukaya category. I’m interested in a different question, which asks what enumerative data can be gleaned from the bounded derived category of coherent sheaves. I’ll share results on giving presentations of derived categories, and if time allows, will describe Kalashnikov’s method to recover Givental’s small J-function and the genus 0 Gromov-Witten potential for CP^1 by viewing it as a toric quiver variety associated to the Kronecker quiver; i.e., from a presentation of the bounded derived category of coherent sheaves.
Date: May 1.
Speaker: Emily Clader (San Francisco State University)
Title: Nontautological Cycles on Moduli Spaces of Smooth Curves
Abstract: The cohomology of the moduli space of stable curves has been widely studied, but in general, understanding the full cohomology ring of this space is too much to ask. Instead, one generally settles for studying the tautological ring, a subring of the cohomology that is simultaneously tractable to study and yet rich enough to contain most cohomology classes of geometric interest. The first known example of an algebraic cohomology class that is *not* tautological was discovered by Graber and Pandharipande, in work that was later significantly generalized by van Zelm to produce an infinite family of non-tautological classes on the moduli space of stable curves. A similar study can be undertaken on the moduli space of smooth curves, but in this case, almost no non-tautological classes were previously known. I will report on joint work with V. Arena, S. Canning, R. Haburcak, A. Li, S.C. Mok, and C. Tamborini (from the 2023 AGNES Summer School), in which we produce non-tautological algebraic classes on the moduli space of smooth curves in an infinite family of cases, including on M_g for all g>15.
Date: September 20.
Speaker: Adam Boocher (University of San Diego)
Title: From Classical Commutative Algebra to Some Special Diophantine Equations
Abstract: One of the first theorems one learns in a commutative algebra class is the "Principal Ideal Theorem" which essentially says that in a system of polynomial equations, the codimension of the corresponding variety is never more than the number of equations. In this talk, I'll share a bit of history and explain how this Theorem is actually the start of a rich story. The focus will be on two rather mysterious conjectures about betti numbers and I will explain an attempt to study them in a special case, using the relatively new techniques of Boij-Soederberg Theory. This approach leads to some interesting diophantine equations, which may shed light on the original conjectures. This is joint work with my two undergraduate students Noah Huang and Harrison Wolf.
Date: October 4.
Speaker: John Little (Holycross)
Title: Rational “multi-lump” solutions of the KP equation from cuspidal algebraic curves
Abstract: We discuss an application of algebraic geometry to the construction of certain solutions of a PDE modeling solitary wave (soliton) phenomena in two space dimensions and time (the Kadomtsev-Petviashvili, or KP equation). The connection between the theta functions associated with the Jacobians of (smooth) algebraic curves and solutions of this equation was a major development of the 1970's; in the 1980's, this provided significant progress toward a solution of the Schottky problem. In this talk we will discuss recent work on obtaining rational "multi-lump" KP solutions from cuspidal singular curves in a parallel way building on results obtained by the speaker together with Daniele Agostini, and Turku Ozlum Celik. Through discussions of the context and detailed explicit examples, we will strive to make this understandable for non-experts in this area.
Date: October 18.
Speaker: Ursula Whitcher (Mathematical Reviews)
Title: The divisors under the rainbow
Abstract: Adinkras are decorated graphs that encapsulate information about the physics of supersymmetry. If we color the edges of an Adinkra with a rainbow of shades in a specific order, we obtain an associated algebraic curve. We use this structure to characterize height functions on Adinkras, then show how to encapsulate the same information using data from our rainbow. We give a complete description of the map in the case of the 5-dimensional hypercube Adinkra. This talk describes joint work with Amanda Francis.
Date: November 1.
Speaker: Gabriel Dorfsman-Hopkins (St. Lawrence University)
Title: A Condensed Approach to Continuous Group Cohomology
Abstract: A common issue one can run into is that topological algebra doesn't play well with homological algebra: for example, the category of topological abelian groups is not an abelian category. This introduces many pathologies, including that continuous group cohomology does not give rise to long exact sequences. Clausen and Scholze (resp, Barwick and Haine) suggest a solution to this kind of problem by extending the category of topological abelian groups to the abelian category of condensed abelian groups (resp. pyknotic abelian groups), which are sheaves of abelian groups on the pro-etale site of a geometric point. We will explain a construction of continuous group cohomology in this condensed setting, give comparisons to classical continuous group cohomology, and explain some consequences and future directions that fall out of this formalism (including, if time allows, to p-adic and perfectoid geometry).
Date: November 22.
Speaker: Edray Goins (Pomona)
Title: Quasi-Critical Points of Toroidal Bely\u{\i} Maps
Abstract: A Bely\u{\i} map $\beta: \mathbb{P}^1(\mathbb{C}) \to \mathbb{P}^1(\mathbb{C})$ is a rational function with at most three critical values; we may assume these values are $\{ 0, \, 1, \, \infty \}$. Replacing $\mathbb{P}^1$ with an elliptic curve $E: \ y^2 = x^3 + A \, x + B$, there is a similar definition of a Bely\u{\i} map $\beta: E(\mathbb{C}) \to \mathbb{P}^1(\mathbb{C})$. Since $E(\mathbb{C}) \simeq \mathbb T^2(\mathbb {R})$ is a torus, we call $(E, \beta)$ a Toroidal Bely\u{\i} pair.
There are many examples of Bely\u{\i} maps $\beta: E(\mathbb{C}) \to \mathbb P^1(\mathbb{C})$ associated to elliptic curves; several can be found online at \texttt{LMFDB}. Given such a Toroidal Bely\u{\i} map of degree $N$, the inverse image $G = \beta^{-1} \bigl( \{ 0, \, 1, \, \infty \} \bigr)$ is a set of $N$ elements which contains the critical points of the Bely\u{\i} map. In this project, we investigate when $G$ is contained in $E(\mathbb{C})_{\text{tors}}$. This is joint work with Tesfa Asmara (Pomona College), Erik Imathiu-Jones (California Institute of Technology), Maria Maalouf (California State University at Long Beach), Isaac Robinson (Harvard University), and Sharon Sneha Spaulding (University of Connecticut). This was work done as part of the Pomona Research in Mathematics Experience (NSA \texttt{H98230-21-1-0015}).
Date: December 6.
Speaker: Youngsu Kim (California State University San Bernardino)
Title: Abelian Sandpile Models
Abstract: In this talk, we introduce abelian sandpile groups and their connections to graph theory and algebra. Sandpile models were invented by Bak, Tang, and Wiesenfeld to study self-organized criticality in dynamical systems. Its algebraic interpretation, called critical or Picard groups, can be defined by the chip-firing game played on certain graphs. These are finitely generated abelian groups, and some of their properties, such as rank and order (of their torsion subgroups), can be deduced from their Smith normal forms. We provide Smith normal forms of critical groups for several classes of directed graphs. This is joint work with J. Jun and M. Pisano.
Date: Feb. 6.
Speaker: David Swinarski (Fordham University)
Title: Some singular curves in Mukai's model of $\overline{M}_7$
Abstract: In the 1990s Mukai introduced birational models of the moduli spaces $\overline{M}_g$ for $g = 7, 8, 9$. For example, the GIT quotient Gr(7,16)//Spin(10) is a birational model of the moduli space of Deligne-Mumford stable genus 7 curves $\overline{M}_7$. The key observation is that a general smooth genus 7 curve can be realized as the intersection of the orthogonal Grassmannian OG(5,10) in $\mathbb{P}^{15}$ with a six-dimensional projective linear subspace. In this AG@PUI talk, I will discuss many of the background ideas in Mukai's construction. This is a warmup to my recent preprint arXiv:2304.12936, where I give examples of singular curves that are GIT semistable in Mukai's construction.
Date: Feb. 20.
Speaker: Dagan Karp (Harvey Mudd College)
Title: Tropical Linear Series
Abstract: In this talk I'll attempt to give a friendly and example-driven introduction to the theory of linear series on tropical curves. While in some respects mirroring the classical study of linear series, in the tropical setting there are many surprises and even basic questions remain open. This work is joint with Chang Chih-Wei, Hernan Iriarte, David Jensen, Sam Payne, and Jidong Wang.
Date: Mar. 5.
Speaker: Harpreet Bedi (Alfred University)
Title: Modulo p equivalence of categories
Abstract: An equivalence between categories of Char 0 and Char p is constructed via a modulo p map. No Witt vectors or perfectoid spaces will be harmed, but the constructions will be similar.
Date: Mar. 19.
Speaker: Javier Gonzalez Anaya (Harvey Mudd College)
Title: Higher-dimensional Losev-Manin spaces and their geometry
Abstract: The classical Losev-Manin space can be interpreted as a toric compactification of the moduli space of points in the affine line modulo translation and scaling. In this talk, we will discuss the geometry of higher-dimensional analogs of these moduli spaces, referred to as higher-dimensional Losev-Manin spaces. These varieties emerge as toric compactifications of the moduli space of distinct labeled points in affine space modulo translation and scaling, and possess their fair share of interesting geometric properties. For instance, they are locally trivial fibrations over a product of projective spaces, with fibers isomorphic to the Losev-Manin space. Additionally, they are isomorphic to the normalization of a Chow quotient. This is joint work with Patricio Gallardo, Jose Gonzalez, and Evangelos Routis.
Date: Apr. 16.
Speaker: Sebastian Bozlee (Fordham University)
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 stable curves such strata have long been constructed and are used throughout the theory of moduli of curves. 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: Apr. 30.
Speaker: David Zureick-Brown (Amherst College)
Title: The Canonical Ring of a Stacky Curve
Abstract: We give a generalization to stacks of the classical theorem of Petri -- i.e., we give a presentation for the canonical ring of a stacky curve. This is motivated by the following application: we give an explicit presentation for the ring of modular forms for a Fuchsian group with cofinite area, which depends on the signature of the group. (The talk will be mostly geometric and will require little understanding of modular forms; additionally, no prior knowledge of stacks will be needed to understand the talk.)
Date: Sep. 12.
Speaker: Kelly Jabbusch (University of Michigan - Dearborn)
Title: The minimal projective bundle dimension and toric 2-Fano manifolds
Abstract: In this talk we will discuss higher Fano manifolds, which are Fano manifolds with positive higher Chern characters. In particular we will focus on toric 2-Fano manifolds. Motivated by the problem of classifying toric 2-Fano manifolds, we will introduce a new invariant for smooth projective toric varieties, the minimal projective bundle dimension, m(X). This invariant m(X) captures the minimal degree of a dominating family of rational curves on X or, equivalently, the minimal length of a centrally symmetric primitive relation for the fan of X. We'll present a classification of smooth projective toric varieties with m(X) ≥ dim(X)-2, and show that projective spaces are the only 2-Fano manifolds among smooth projective toric varieties with m(X) equal to 1, 2, dim(X)-2, dim(X)-1, or dim(X). This is joint work with Carolina Araujo, Roya Beheshti, Ana-Maria Castravet, Svetlana Makarova, Enrica Mazzon, and Nivedita Viswanathan.
Date: Sep. 26.
Speaker: Jaiung Jun (SUNY New Paltz)
Title: Schemes over the natural numbers
Abstract: In 2006, Baker and Norine proved a Riemann-Roch theorem for finite graphs. To generalize this result to higher dimension, one is naturally led to study the scheme theory over the tropical semifield (or more generally schemes over natural numbers). I will introduce basic notions and examples for schemes over the natural numbers. Then, I will discuss several properties of line bundles and vector bundles in this setting. This is joint work with James Borger, Kalina Mincheva, and Jeffrey Tolliver.
Date: Oct. 10.
Speaker: Jason Lo (California State University, Northridge)
Title: Stability of line bundles and the deformed Hermitian-Yang-Mills equation on elliptic surfaces
Abstract: In the 1980s, Donaldson and Uhlenbeck-Yau established a correspondence between the existence of solutions to the Hermitian-Yang-Mills equation associated to a vector bundle on a compact Kahler manifold, and the Mumford-Takemoto stability (i.e. slope stability) of the vector bundle. Motivated by the recent developments of deformed Hermitian-Yang-Mills (dHYM) equations and Bridgeland stability conditions, Collins-Yau asked if a similar relation might hold between the dHYM equation and the Bridgeland stability of line bundles. In this talk, we will present a partial result on elliptic surfaces. This talk is based on joint work with Tristan Collins, Yun Shi, and Shing-Tung Yau.
Date: Oct. 24.
Speaker: Han-Bom Moon (Fordham University)
Title: Algebraic geometry over finite fields
Abstract: I will share my experience on an REU project during last summer. We explored two related projects, which can be described in an elementary way but are about the number of rational points on schemes defined over a finite field. The results are joint work with six students, Alana Campbell, Flora Dedvukaj, Ritik Jain, Donald McCormick III, Joshua Morales, and Peter Wu.
Date: Nov. 7.
Speaker: Mee Seong Im (United States Naval Academy)
Title: Correspondence between automata and one-dimensional Boolean topological theories and TQFTs
Abstract: Automata are important objects in theoretical computer science. I will describe how automata emerge from topological theories and TQFTs in dimension one and carrying defects. Conversely, given an automaton, there is a canonical Boolean TQFT associated with it. In those topological theories, one encounters pairs of a regular language and a circular regular language that describe the theory.
Date: Dec. 5.
Speaker: Nicola Tarasca (Virginia Commonwealth University)
Title: Coinvariants of metaplectic representations and abelian varieties
Abstract: Spaces of coinvariants have classically been constructed by assigning representations of affine Lie algebras, and more generally, vertex operator algebras, to pointed algebraic curves. Removing curves out of the picture, I will construct spaces of coinvariants at abelian varieties with respect to the action of an infinite-dimensional Lie algebra. I will show how these spaces globalize to twisted D-modules on moduli of abelian varieties, extending the classical picture from moduli of curves. This is based on the preprint arXiv:2301.13227.