All semesters: Spring 2025, Fall 2024 , Spring 2024, Fall 2023, Spring 2023, Fall 2022
Tuesdays 4-6, Evans 939
| date | speaker | title | abstract |
|---|---|---|---|
| 1/14 | Daniel Erman | Long Live the King (Conjecture) | The King Conjecture proposed that every toric variety has a full, strong exceptional collection of line bundles. While the conjecture turned out to be false, it has continued to inspire a huge amount of research on derived categories of toric varieties. I will explain how King’s Conjecture can be remedied, and proven, if one incorporates a birational geometry perspective. |
| Eric Larson | Normal bundles of rational curves in Grassmannians | Let \(C\) be a general rational curve of degree $d$ in a Grassmannian \(G(k, n)\). The natural expectation is that its normal bundle is balanced, i.e., isomorphic to \(\bigoplus \mathcal{O}(e_i)\) with all \(|e_i - e_j| \leq 1\). In this talk, I will describe several counterexamples to this expectation, propose a suitably revised conjecture, and describe recent progress towards this conjecture. | |
| 1/21 | Reed Jacobs | Borel-Weil Theory through Examples | Let \(V\) be the standard representation of \(\operatorname{SL}_2(\mathbb{C})\) acting on \(\mathbb{C}^2\).
One of the first theorems proved in a Lie theory course is that all the irreducible representations of \(\operatorname{SL}_2(\mathbb{C})\) are given by the symmetric powers \(V(n) := \operatorname{Sym}^n(V)\).
\(V(n)\) is the vector space of homogeneous degree \(n\) polynomials in \(2\) variables; this is also the global sections of the holomorphic line bundle \(\mathcal{O}(n)\) on \(\mathbb{C}\mathbb{P}^1\)!
This is the simplest example of Borel-Weil-Bott theory, which says all representations of complex semisimple Lie groups arise from the cohomology of holomorphic line bundles on a space constructed from the Lie group. I will review just enough Lie theory to state this, explain the construction, and do some simple examples. |
| Ben Church | Curves on complete intersections and measures of irrationality | Given a projective variety \(X\), it is always covered by curves obtained by taking the intersection with a linear subspace. We study whether there exist curves on \(X\) that have smaller numerical invariants than those of the linear slices. If \(X\) is a general complete intersection of large degrees, we show that there are no curves on \(X\) of smaller degree, nor are there curves of asymptotically smaller gonality. This verifies a folklore conjecture on the degrees of subvarieties of complete intersections as well as a conjecture of Bastianelli--De Poi--Ein--Lazarsfeld--Ullery on measures of irrationality for complete intersections. This is joint work with Nathan Chen and Junyan Zhao. | |
| 1/28 | Mahrud Sayrafi | Oda's Problem for Toric Projective Bundles | Given an ample line bundle \(\mathcal L\) on a smooth projective toric variety \(X\), whether the complete linear series of \(\mathcal L\) induces a projectively normal embedding is an interesting open problem in the intersection of multiple areas of mathematics. I will explain a connection with syzygies of truncations and the derived category of \(X\) and prove projective normality for a class of toric varieties. |
| Eugene Gorsky | Compactified Jacobians and affine Springer fibers | Given a plane curve singularity \(C\), one can define its compactified Jacobian as a certain moduli space of sheaves on \(C\). In the talk, I will define compactified Jacobians and review their properties, results and conjectures about them. | |
| 2/4 | Vivian Kuperberg | Sums of odd-ly many fractions | In this talk, I will discuss new bounds on constrained sets of fractions. Specifically, I will discuss the answer to the following question, which arises in several areas of number theory: for an integer \(k \ge 2\), consider the set of \(k\)-tuples of reduced fractions \(\frac{a_1}{q_1}, \dots, \frac{a_k}{q_k} \in I\), where \(I\) is an interval around \(0\). How many \(k\)-tuples are there with \(\sum_i \frac{a_i}{q_i} \in \mathbb Z\)? When \(k\) is even, the answer is well-known: the main contribution to the number of solutions comes from ``diagonal'' terms, where the fractions \(\frac{a_i}{q_i}\) cancel in pairs. When \(k\) is odd, the answer is much more mysterious! In work with Bloom, we prove a near-optimal upper bound on this problem when \(k\) is odd. I will also discuss applications of this problem to estimating moments of the distributions of primes and reduced residues. |
| 2/11 | |||
| 2/18 | Nate Gallup | Semigroup Graded Stillman's Conjecture | Stillman's conjecture (which has at least four proofs, see work of Ananyan-Hochster, Erman-Sam-Snowden, and Draisma-Lasoń-Leykin) states that there is a bound on the projective dimension of any homogeneous ideal in a standard graded polynomial ring which depends only on the number and degrees of the generators of the ideal, and not on the number of variables in the polynomial ring. We show that a family of polynomial rings graded by an abelian monoid supports a Stillman bound if and only if the monoid has bounded factorization. I will discuss our proof, along with extensions of Erman-Sam-Snowden's ultraproduct techniques to monoid-graded algebras. This is joint work with John Cobb (Auburn) and John Spoerl (UW Madison). |
| 2/25 | Noah Olander | Henselian pairs and weakly étale morphisms | The class of weakly étale morphisms of schemes is used by Bhatt and Scholze to define the pro-étale site of a scheme. We will review this notion and propose a new definition of weakly étale morphism which is analogous to the characterization of étale morphisms via a lifting property. We will use a result of Gabber on the cohomology of Henselian pairs to deduce the equivalence of the two definitions. If time permits, we will discuss an example of a weakly étale morphism which does not lift along a surjective ring map. This is joint work with Johan de Jong. |
| 3/4 | Hannah Larson | Tautological classes on strata of differentials | In this talk, I will introduce the strata of differentials \(P(\mu)\) and their tautological ring. For \(\mu\) a partition of \(2g - 2\), we define \(P(\mu)\) to be the moduli space of pointed genus \(g\) curves \((C, z_1, \dots , z_n)\) satisfying \(\sum \mu_i z_i \sim K_C\). I'll define the "tautological" classes in the Chow ring of \(P(\mu)\) as those that arise from certain universal line bundles for the moduli problem. Next, I'll explain several known relations among these classes, due to Dawei Chen, which imply that the tautological ring of \(P(\mu)\) is particularly simple. In fact, there is just one unanswered, concrete question that would determine the tautological ring. I'll finish by sharing my thoughts and a conjecture related to this problem, which is the topic of ongoing joint work with Dawei Chen. |
| 3/11 | Christopher O'Neill | Classifying numerical semigroups using polyhedral geometry | A numerical semigroup is a subset of the natural numbers that is closed under addition. There is a family of polyhedral cones \(C_m\), called Kunz cones, for which each numerical semigroup with smallest positive element m corresponds to an integer point in \(C_m\). It has been shown that if two numerical semigroups correspond to points in the same face of \(C_m\), they share many important properties, such as the number of minimal generators and the Betti numbers of their defining toric ideals. In this way, the faces of the Kunz cones naturally partition the set of all numerical semigroups into "cells" within which any two numerical semigroups have similar algebraic structure. In this talk, we discuss how studying Kunz cones can inform the classification of numerical semigroups, using complete intersection numerical semigroups as a case study. No familiarity with numerical semigroups or polyhedral geometry will be assumed for this talk. |
| Speaker 2 | Title 2 | Abstract 2 | |
| 3/18 | Joe Harris | Title 1 | Abstract 1 |
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| 3/25 | No seminar -- spring break | ||
| 4/1 | Shubham Sinha | Title 1 | Abstract 1 |
| Brian Yang | Title 2 | Abstract 2 | |
| 4/8 | Kabir Kapoor | Title 1 | Abstract 1 |
| Speaker 2 | Title 2 | Abstract 2 | |
| 4/15 | Cameron Chang | Title 1 | Abstract 1 |
| Speaker 2 | Title 2 | Abstract 2 | |
| 4/22 | Lizzie Pratt | Title 1 | Abstract 1 |
| Bernd Sturmfels | The Likelihood Correspondence | We report on the article arXiv:2503.0253 with Thomas Kahle, Hal Schenck and Max Wiesmann. An arrangement of hypersurfaces in projective space is SNC if and only if its Euler discriminant is nonzero. We study the critical loci of all Laurent monomials in the equations of the smooth hypersurfaces. These loci form an irreducible variety in the product of two projective spaces, known in algebraic statistics as the likelihood correspondence and in particle physics as the scattering correspondence. We establish an explicit determinantal representation for the bihomogeneous prime ideal of this variety. | |
| 4/29 -- last meeting of the semester | Oliver Pechenik | Title 1 | Abstract 1 |
| Speaker 2 | Title 2 | Abstract 2 | |