All semesters: Spring 2025, Fall 2024 , Spring 2024, Fall 2023, Spring 2023, Fall 2022
Tuesdays 4-6, Evans 939
John Nolan kindly agreed to share his live-TeXed notes publicly.
| date | speaker | title | abstract |
|---|---|---|---|
| 8/27 | David Eisenbud | Resolutions over Golod Rings | Recent work of mine with Dao, Cuong, Polini, Takahashi and Ulrich has uncovered surprising new structure in the resolution of the residue field of a Golod ring. I will explain what Golod rings are, why they have been studied, and what we now know and suspect about the structure of their resolutions. |
| Sterling Saint Rain | Sullivan Minimal Models and Free Torus Actions | This is an expository talk introducing differential graded algebras and Sullivan minimal models, as well as outlining their connection through rational homotopy theory to the existence of free torus actions on reasonable (i.e. finite CW-complex) topological spaces. | |
| 9/3 | Bernd Sturmfels | Maximal Mumford Curves from Planar Graphs | We introduce a recent project (arXiv:2404.11838) with Mario Kummer and Raluca Vlad in real non-archimedean geometry. A curve of genus \(g\) is maximal Mumford (MM) if it has \(g+1\) ovals and \(g\) tropical cycles. We construct full-dimensional families of MM curves in the Hilbert scheme of canonical curves. This rests on first-order deformations of graph curves whose graph is planar. |
| Feiyang Lin | Abelian covers | An abelian cover is a finite morphism \(X \to Y\) which is the quotient map for a generically faithful action of a finite abelian group G. Unlike the situation of general finite covers, where there is only a structure theorem for degree at most \(5\), there is a practical structure theorem of abelian covers due to Pardini in 1991, in the situation where the target is smooth and proper, and the source is normal. I will explain this structure theorem and give many examples along the way. | |
| 9/10 | -- | -- | No Seminar |
| 9/17 | Peter Haine | Reconstructing schemes from their étale topoi | In Grothendieck’s 1983 letter to Faltings that initiated the study of anabelian geometry, he conjectured that a large class of schemes can be reconstructed from their étale topoi. In this talk, I’ll explain a number of formulations of Grothendieck’s conjecture, and how one might come to make this conjecture. I’ll also discuss joint work with Magnus Carlson and Sebastian Wolf, generalizing work of Voevodsky, that proves Grothendieck’s conjecture. Specifically, we show that over a finitely generated field k of characteristic 0, seminormal finite type k-schemes can be reconstructed from their étale topoi. Over a finitely generated field k of positive characteristic and transcendence degree \(\geq 1\), we show that perfections of finite type k-schemes can be reconstructed from their étale topoi. |
| Hannah Larson | Chow rings of moduli spaces | The Chow ring of a variety captures information about its subvarieties and how those subvarieties intersect each other. I'll start by defining the Chow ring and describing some of its basic properties. Then I'll use the example of projective space to introduce the idea of "tautological classes", which are distinguished classes in the Chow ring of a moduli space. Finally, I'll define tautological classes on the moduli space of curves and tell you what is known (and not known) about them. | |
| 9/24 | Christian Gaetz | Combinatorics of singularities of Schubert varieties and torus orbit closures therein | The singularities of Schubert varieties in the flag variety have deep connections to geometric representation theory via Kazhdan-Lusztig theory. I will describe joint work with Yibo Gao in which we resolve a conjecture of Billey-Postnikov by quantifying these singularities by determining the first homological degree in which Poincare duality fails. Equivalently, this determines the smallest power of \(q\) appearing in a Kazhdan-Lusztig polynomial. Our methods rely on the combinatorics of permutation patterns. I'll also explain how this can be combined with my other work on generic torus orbit closures in Schubert varieties in order to determine when these are smooth, thereby proving a conjecture of Lee-Masuda-Park. |
| 10/1 | Daigo Ito | A new proof of the Bondal-Orlov reconstruction theorem | In 1997, Bondal and Orlov showed that a smooth (anti-)Fano variety \(X\) can be reconstructed from the triangulated category structure of its derived category. On the other hand, in 2004, Balmer showed that any variety \(X\) can be reconstructed from its derived category together with the monoidal structure given by the derived tensor product. This indicates that for a smooth (anti-)Fano variety, Balmer's reconstruction should be able to be done without actually using the monoidal structure and from this perspective, we will provide a new proof of the Bondal-Orlov reconstruction theorem. Furthermore, we offer new insights and institutions on derived categories from the techniques we developed as well as generalizations of several other reconstruction theorems. |
| Noah Olander | Fully faithful functors and dimension | Can one embed the derived category of a higher dimensional variety into the derived category of a lower dimensional variety? The expected answer was no. We give a simple proof and prove new cases of a conjecture of Orlov along the way. | |
| 10/8 | Hannah Larson | Chow rings of moduli spaces of pointed hyperelliptic curves | In this talk, I describe the geometry of the moduli space \(\mathcal{H}_{g,n}\) of n-pointed, genus g hyperelliptic curves. As n grows relative to g, work of Casnati, Barros--Mullane, and Schwarz shows that \(\mathcal{H}_{g,n}\) goes from being rational (the simplest kind of variety) to general type (quite complicated). This suggests that we have hope of probing finer invariants of \(\mathcal{H}_{g,n}\) when \(n\) is small relative to \(g\). The Chow ring of \(\mathcal{H}_{g,n}\) is one such invariant. I will describe an inductive procedure for stratifying \(\mathcal{H}_{g,n}\) into nice pieces, which allows us to calculate its rational Chow ring when \(n\) is less than or equal to \(2g + 6\). This is joint work with Samir Canning. |
| 10/15 | Smita Rajan | Kinematic Varieties for Massless Particles | We study algebraic varieties that encode the kinematic data for \(n\) massless particles in \(d\)-dimensional spacetime subject to momentum conservation. Their coordinates are spinor brackets, which we derive from the Clifford algebra associated to the Lorentz group. This was proposed for \(d = 5\) in the recent physics literature. Our kinematic varieties are given by polynomial constraints on tensors with both symmetric and skew symmetric slices. This is joint work with Bernd Sturmfels and Svala Sverrisdóttir. |
| John Nolan | Toric Varieties as Quotients of Affine Spaces | A toric variety is a normal algebraic variety \(X\) equipped with the action of an algebraic torus \(T\) and a \(T\)-equivariant dense open immersion \(T \hookrightarrow X\). In 1992, Cox showed that reasonably nice (e.g. smooth) toric varieties have canonical descriptions as quotients of affine space (minus a union of coordinate subspaces) by tori. I will present the basic theory of toric varieties alongside Cox's construction, using projective space as a running example. If time permits (it won't), I will also discuss Ballard-Favero-Katzarkov's application of related machinery to constructing full exceptional collections on smooth projective toric varieties. | |
| 10/22 | Will Fisher | Introduction to Hochschild Homology | We will define Hochschild homology and introduce basic computational techniques such as the HKR isomorphism. Towards the end we will briefly discuss more modern perspectives including THH and Bokstedt's computation and the relation to loop spaces. This talk should be accessible to all listeners. |
| 10/29 | Catherine Cannizzo | Homological Mirror Symmetry for Theta Divisors | Symplectic geometry is a relatively new branch of geometry. However, a string theory-inspired duality known as “mirror symmetry” reveals more about symplectic geometry from its mirror counterparts in complex geometry. M. Kontsevich conjectured an algebraic version of mirror symmetry called “homological mirror symmetry” (HMS) in his 1994 ICM address. HMS results were then proved for symplectic mirrors to Calabi-Yau and Fano manifolds. Those mirrors to general type manifolds have been studied in more recent years, including my research. In this talk, we will introduce HMS through the example of the 2-torus \(T^2\). We will then outline how it relates to HMS for a hypersurface of a 4-torus \(T^4\) and generalize to hypersurfaces of higher dimensional tori, otherwise known as “theta divisors.” This is joint work with Haniya Azam, Heather Lee, and Chiu-Chu Melissa Liu. |
| 11/5 | Martin Olsson | Ample vector bundles and projective geometry of stacks | I will report on joint work with Bragg and Webb on the following two questions:
(1) What is the analogue of a projective embedding for an algebraic stack? (2) How do we describe embeddings into such stacks in terms of vector bundles and sections? I will explain how exploration of these two questions leads to a notion of ample vector bundle on an algebraic stack. As an application we construct moduli of stacky curves. |
| 11/12 | Nathaniel Gallup | The Grothendieck Ring of Certain Non-Noetherian Multigraded Algebras via Hilbert Series | Inspired by work of Knutson, Miller, and Sturmfels, and with the goal of defining Grothendieck polynomials for infinite matrix Schubert varieties, we introduce a new category of graded “BDF modules” (containing the syzygies of infinite matrix Schubert varieties, which need not be finitely generated) over a new type of \(\Gamma\)-graded “PDCF algebra” (which includes infinite polynomial rings). If \(R\) is a PDCF \(\Gamma\)-graded \(k\)-algebra, we prove that every projective BDF \(\Gamma\)-graded \(R\)-module is free, and that when \(R\) is regular, a version of the Hilbert Syzygy Theorem holds: every BDF \(R\)-module has a resolution of free BDF modules which, though not finite, has the property that each graded piece is eventually zero. We use this to show that the natural map from the Grothendieck group of projective BDF \(R\)-modules to the Grothendieck group of all BDF \(R\)-modules is an isomorphism. We describe this Grothendieck group explicitly by using Hilbert series to give an isomorphism with a certain space of formal Laurent series. Finally we give a version of Serre's formula for the product in the Grothendieck group, making it into a ring. |
| 11/19 | David Eisenbud | Some one-dimensional local rings | One-dimensional local rings, also known as curve singularities, are often mistakenly overlooked as "trivial" in algebraic geometry, but there is a rich theory with interesting open problems. I'll explain some of the theory around one interesting class that has recently arisen in my research: Arf rings, named after the Turkish mathematician Cahit Arf, and studied in depth by Joe Lipman. |
| Cameron Chang | Applications of Molien's Formula | Given a finite group \(G\subset \operatorname{GL}(V)\), the ring of invariants \(\operatorname{Sym}(V)^G\) is the subalgebra of the polynomial ring \(\operatorname{Sym}(V)\) fixed under the \(G\)-action. In 1897, Theodore Molien wrote down a beautiful formula for the generating function of \(\dim \operatorname{Sym}^k(V)^G\). I will explain this formula and apply it to as many beautiful applications as I can before time runs out, such as: every irreducible representation is contained in \(\operatorname{Sym}(V)\), the only groups where \(\operatorname{Sym}(V)^G\) is a polynomial ring are pseudo-reflection groups, the Polya-Redfield counting formula, and more! | |
| 11/26 | Serkan Hoşten | Matroid Stratification of the Maximum Likelihood Degree of Segre Products | In this talk, we will look at the maximum likelihood (ML) degree of Segre products, also known as independence models. For \(\mathbb{P}^n \times \mathbb{P}^m\), we show that the ML degree is an invariant of an associated realizable matroid. For Segre products with more terms, we investigate the connection between the vanishing of factors of its principal \(A\)-determinant and its ML degree. Similarly, for toric varieties defined by the second hypersimplex, we determine its principal \(A\)-determinant and give computational evidence towards a conjectured lower bound of its ML degree. |
| Dustin Ross | Ehrhart Fans | For any smooth complete variety, the Euler characteristic of a vector bundle (the alternating sum of dimensions of cohomology groups) gives a linear map from the \(K\)-ring of vector bundles to the integers. One shouldn’t expect an Euler characteristic to exist for incomplete varieties, because the cohomology groups become infinite dimensional. Yet, in recent work of Larson, Li, Payne, and Proudfoot, an analogue of an Euler characteristic is defined for a class of incomplete toric varieties associated to matroids. Motivated by lattice point counting in polytopes, we introduce a new class of toric varieties whose \(K\)-rings admit a canonical linear map to the integers, creating a general framework for studying Euler characteristics of matroids alongside Euler characteristics of smooth complete toric varieties. This is joint and ongoing work with Melody Chan, Emily Clader, and Carly Klivans. | |
| 12/3 | Dawei Chen | Geometry of Subcanonical Points | If \((2g-2)p\) is a canonical divisor on a curve of genus \(g\), then \(p\) is called a subcanonical point. In this talk, I’ll give an introduction to the interesting geometry of subcanonical points, focusing on their moduli spaces, Weierstrass semigroups, deformations of monomial singularities, and some open problems. |
| 12/10 | Adam Boocher | From Classical Commutative Algebra to Some Diophantine Equations | In a first course in commutative algebra one might encounter the "Principal Ideal Theorem" or the "Auslander-Buchsbaum Formula". It turns out that these are both implied by a longstanding conjecture about lower bounds for ranks of syzygies - the Buchsbaum-Eisenbud-Horrocks Rank Conjecture. In this talk I'll discuss historical progress on the conjecture as well as a related (and rather mysterious) conjecture about even larger bounds. Later, I'll discuss some recent work about what happens if one looks at the special case of pure modules, using techniques from 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. |
| Tejas Rao | CM Elliptic Curves and Primality Proving | This is an expository talk introducing complex multiplication of elliptic curves: we will discuss the basic theory of CM and the group of points of an elliptic curve, show how to construct CM elliptic curves over finite fields, and explore an application to primality proving given by Atkin and Morain. |