Time and Place Tuesday 4-6 unless otherwise noted
All semesters: Spring 2025, Fall 2024 , Spring 2024, Fall 2023, Spring 2023, Fall 2022
| date | speaker | title | abstract |
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| August 25 | Speaker 1 | Title 1 | Abstract 1 |
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| September 1 | Speaker 1 | Title 1 | Abstract 1 |
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| September 8 | Reginald Anderson (4:10-4:30) | Stability conditions on abelian categories | We recall the definition of a stability condition on an abelian category. |
| Reginald Anderson (4:45-5:35) | Rationality of Pandharipande-Thomas descendent generating series for superpositive curve classes on smooth projective complex 3-folds | Let X be a nonsingular quasi-projective 3-fold, and fix beta as an element of the second integral homology of X. In 2007, Pandharipande-Thomas conjectured that any descendent generating series of Pandharipande-Thomas stable pair invariants with curve class beta forms the Laurent expansion over the rational numbers of a rational function in a single formal variable q. In 2017, Pandharipande conjectured that moreover, this series is the Laurent expansion of a rational function with poles only at q=0 and at certain roots of unity bounded in order by the divisibility of beta in second integral homology of X, modulo torsion. While counterexamples are known for the second conjecture in Picard rank greater than 1, we prove rationality of the Pandharipande-Thomas descendent generating series for the case of superpositive curve classes on connected smooth complex projective 3-folds and show poles only at q=0 and at roots of unity. We also prove analogous conjectures in G-equivariant cohomology for the case of a linear algebraic complex group G acting on X which acts trivially on the second integral homology of X and on a certain quotient of the first Chow group. Our proof approach uses Joyce's theory of wall-crossing from Donaldson-Thomas invariants to Pandharipande-Thomas stable pair invariants. This is based on joint work with Dominic Joyce at Oxford University in arXiv: 2604.05664. | |
| September 15 | Smita Rajan | Universal Formulas for de Jonquieres Loci and Equivariant Contact Problems | Given a family of genus \(g\) curves \(f:\Psi\to X\) and a relative degree \(d\) line bundle \(\mathcal{L}\) on \(\Psi\) such that \(f_*\mathcal{L}\) is a vector bundle, the projectivization \(\mathbb{P}(f_*\mathcal{L})\) is stratified by \(\textit{relative de Jonqui\'eres loci}\), which parameterize sections of \(\mathcal{L}\) with prescribed vanishing orders. We give universal formulas for the classes of these loci in terms the relative \(\mathcal{O}(1)\) on \(\mathbb{P}(f_*\mathcal{L})\) and tautological classes pulled back from the universal Picard stack \(\mathrm{Pic}_g^d\). When \(f\) is the family of lines in \(\mathbb{P}^r\), we use the universal polynomials to compute \(\GL_{r+1}\)-equivariant classes of loci of hypersurfaces and complete intersections admitting lines with prescribed contact orders. |
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| September 22 | Younghan Bae | Cycles on moduli space of curves and line bundles | Classical Abel-Jacobi theory relates algebraic cycles on the m-th symmetric product of a smooth projective curve and those on the Jacobian. In this talk, I will generalize this picture over the moduli space of stable curves and construct a bi-graded algebra isomorphism between (i) the Chow ring of the universal compactified Jacobian, and (ii) a quotient of the Chow ring of moduli space of stable curves with arbitrary number of additional markings. Moreover, I will explain that structures on the cohomology of compactified Jacobians, such as the Decomposition Theorem and the relative Hard Lefschetz theorem, admit motivic lifts. This isomorphism together with the structural results can be used to study the Chow ring of both spaces. This is a joint work in progress with A. Pixton. |
| Speaker 2 | Title 2 | Abstract 2 | |
| September 29 | Thomas Brazelton | Osculating Schubert calculus and twisted A1-degrees | Given a square system of real polynomials, we should never expect all solutions to the system to be real. Remarkably, a family of such solutions emerges in geometry from simple Schubert problems osculating rational normal curves (this is a deep and recent theorem previously known as the Shapiro-Shapiro conjecture). This has been a central topic in real algebraic geometry the past few decades, and reduces in many cases to the study of a map of real algebraic manifolds called the Wronski map. This map is not oriented in many cases, which resists our attempts to approach it through the lens of quadratically enriched enumerative geometry. We introduce the idea of twisted A1-degrees and A1-chambers, which allow us to repair conservation of number in many cases when orientability hypotheses may fail. We apply these ideas to the study of the Wronski map, exploring its twisted degrees, which unifies work from many authors. This is based on joint work with Jake Levinson. |
| Josephine Hlavinka | Chow Vanishing and Motives of Cluster Varieties | Progress over the last decade has shown that the cohomologies of full rank, locally acyclic cluster varieties enjoy remarkable combinatorial structures. Crucial in this program has been the observation that these cohomology groups are, in certain Hodge theoretic senses, about as simple as possible. In this talk we will discuss a motivic lift of this observation. In particular we establish that the Voevodsky motives of really full rank sink-recurrent cluster varieties are mixed Tate and split over Q (strengthening results of Lam and Speyer), and we prove that their integral Chow groups are all trivial. | |
| October 6 | Candace Bathea | Title 1 | Abstract 1 |
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| October 13 | Kirsten Wickelgren | Title 1 | Abstract 1 |
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| October 20 | Anna Abasheva | Title 1 | Abstract 1 |
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| October 27 | Speaker 1 | Title 1 | Abstract 1 |
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| November 3 | Speaker 1 | Title 1 | Abstract 1 |
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| November 10 | Speaker 1 | Title 1 | Abstract 1 |
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| November 17 | Ronnie Cheng | Title 1 | Abstract 1 |
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| November 24 | Speaker 1 | Title 1 | Abstract 1 |
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| December 1 | Bernd Sturmfels | Title 1 | Abstract 1 |
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| December 8 | Speaker 1 | Title 1 | Abstract 1 |
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