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 | Speaker 1 | Title 1 | Abstract 1 |
| Speaker 2 | Title 2 | Abstract 2 | |
| 2/18 | Nate Gallup | Title 1 | Abstract 1 |
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| 2/25 | Noah Olander | Title 1 | Abstract 1 |
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| 3/4 | Speaker 1 | Title 1 | Abstract 1 |
| Tejas Rao | Title 2 | Abstract 2 | |
| 3/11 | Christopher O'Neill | Title 1 | Abstract 1 |
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| 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 |
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| 4/8 | Speaker 1 | Title 1 | Abstract 1 |
| Speaker 2 | Title 2 | Abstract 2 | |
| 4/15 | Speaker 1 | Title 1 | Abstract 1 |
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| 4/22 | Speaker 1 | Title 1 | Abstract 1 |
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| 4/29 -- last meeting of the semester | Oliver Pechenik | Title 1 | Abstract 1 |
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