All semesters: Spring 2025, Fall 2024 , Spring 2024, Fall 2023, Spring 2023, Fall 2022
Tuesdays 4-6, Evans 939
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9/2 |
UNUSUAL Time/Place: 5pm in Evans 1015: JM Landsberg (Texas A&M; Simons Institute) |
Tensors of minimal border rank |
I will describe a problem motivated by computer science that quickly leads to questions about smoothability of zero dimensional schemes and new and interesting algebraic structures. In geometric language, the \(m\)-th secant variety to the Segre variety \(Sec_m(Seg(\mathbb{P}^{m-1} \times \mathbb{P}^{m-1}\times \mathbb{P}^{m-1}))\) is an orbit closure and one would like to know what points are on the boundary, or more coarsely, what are the components of the boundary? This is joint work with F. Gesmundo, J. Jelisiejew, T. Mandziuk, and A. Pal. |
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9/9 |
David Eisenbud |
Summands of syzygies: a family of examples |
A large class of interesting rings called Golod rings includes things like \(R = S/I^d\) for \(d >1\), where \(S\) is a regular local (or graded polynomial) ring containing a field \(k\) of characteristic 0 and \(I\) is a (homogeneous) ideal. Hai Long Dao and I were surprised to discover (first experimentally) that over a Golod ring, some sequence of syzygy modules of \(k\) contain \(k\) as a direct summand, and we gave a complete family of examples of the possible sequences. To do this we needed to understand a natural minimal \(S\)-free resolution of \(I^2\) when \(I\) is generated by a regular sequence. In this talk I'll explain this problem and its solution using a simple part of the representation theory of the general linear group. |
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9/16 |
Hannah Larson |
Brill--Noether theory of special curves |
Brill--Noether theory studies the maps of curves C to projective spaces. The classical Brill--Noether theorem (established by work of Eisenbud, Fulton, Geiseker, Griffiths, Harris, Lazarsfeld) describes the geometry of this space of maps when C is a general curve. However, the theorem fails for special curves, notably curves that are already equipped with some unexpected map to a projective space. The first case of this is when C is a low-degree cover of the projective line. For general such covers, the Hurwitz--Brill--Noether theorem (joint with E. Larson and I. Vogt) provides a suitable analogue. I'll also present results (joint with S. Vemulapalli) regarding the next natural case: when C is equipped with an embedding in the projective plane. |
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9/23 |
Frank-Olaf Schreyer |
Smooth surfaces in \(\mathbb{P}^4\) |
Hartshorne conjectured that the degree of a smooth rational surface in \(P4\) is bounded. This conjectured was answered positively by Ellingsrud and Peskine (1989): There only finitely many components of the Hilbert scheme of surfaces in \(P4\), whose general element correspond to a smooth surface, which is not of general type. Around that time there was a flourish of activities to construct and classify such surfaces, e.g. by Okonek, Ranestad, Decker-Ein-Schreyer, Popescu, Abo. The existence proofs in [DES] used Macaulay (classic) in an essential way. Now more then 25 years later, Macaulay2 is much faster and we should be able to go further. In the talk I will explain our three basic construction methods: Hilbert-Burch morphism with vector bundles random searches over finite fields Tate resolutions and will illustrate each of these techniques in striking examples. |
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9/30 |
Hannah Friedman |
Counting Homogeneous Einstein Metrics |
The problem of finding Einstein metrics on a compact homogeneous space reduces to solving a system of Laurent polynomial equations. We prove that the number of isolated solutions of this system is bounded above by the central Dellanoy numbers and we describe the discriminant locus where the number of isolated solutions drops in terms of the principal \(A\)-determinant. |
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10/7 |
Noah Olander |
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10/14 |
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10/21 |
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10/28 |
Sam Grushevsky |
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11/4 |
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11/11 |
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11/18 |
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11/25 |
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12/2 |
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12/9 |
Yuhan Liu |
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12/16 |
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