Talks during the Panorama lecture series.

David Eisenbud, A Survey of Boij-Soederberg Theory.

Abstract: The Hilbert polynomial is an invariant of a graded module or a coherent sheaf on projective space that plays a central role in commutative algebra and algebraic geometry. It has two natural refinements: the Betti table of the module, and the cohomology table of the sheaf. Boij-Soederberg theory, which has grown up over the last five years, shows that these two refinements are, in a precise sense, dual to one another. The duality pairing gives substantial new information about each of the two invariants. I will explain these ideas, some of their applications, and some of the many open problems of the field.

Click here here for a reading list on Boij-Soederberg theory.

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Srikanth Iyengar, The Betti table of a Koszul algebra.

Abstract: Let S be a polynomial ring over a field and R a quotient of S, defined by an ideal of homogenous forms of degree at least two. This talks concerns some constraints on the Betti table of R viewed as an S-module, when R is a Koszul algebra. This is based on joint work with Luchezar Avramov and Aldo Conca.

Click here here for the notes.

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Jaya Iyer, Linear systems on hyperelliptic varieties.

Abstract: We will discuss an approach to investigate linear systems on hyperelliptic varieties. Some results will be announced. This is joint work with S. Chintapalli.

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Bangere Purnaprajna, Syzygies and geometry.

Abstract: We will survey some results on syzygies of embeddings of projective varieties and their geometry.

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S. Ramanan, An introduction to syzygies.

Abstract: TBA.

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Kavita Sutar-Deshpande, Orbit closures for source-sink quivers and their resolutions.

Abstract: Quiver representations are an important tool in the study of representations of finite-dimensional algebras. Given a quiver Q, the isomorphism classes of representations of Q correspond to the orbits of the action of a suitable algebraic group on the set of representations of Q. The study of (Zariski) closures of these orbits is of interest to algebraic geometers and representation theorists alike.

One aspect of studying orbit closures is the calculation of their defining ideals. In my talk I will demonstrate how one can construct a minimal free resolution of these ideals for a certain class of quivers. In addition to giving us an explicit description of the minimal generators of the ideal, such a resolution also makes it possible to say if the orbit closure has other geometric properties such normality, Cohen-Macaulay etc.

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Hema Srinivasan, Minimal Resolutions of ideals defining monomial curves.

Abstract: Click Here

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Preparatory workshop

Sudhir Ghorpade. Betti numbers of Stanley-Reisner rings and Generalized Hamming weights of codes and matroids. No prior knowledge of codes and matriods will be assumed.

Krishna Hanumanthu. Coherent sheaves on projective space. Cohomology tables. Line bundles and vector bundles on projective space. Projection formula. Higher direct images of a sheaf under proper push-forward. Ample line bundles, embeddings of projective varieties.


Manoj Kummini. Regular local rings, Auslander-Buchsbaum-Serre theorem, Cohen-Macaulay modules, polynomial rings, Hilbert functions, minimal free resolutions, Betti tables. regularity, projective dimension etc. Canonical modules. Schur functors in characteristic 0.

Pramathanath Sastry. Homological algebra, derived categories, local and global duality.

Macaulay2 We will use the computer algebra program Macaulay2 to compute examples. Click here for the M2 examples file.