# Algebra Seminar 2016

This seminar is held on Tuesdays at 4pm in 210 Deady.

### Spring Quarter, 2017

- April 4,
**Alexander Petrov**
- May 2,
**Macky Shotaro** (Stanford)

**Special Time**:11am

Koszul duality for Kac–Moody groups and characters of tilting modules
**Abstract**: I will introduce a monoidal category of “free-monodromic tilting sheaves” on a Kac–Moody flag variety. The main result is that this category is equivalent to the monoidal category of parity sheaves on the Langlands dual Kac–Moody flag variety. This result implies the Riche–Williamson conjecture on characters of tilting modules of reductive groups. This is joint work with P. N. Achar, S. Riche, and G. Williamson.

- May 23,
**Jarod Alper** (Washington)

Quotients of algebraic varieties
**Abstract**: In this talk, we will address the following question: given an algebraic group G acting on a variety X, when does the quotient X/G exist? We will provide an answer to this question in the case that G is reductive by giving necessary and sufficient conditions for the quotient to exist. We will discuss various applications to equivariant geometry, moduli problems and Bridgeland stability.

- May 30,
**Richard Green** (Colorado)

**Special Time:**10am in Deady 210

The nil Temperley–-Lieb algebra of type affine C
**Abstract**: The nil Temperley–Lieb algebra of type affine C is a certain infinite dimensional associative algebra. It can be defined either in terms of generators and relations, or as an algebra of creation and annihilation operators on certain particle configurations. The algebra turns out to have an interesting combinatorial structure, and I will discuss the extent to which this can be used to understand the representation theory of the algebra.

### Winter Quarter, 2017

- February 21,
**Agustín García Iglesias** (Universidad Nacional de Cordoba, Argentina)

Liftings of Nichols algebras via cocycle deformation
**Abstract**: Let V be a Yetter-Drinfeld module over a cosemisimple Hopf algebra H and assume that the Nichols algebra B(V) is finite-dimensional. We present a strategy, based on cocycle deformations, to compute all liftings of V; that is all Hopf algebras L with gr(L) = B(V)#H, where gr(L) stands for the graded algebra associated to the coradical filtration of L. When V is of diagonal type, this method is exhaustive and reduces the problem to a (hard) algorithmic computation. After the works of Heckenberger and Angiono, this corresponds to the final step in the Lifting Program developed by Andruskiewitsch and Schneider to classify all pointed Hopf algebras with abelian group of group-like elements.

### Fall Quarter, 2016

- September 27, Organizational Meeting
- October 4,
**Stefan Tohaneanu ** (U Idaho)

On some Rees algebras of hyperplane arrangements

(joint work with Mehdi Garrousian and Aron Simis)
**Abstract**: Consider n linear forms in C[x_1,…,x_k]. The Orlik-Terao algebra of the hyperplane arrangement A defined by these forms is the algebra generated by the reciprocals of the linear forms. Using methods specific to algebraic geometry, Schenck shows, for k=3, that this algebra is the coordinate ring of the projection on the second factor of the blowup of P^2 at the singular locus of \mathcal A, counted with multiplicities. In commutative algebraic terms this translates into the fact that the Orlic-Terao algebra is the special fiber ring of the ideal I_{n-1}(A) generated by the (n-1) fold products of the n given linear forms. And this is true for any k. We are interested in the Rees algebra of the ideal I_{n-1}(A). We show that this ideal is fiber-type, and we conjecture that any generalized star configuration has fiber-type defining ideal. This conjecture assumes that the conjecture stated in the Colloquium talk is true. Nonetheless, avoiding dealing with that conjecture, equally interesting is to analyze the special fiber rings of any generalized star configuration. This translates in investigating the algebra generated by reciprocals of fold products of linear forms.

- October 11,
** Drew Johnson ** (UO)

Strange duality, quot schemes, and multiple point formulas for del Pezzo surfaces
**Abstract**: Strange duality is a conjectural perfect pairing between spaces of sections of theta (or determinant) bundles on moduli spaces of sheaves. The case of del Pezzo surfaces has a particularly nice, symmetric set up. We adapt an argument from Marian and Oprea’s proof of strange duality for curves to some special cases on del Pezzo surfaces. This argument involves constructing finite quot schemes whose points give candidate dual bases for the spaces of section of the theta bundles. In order to obtain the expected cardinality of these quot schemes, we make a new construction that utilizes the topological multiple point formulas of Marangell and Rimanyi, which are valid for up to 7 points. On the other hand, we use a result of Ellingsrud, Gottche, and Lehn to compute the dimension of a space of sections. The results of these two very different-looking computations match! We also investigate conditions under which a suitable quot scheme can be shown to exist and prove some new results about the global generation of general bundles.

- October 18,
**Agustín García Iglesias** (Universidad Nacional de Cordoba, Argentina)

Liftings of Nichols algebras via cocycle deformation
**Abstract**: Let V be a Yetter-Drinfeld module over a cosemisimple Hopf algebra H and assume that the Nichols algebra B(V) is finite-dimensional. We present a strategy, based on cocycle deformations, to compute all liftings of V; that is all Hopf algebras L with gr(L) = B(V)#H, where gr(L) stands for the graded algebra associated to the coradical filtration of L. When V is of diagonal type, this method is exhaustive and reduces the problem to a (hard) algorithmic computation. After the works of Heckenberger and Angiono, this corresponds to the final step in the Lifting Program developed by Andruskiewitsch and Schneider to classify all pointed Hopf algebras with abelian group of group-like elements.

- October 25,
**Martin Helmer ** (UC Berkeley)

Nearest Points on Toric Varieties
**Abstract**: We determine the Euclidean distance degree of a projective toric variety. This extends the formula of Matsui and Takeuchi for the degree of the A-discriminant in terms of Euler obstructions. Our primary goal is the development of reliable algorithmic tools for computing the points on a real toric variety that are closest to a given data point. This also leads to expressions for the polar degrees and Chern-Mather class of a projective toric variety. This is joint work with Bernd Sturmfels.

- November 1,
**Dmitry Vaintrob** (Institute for Advanced Study)

Coherent-constructible correspondence for toric varieties
- November 8,
**Peter Samuelson ** (University of Iowa)

The trace of the quantum Heisenberg category
**Abstract**: Khovanov and Licata-Savage (for q=1 and generic q, respectively) introduced a monoidal category whose K-theory is the Heisenberg algebra. Roughly, objects are compositions of induction and restriction functors for Hecke algebras, morphisms are certain natural transformations, and the tensor product is functor composition. We give an algebra presentation of the trace (or Hochschild homology) of this category and show it is isomorphic to a central extension of the elliptic Hall algebra E of Burban and Schiffmann, which is the “universal” Hall algebra of an elliptic curve. (This is joint work with Cautis, Lauda, Licata, and Sussan.)

- November 15,
**Jose Simenthal-Rodriguez** (Northeastern University)

Harish-Chandra bimodules for rational Cherednik algebras
**Abstract**: Associated to a pair of algebras quantizing the same graded Poisson algebra there is a category of Harish-Chandra bimodules. These have been studied with some detail in the context of universal enveloping algebras, finite W-algebras and hypertoric enveloping algebras, among others. I will introduce this concept in the setting of rational Cherednik algebras, with an emphasis on the relationship between Harish-Chandra bimodules and category O, which can be more clearly seen in type A.

- November 22,
**Nicolas Addington** (UO)

A computer-based proof of a result of Voisin and Ranestad
**Abstract**: In the moduli space of cubic 4-folds, there is a certain divisor that one would expect to be a Noether-Lefschetz divisor, but Voisin and Ranestad have shown that it isn’t. This is surprising, and their proof is very difficult. I will discuss an independent proof, using the Weil conjectures and a computer.

- November 29,
**Benjamin Schmidt** (UT Austin)

Classical Algebraic Geometry and Derived Categories
**Abstract**: Algebraic geometry is full of sophisticated techniques. From time to time, we should ask the question how all of this relates back to the classical roots of the field. In this talk, I will explain the concept of a Bridgeland stability condition in the derived category of a smooth projective variety. It was introduced in order to provide a mathematical framework for questions arising from string theory. However, it also relates to various concrete question such as the geometry of moduli spaces of points on surfaces, the geometry of moduli spaces of space curves, and bounds on the genus of space curves under geometric constraints. If you are surprised how any of these things could possibly relate to derived categories, you should attend my talk.

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