# Algebra Seminar 2017

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

### Spring Quarter, 2018

- April 3,
**Felix Janda** (U Michigan)

**Special Time:** 3pm

On higher genus Gromov-Witten invariants of quintic threefolds
**Abstract**: The Gromov-Witten invariants of a smooth variety X are virtual counts of algebraic curves in X. In the case that X is a Calabi-Yau threefold, there is significant interest from string theorists in computing these invariants, and many results and open questions about Gromov-Witten invariants are motivated from physics. In my talk, I will survey some recent developments in the case of the (in some sense) most simplest Calabi-Yau threefolds: quintic hypersurfaces in projective 4-space.

- April 10,
**Ben Antieau** (UIC)

The topological period-index problem
**Abstract**: I will survey recent progress on the topological period-index problem for Brauer groups and its connection to the algebraic period-index problem.

- April 17,
**Robert Laugwitz** (Rutgers)

A categorification of cyclotomic integers at non-prime orders
**Abstract**: I am reporting on recent joint work with You Qi (Caltech). Using the framework of Hopfological algebra, developed by Khovanov–Qi, and used by Elias–Qi to categorify versions of quantum

at a prime root of unity, we construct a monoidal triangulated category whose Grothendieck ring is isomorphic to the ring of cyclotomic integers. The construction removes the restriction that the order has to be a prime number as required in the original work of Khovanov.

- April 24,
**Zijun Zhou** (Stanford)

Quantum K-theory of hypertoric varieties
**Abstract**: Okounkov’s quantum K-theory can be defined for any hyperkähler quotient via counting parameterized quasimaps with genus zero domain. This construction yields many interesting results in geometric representation theory. In the case of hypertoric varieties, computations can be done explicitly due to the good torus action and nice combinatoric structure. In particular, the information of K-theoretic quantum multiplication can be extracted from the saddle point equations of an integral presentation of the vertex function. If time permits, I will also talk about the phenomenon of symplectic duality in this context.

- May 8,
**Leonardo Patimo** (MPIM Bonn, MSRI)

Moment graphs, Kazhdan-Lusztig polynomials and combinatorics
**Abstract**: Kazhdan-Lusztig polynomials contain crucial data for the representation theory of reductive groups and the geometry of flag varieties. In this talk I will review Braden-MacPherson-Fiebig’s approach to KL polynomials using moment graphs. This provides an incarnation of the Hecke category where KL polynomials (or their positive characteristic analogue) can be computed using elementary algebra. I will conclude by using moment graphs to give a purely combinatorial interpretation of the coefficient of q for KL polynomials in Type A.

- May 17,
**Jeongseok Oh** (UC Berkeley)

**Special Day/Time**: 2pm in 210 Deady

Localized Chern characters for 2-periodic complexes
**Abstract**: For a two-periodic complex of vector bundles, Polishchuk and Vaintrob have constructed its localized Chern character. We explore some basic properties of this localized Chern character. In particular, we show that the cosection localization defined by Kiem and Li is equivalent to a localized Chern character operation for the associated two-periodic Koszul complex, strengthening a work of Chang, Li, and Li. We apply this equivalence to the comparison of virtual classes of moduli of ε-stable quasimaps and moduli of the corresponding LG ε-stable quasimaps, in full generality. The talk is based on joint work with Bumsig Kim.

- May 22,
**Robert Muth** (Tarleton)

Schurifying superalgebras
**Abstract**: I will describe how to `schurify’ a superalgebra. Some important algebras arise in this way; the classical Schur algebra is the schurification of its ground field, and RoCK blocks of Hecke algebras were shown by Evseev and Kleshchev to be Morita equivalent to schurified zigzag algebras. Many nice properties of superalgebras are preserved under schurification; in particular, if a superalgebra is quasihereditary/cellular/symmetric, then (under certain constraints), its schurification inherits that structure as well. I will present some examples and describe some supertableau combinatorics related to standard bases and decomposition numbers for schurified superalgebras. This is joint work with Alexander Kleshchev.

- May 29,
**Thorge Jensen** (MPIM Bonn, MSRI)

The ABC of p-Cells
**Abstract**: The Hecke category is a categorification of the Hecke algebra that plays an important role in (geometric) representation theory. Using this categorification, I will introduce a positive characteristic analogue of the famous Kazhdan-Lusztig basis of the Hecke algebra, called the p-canonical or p-Kazhdan-Lusztig basis. If time permits, I will mention connections between the p-Kazhdan-Lusztig basis and the representation theory of reductive algebraic groups. Motivated by the very rich theory of Kazhdan-Lusztig cells, I study cells with respect to the p-Kazhdan-Lusztig basis. Throughout the talk, I will use SL_2 as a running example. In the end, I will give a complete description of p-cells in finite type A and mention some interesting results in finite types B and C.

### Winter Quarter, 2018

- January 9,
**Jonathan Kujawa** (U Oklahoma)

Twisted Heisenberg categories of level zero and beyond
**Abstract**: Paralleling the recent work of Brundan, we use diagrammatic categories to define the twisted Heisenberg category of arbitrary integral level. At level zero this category describes endofunctors of representations of the Lie superalgebra of type Q. At higher/lower levels it describes endofunctors of representations of the Sergeev superalgebra (= spin representations of the symmetric group) and related cyclotomic quotient superalgebras. This is joint work with Jonny Comes.

- January 16,
**Karina Batistelli** (U Cordoba, Argentina)

QHWM of the “orthogonal” and “symplectic” types Lie subalgebras of the matrix quantum pseudodifferential operators
- January 30,
**Katrina Honigs** (Univ. of Utah)

Derived Equivalence and Albanese Varieties
**Abstract**: A conjecture of Orlov states that derived equivalent smooth, projective varieties have isomorphic effective Chow motives with rational coefficients. This conjecture implies that derived equivalent smooth, projective varieties over finite fields have equal zeta functions. In this talk I will discuss cases in which this conjectural implication has been proven, particularly a proof for three-dimensional varieties that uses an adaptation to positive characteristic of a result of Popa and Schnell showing that derived equivalent complex smooth, projective varieties have isogenous Albanese varieties.

- February 6,
**Daniel Bragg** (U Washington)

Supersingular Twistor Space
**Abstract**: We will describe how the crystalline cohomology of a supersingular K3 surface gives rise to certain one-parameter families of K3 surfaces, which we call supersingular twistor spaces. Our construction relies on the unique behavior of the Brauer group of a supersingular K3 surface, as well as techniques coming from the study of the derived category and Fourier–Mukai equivalences. As applications, we find new proofs of Ogus’s crystalline Torelli theorem and Artin’s conjecture on the unirationality of supersingular K3 surfaces. These results are new in small characteristic.

- February 20,
**Jørgen Rennemo** (Oslo)

The Torelli theorem for cubic 4-folds via derived categories
**Abstract**: The Torelli theorem for cubic 4-folds, first shown by Voisin, says that a smooth cubic 4-fold X is determined up to isomorphism by its primitive middle cohomology, considered as a polarised Hodge structure. I’ll explain a new proof of this theorem which builds on recent results regarding a K3-like subcategory of the derived category of X. This is joint work with Daniel Huybrechts.

- February 27,
**Nicolas Libedinsky** (Universidad de Chile)

Towards a physical interpretation of modular representation theory
**Abstract**: In joint work with David Plaza we conjecture, in type A, that the Hecke category (that controls most of modular representation theory) is equivalent to some “Blob category” that we introduce. We prove that the morphism spaces in both categories have the same graded degrees. The blob category is formed out by “generalized blob algebras” in positive characteristic. The blob algebra in characteristic zero can be seen as a transfer matrix algebra of some statistical mechanics systems. If one could interpret the generalized blob algebras in positive characteristic in the same terms, that would be very nice, as one could obtain physical intuitions to this extraordinarily complicated problem.

- March 13,
**Tasos Moulinos** (UIC)

Derived Azumaya Algebras and Twisted K-Theory

### Fall Quarter, 2017

- September 26, Organizational Meeting
- October 3,
**Noah Snyder** (U Indiana)

Simple Skein Theories
**Abstract**: The famous Jones, HOMFLY, and Kauffman knot polynomials are defined by skein relations between tangles (i.e. links with boundary). Using these skein relations, one can see that the vector space of all tangles with 2n-boundary points modulo these skein relations is finite dimensional. Furthermore, these knot polynomials are characterized by being the unique ones where the space of 2n-tangles have small dimension for small n. But there’s nothing essential here about looking at tangles, instead one could look skein theories for planar trivalent graphs, or knotted trivalent graphs, or any number of other possibilities. I will survey the current landscape of known simple skein theories, which include many striking examples like the exceptional Lie algebra and the Haagerup subfactor, and explain some of the techniques used in such classifications.

- October 10,
** Ben Elias ** (UO)

The Bergman diamond lemma for symmetric groups

**Abstract**: (A major goal of this talk is to introduce the Bergman diamond lemma to graduate students. This is a really useful tool whenever you’re working with an algebra given by generators and relations. The talk will be very accessible!)

If you are given an algebra by generators and relations, it is not obvious how big it is (it could be zero!). Often it is easy to show that a set of words (in the generators) spans the algebra, but linear independence is trickier. The Bergman diamond lemma is a general tool which takes a presentation of an algebra and determines, using relatively little work, whether the spanning set of “irreducible monomials” is linearly independent. This is great when it applies, but when the irreducible monomials are not a basis, you’re stuck and typically have to change your presentation of the algebra.

A major example where the Bergman diamond lemma does not apply is the Coxeter presentation of the group algebra of the symmetric group. We introduce a variant of the diamond lemma which does apply to this presentation, as well as similar presentations (like for Hecke algebra or Khovanov-Lauda-Rouquier algebras). In order to apply the lemma, one must prove the existence of a compatible orientation on the graph of expressions of a permutation, which is of independent interest. We show that a natural extension of the Manin-Schechtmann orientation on the reduced expression graph will suffice.

- October 17,
**Nick Proudfoot** (UO)

The quantum Hikita conjecture
**Abstract**: The Hikita conjecture relates the cohomology ring of a conical symplectic resolution to the quantized coordinate ring of another such resolution. I will explain this conjecture, and present a new version of the conjecture involving the quantum cohomology ring. There will be an emphasis on explicit examples.

- October 24,
**Karl Schmidt** (UO)

Factorizable g-module algebras
**Abstract**: The aim of this talk is to introduce and study a large class of $\mathfrak{g}$-module algebras which we call factorizable, generalizing the Gauss factorization of (square or rectangular) matrices. This class includes appropriate localizations of coordinate algebras of corresponding reductive groups $G$, their parabolic subgroups, basic affine spaces, and many others. It turns out that tensor products of certain factorizable algebras are also factorizable. We also have quantum versions of all these constructions in the category of $U_q(\mathfrak{g})$-module algebras. Quite surprisingly, our quantum factorizable algebras are naturally acted on by the quantized enveloping algebra $U_q(\mathfrak{g}^*)$ of the dual Lie bialgebra $\mathfrak{g}^*$ of $\mathfrak{g}$

- October 31,
**Yang Zhou** (Stanford)

Higher genus wall-crossing in weighted FJRW theory
**Abstract**: Given a Fermat quasi-homogeneous polynomial W of degree r, the Fan-Jarvis-Ruan-Witten theory produces cohomology classes of M_{g,n} in a systematic way, so that they form a cohomology field theory. Conjectually the FJRW invariants are related to curve-counting on the hypersurface defined by W.

The FJRW invariants are defined via intersection theory on moduli of stable r-spin curves. Varying the stability condition give different invariants called the weighted FJRW invariants, related to the original invariants by wall-crossing formulas. In this talk we prove a wall-crossing formula in all genera, for narrow insertions. This is analogous to the higher genus quasi-map wall-crossing formula proved by Ciocan-Fontanine and Kim. It generalizes the genus-$0$ result by Ross-Ruan and the genus-1 result by Guo-Ross. The higher-genus wall-crossing formula is also proved by Clader, Janda and Ruan.

- November 7,
**Seth Shelley-Abrahamson** (MIT)

The Dunkl Weight Function for Representations of Rational Cherednik Algebras
**Abstract**: Let W be a finite Coxeter group and let V be an irreducible representation of W. I will discuss the “Dunkl weight function”, an analytic family of functions/tempered distributions on the real reflection representation of W taking values in Hermitian forms on V. In particular, I will show how these functions arise naturally in the setting of representations of rational Cherednik algebras, where they give rise to integral representations for the invariant Hermitian forms on the “Verma modules” in the associated category O. I will explain how the Dunkl weight function provides a bridge between the study of invariant Hermitian forms on representations of rational Cherednik algebras and Hecke algebras, and I will state some related conjectures having to do with Jantzen filtrations and signatures.

- November 14,
**Andrei Smirnov** (UC Berkeley)

Quantum difference equations for Nakajima varieties
**Abstract**: Let QH(X) be a quantum cohomology ring of some variety X. The operation of quantum multiplication defines a flat connection on H^2(X) also known as quantum differential equation. In this talk I will discuss the generalization of this picture to the quantum K-theory of X given by a quiver variety. The corresponding differential equation is now substituted by a difference equation, which can be considered as a “flat difference connection” on a lattice ( Picard group of X ). I will explain the relation of this difference connection with K-theoretic J-function of Givental, qKZ-equations, monodromy problem (for quantum connection), and dynamical Weyl group.

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