# Algebra Seminar

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

### Spring Quarter, 2019

- April 2,
**Nicolas Addington**(UO)

Rational points and derived categories**Abstract**: For smooth projective varieties over Q, is the existence of a rational point preserved under derived equivalence? First I’ll discuss why this question is interesting, and what is known. Then I’ll show that the answer is no, giving two counterexamples: an abelian variety and a torsor over it, and a pair of moduli spaces of sheaves on a K3 surface. Joint with Ben Antieau and Sarah Frei. - April 3,
**Hsian-Hua Tseng**(Ohio State)

**Special Time:**3pm in 303 Deady

The descendant Hilb/Sym correspondence for the plane**Abstract**: Let be a nonsingular surface. A version of the crepant resolution conjecture predicts that the descendant Gromov-Witten theory of , the Hilbert scheme of points on , is equivalent to the descendant Gromov-Witten theory of , the -fold symmetric product of . In this talk we discuss how this works when is . We explicitly identify a symplectic transformation equating the two descendant Gromov-Witten theories. We also establish a relationship between this symplectic transformation and the Fourier-Mukai transformation which identifies the (torus-equivariant) -groups of and . This is based on joint work with R. Pandharipande. - April 9,
**Boris Shapiro**(Stockholm)

**Special Time:**11am

Around Waring problem for polynomials**Abstract**: We will discuss what is currently known about the following question.Problem. For a given triple (n, k , d ) of positive integers, find the minimal number ♯(n, k , d ) such that every (resp. almost every) complex-valued homogeneous polynomial of degree kd in (n + 1) variables can be represented as a sum of at most ♯(n,k,d) many k-th powers of polynomials of degree d.

This question goes back to the pioneering works of J.J. Sylvester in the 1860s and is currently a very active area of research. Several conjectures will be posed. No preliminary knowledge of the topic is required.

- April 16,
**Mai Gehrke**(Nice)

Bands, sheaves, and skew lattices**Abstract**: Bands are idempotent semigroups. These are very simple algebraic structures with a surprisingly rich theory and a strong connection to a number of more geometric ideas. In particular, they carry an induced order and they each have a free-est quotient which is a semilattice. We introduce two subvarieties of bands, namely the regular and the further subvariety of normal bands. In on-going joint work with Clemens Berger, we have shown that the comprehensive factorisation for posetal categories of [3], which is something like the epi-monic decomposition of maps between sets, lifts to regular bands, giving rise to the representation of normal bands as presheaves over semilattices. This provides a wider perspective on an early result of Kimura about normal bands [4].

Presheaves have geometric flavor, and we consider those normal bands that correspond to sheaves over certain topological spaces known as spectral spaces. That is, we consider the special cases where the presheaf has a finitary patch property and the semilattice is either a distributive lattice and a Boolean algebra. In the latter case, these geometrically defined structures are equivalent to certain universal algebras, namely the skew Boolean algebras of Leech, see [5,6]. In order to understand the distributive version of this, we are lead to study what we call the `patch monad’. This is the monad on sheaves on a spectral space given by the relationship between the spectral topology and the corresponding constructible or `patch’ topology on the space. This provides a new, more transparent approach to the main result of [1], showing that strongly distributive skew lattices are equivalent to sheaves on Priestley spaces.[1] A. Bauer, K. Cveto-Vah, M. Gehrke, S. J. van Gool, G. Kudryavtseva, A non-commutative Priestley duality, Topology Appl. 160 (2013), 1423–1438.

[2] A. Bauer, K. Cvetko-Vah, Stone duality for skew Boolean algebras with intersections, Houston J. Math. {\bf 39}(1) (2013), 73–109.

[3] C. Berger and R. M. Kaufmann, Comprehensive factorisation systems, Tbilisi Math. J. 10(3) (2017), 255–277.

[4] N. Kimura, The structure of idempotent semigroups I, Pacific J. Math. 8 (1958), 257–275.

[5] G. Kudryavtseva, A refinement of Stone duality to skew Boolean algebras, Algebra Universalis 67 (2012), 397–416.

[6] J. Leech, Normal Skew Lattices, Semigroup Forum 44 (1992), 1–8. - April 23,
**Melissa Liu**(Columbia)

**Special Time:**11am in 210 Deady

GW theory, FJRW theory, and MSP fields**Abstract**: Gromov-Witten (GW) invariants of the quintic Calabi-Yau 3-fold are virtual counts of stable maps to the quintic 3-fold. Fan-Jarvis-Ruan-Witten (FJRW) invariants of the Fermat quintic polynomial are virtual counts of solutions to the Witten equation associated to the Fermat quintic polynomial. In this talk, I will describe the theory of Mixed-Spin-P (MSP) fields interpolating GW theory of the quintic 3-fold and FJRW theory of the Fermat quintic polynomial, based on joint work with Huai-Liang Chang, Jun Li, and Wei-Ping Li, and the theory of N-MSP fields based on recent work of Huai-Liang Chang, Shuai Guo, Jun Li, and Wei-Ping Li. - April 30,
**Isabel Vogt**(MIT) - May 14,
**Yuri Zahrin**(Penn State)

Jordan properties of automorphism groups of complex algebraic varieties and real manifolds

### Winter Quarter, 2019

- February 14,
**Si Li**(Tsinghua University)

Landau-Ginzburg models and Hodge-to-de Rham degeneration**Abstract**: Let X be a non-compact Calabi-Yau manifold and f be a holomorphic function on X with compact critical locus, satisfying a general asymptotic condition. We establish a version of twisted L2 Hodge theory for the pair (X,f) and prove the corresponding Hodge-to-de Rham degeneration property. It can be viewed as a generalization of Kyoji Saito’s higher residue theory and primitive forms for isolated singularities, putting Landau-Ginzburg Bmodel of the pair (X,f) into the same setting as compact Calabi-yau manifolds. This is joint work with Hao Wen. - February 19,
**Eric Larson**(Stanford)

The Maximal Rank Conjecture**Abstract**: Curves in projective space can be described in either parametric or Cartesian equations. We begin by describing the Maximal Rank Conjecture, formulated originally by Severi in 1915, which prescribes a relationship between the “shape” of the parametric and Cartesian equations — that is, which gives the Hilbert function of a general curve of genus g, embedded in P^r via a general linear series of degree d. We then explain how recent results on the interpolation problem can be used to prove this conjecture. - March 5,
**Michael Harris**(Columbia University)

Modularity and potential modularity theorems in the function field setting**Abstract**: Let G be a reductive group over a global field of positive characteristic. In a major breakthrough, Vincent Lafforgue has shown how to assign a Langlands parameter to a cuspidal automorphic representation of G. The parameter is a homomorphism of the global Galois group into the Langlands L-group $^LG$ of G. I will report on my joint work with Böckle, Khare, and Thorne on the Taylor-Wiles-Kisin method in the setting of Lafforgue’s correspondence. New (representation-theoretic and Galois-theoretic) issues arise when we seek to extend the earlier work of Böckle and Khare on the case of GL(n) to general reductive groups. I describe hypotheses on the Langlands parameter that allow us to apply modularity arguments unconditionally, and I will state a potential modularity theorem for a general split group.(And I’ll try to make it comprehensible to group theorists.)

- March 12,
**Rina Anno**(Kansas State U.)

-functors and noncommutative cyclic covers.**Abstract**: This talk is based on joint work with T. Logvinenko. -functors were introduced by N. Addington in 2011 as exact functors F with a right adjoint R for which, among other conditions, RF was a direct sum of powers of an autoequivalence H. We propose a more general definition for a functor, where RF is a repeated coextension of Id by powers of H.Let the map of algebraic varieties be a cyclic n+1-fold cover ramified in a divisor D on X. Then the derived functors of inverse and direct image are -functors; the inverse image is split (RF is a direct sum) while the direct image is not. I am going to discuss this example and show that all -functors stem from an algebraic construction (a DG algebra in the category of bimodules over another DG algebra) that resembles it. The degree of resemblance, i.e. whether any such algebra is a non-commutative cyclic cover in the sense of D. Chan (or rather a suitable generalization thereof), is currently an open question.

### Fall Quarter, 2018

- October 2,
**Justin Sawon**(UNC)

Lagrangian fibrations by Prym varieties**Abstract**: The Hitchin systems are Lagrangian fibrations on moduli spaces of Higgs bundles. Their compact counterparts are Lagrangian fibrations on compact holomorphic symplectic manifolds, such as the integrable systems of Beauville-Mukai, Debarre, Arbarello-Ferretti-Saccà, Markushevich-Tikhomirov, and Matteini. The GL-Hitchin system and the Beauville-Mukai system are both fibrations by Jacobians of curves. Thus they are isomorphic to their own dual fibrations. Moreover, Donagi-Ein-Lazarsfeld showed that the Beauville-Mukai system can be degenerated to a compactification of the GL-Hitchin system. The other Hitchin systems and the other compact integrable systems mentioned above are fibrations by Prym varieties. In this talk, we explore these Prym fibrations and the relations between them. In particular, we describe degenerations of some compact examples to compactifications of the Sp-Hitchin systems. We also describe dual fibrations of certain Lagrangian fibrations by Prym varieties. - October 8,
**Ivan Loseu**(Northeastern)

**Special Time:**3pm in 210 Deady

Goldie ranks of primitive ideals and indexes of equivariant Azumaya algebras**Abstract**: We are interested in two numerical invariants of primitive ideals (=the annihilators of irreducible modules) in the universal enveloping algebras.

The first invariant, the Goldie rank, is classical, it measures how many zero divisors the quotient by the ideal has. The second invariant, the dimension of the finite dimensional irreducible representation of a W-algebra corresponding to the primitive ideal, is more recent. It is expected (and is known in many important cases) that the latter invariant is computable via a Kazhdan-Lusztig type formula. Also the two invariants are known to be very closely related thanks to the work of Premet and the speaker. In this talk we discuss the joint work of Ivan Panin and the speaker, arXiv:1802.05651, where we find a lower bound on the ratio of the dimension of the W-algebra module by the Goldie rank. This lower bound is the index of a suitable equivariant Azumaya algebra on a cover of the nilpotent orbit corresponding to the ideal. We explain how to compute the index in the elementary representation theoretic terms. We hope that our lower bound is exact at least for the classical Lie algebras. No prior knowledge of primitive ideals, W-algebras or Azumaya algebras is required. - October 9,
**Rahul Singh**(Northeastern)

The Conormal Variety of a Schubert Variety**Abstract**: Let N be the conormal variety of a Schubert variety X.

In this talk, we discuss the geometry of N in two cases, when X is cominuscule, and when X is a divisor.

In particular, we present a resolution of singularities and a system of defining equations for N, and also describe certain cases when N is normal, Cohen-Macaulay, and Frobenius split.

Time permitting, we will also illustrate the close relationship between N and orbital varieties, and discuss consequences of the above constructions for orbital varieties. - October 16,
**Dick Hain**(Duke)

Hodge Theory and the Goldman–Turaev Lie Bialgebra**Abstract**: The Goldman–Turaev Lie bialgebra of a compact oriented surface X is the free Z-module generated by the conjugacy classes in the fundamental group of X — equivalently, by the closed geodesics in X when X is hyperbolic. The bracket, constructed by Goldman in the 1980s, is related to the Poisson geometry of the moduli space of GL_N bundles over X. The cobracket was defined by Turaev (1970s and 1990s). In its most refined version, it requires a framing of X.The Kashiwara–Vergne problem is, in some sense, a vast generalization of the Baker-Campbell-Hausdorff formula. Solutions in all genera were recently constructed by Alexeev, Kawazumi, Kuno and Naef using classical and elliptic associators. Solutions of the KV problem in each genus form a torsor under a group, referred to as the Galois group of the problem. Determining this group is an important problem. It should be related to relative completions of mapping class groups and to what is called the “motivic Lie algebra” — the free Lie algebra generated by the set {s_3, s_5, s_7, s_9, ….} indexed by odd integers > 1.

In this talk, I will construct the Goldman bracket and Turaev cobracket, using elementary surface topology, and then recall the KV problem. I will then explain how one can use Hodge theory to reprove the existence of solutions of the KV problem for all framed surfaces. This approach yields a large and relatively explicit Galois group which is closely related to the motivic Lie algebra and to relative completions of mapping class groups.

- October 23,
**Jen Berg**(Rice)

Odd ordered transcendental obstructions to the Hasse principle on K3 surfaces**Abstract**: K3 surfaces are 2-dimensional analogues of elliptic curves, but lack a group structure. Moreover, they need not have rational points. However, in 2009 Skorobogatov conjectured that the Brauer group (a torsion abelian group which encodes reciprocity laws) should account for all failures of the local-to-global principle for rational points on K3 surfaces. In this talk I will describe the geometric origin of certain 3-torsion classes in the Brauer group of a K3 surface. We utilize this geometric description to show that these classes can in fact obstruct the existence of rational points. This is joint work with Tony

Varilly-Alvarado. - November 6,
**Nicolle S Gonzalez**(U. Southern California)

Categorical Bernstein Operators and the Boson-Fermion correspondence**Abstract**: Bernstein operators are vertex operators that create and annhilate Schur polynomials. These operators play a significant role in the mathematical formulation of the Boson-Fermion correspondence due to Kac and Frenkel. The role of this correspondence in mathematical physics has been widely studied as it bridges the actions of the infinite Heisenberg and Clifford algebras on Fock space. Cautis and Sussan conjectured a categorification of this correspondence within the framework of Khovanov’s Heisenberg category. I will discuss how to categorify the Bernstein operators and settle the Cautis-Sussan conjecture, thus proving a categorical Boson-Fermion correspondence. - November 13,
**Victor Ostrik**(UO)

Negligible and non-negligible tilting modules**Abstract**: Tilting modules form an interesting family of representations of a reductive group over a field of positive characteristic p. One big open problem is compute their characters and, in particular, dimensions. In this talk we discuss how to compute their dimensions modulo p. This is a joint work with Pavel Etingof. - November 20,
**Zach Hamaker**(Michigan)

Involution Coxeter combinatorics**Abstract**: The combinatorics of Coxeter groups has long been a rich area of study with important applications to representation theory and geometry. Many of the key ideas in this realm have natural analogues when we restrict our attention to involutions in Coxeter groups. Based on pioneering work of Richardson and Springer on K-orbits, I will survey many results translated through the lens of involution words, which are the natural analog of reduced words for involutions. This will include new formulas for equivariant cohomology representatives of some spherical varieties and several combinatorial results awaiting geometric or representation-theoretic explanations. These results only scratch the surface, and many open problems remain! This is joint work with Eric Marberg and Brendan Pawlowski. - November 27, Algebra classes organizational meeting
- November 29,
**Lucia Morotti**(Leibniz Universitat Hannover)

**Special Time:**4pm in 210 Deady

Irreducible tensor products of representations of symmetric and related groups**Abstract**: Given two irreducible representations of dimension greater than 1, their tensor product is in general not irreducible. For example, for symmetric groups such irreducible tensor products are only possible in characteristic 2. For alternating groups and for covering groups of symmetric and alternating groups however there are examples of such irreducible tensor products in arbitrary characteristic.In this talk I will present a (partial) characterisation of irreducible tensor products of symmetric, alternating groups and their covering groups.

- December 19,
**Amit Hazi**(University of Leeds)

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