Palace of Catalan Music (S. Adams/GETTY)

## Geometry, Physics, and Representation Theory Seminar

### RTG Group Seminar

#### Fall 2020, Northeastern

 Meeting weekly on Thursdays 2:50-3:50pm via Zoom. If you are not at Northeastern, but would like to recieve announcements, join the mailing list. Organizers: Peter Crooks, Iva Halacheva, Ignacio Barros, Vance Blankers, Christopher Beasley, Valerio Toledano Laredo. If you have a question or would like to speak at the seminar, please email v.blankers [at] northeastern.edu.

### Schedule

 Date Speaker Title Sept 3 Maxence Mayrand (University of Toronto) Symplectic reduction along a submanifold and the Moore-Tachikawa TQFT abstract ± In 2011, Moore and Tachikawa conjectured the existence of certain complex Hamiltonian varieties which generate two-dimensional TQFTs where the target category has complex reductive groups as objects and holomorphic symplectic varieties as arrows. It was solved by Ginzburg and Kazhdan using an ad hoc technique which can be thought of as a kind of "symplectic reduction by a group scheme." We clarify their construction by introducing a general notion of "symplectic reduction by a groupoid along a submanifold," which generalizes many constructions at once, such as standard symplectic reduction, preimages of Slodowy slices, the Mikami-Weinstein reduction, and the Ginzburg-Kazhdan examples. This is joint work with Peter Crooks. video ± Sept 10 Svetlana Makarova (University of Pennsylvania) Moduli spaces of stable sheaves over quasipolarized K3 surfaces and Strange Duality abstract ± In this talk, I will show a construction of relative moduli spaces of stable sheaves over the stack of quasipolarized K3 surfaces of degree two. For this, we use the theory of good moduli spaces, whose study was initiated by Alper. As a corollary, we obtain the generic Strange Duality for K3 surfaces of degree two, extending the results of Marian and Oprea on the generic Strange Duality for K3 surfaces. video ± Sept 17 Lisa Jeffrey (University of Toronto) Flat connections and the $SU(2)$ commutator map abstract ± This talk is joint work with Nan-Kuo Ho, Paul Selick and Eugene Xia. We describe the space of conjugacy classes of representations of the fundamental group of a genus 2 oriented 2-manifold into $G:= SU(2)$. We identify the cohomology ring and a cell decomposition of a space homotopy equivalent to the space of commuting pairs in $SU(2)$. We compute the cohomology of the space $M:=\mu^{-1}(-I)$, where $\mu:G^4 \to G$ is the product of commutators. We give a new proof of the cohomology of $A:= M/G$, both as a group and as a ring. The group structure is due to Atiyah and Bott in their landmark 1983 paper. The ring structure is due to Michael Thaddeus 1992. We compute the cohomology of the total space of the prequantum line bundle over $A$. We identify the transition functions of the induced $SO(3)$ bundle $M\to A$. To appear in QJM (Atiyah memorial special issue). arXiv:2005.07390 References: [1] M.F. Atiyah, R. Bott, The Yang-Mills equations over Riemann surfaces, Phil. Trans. Roy. Soc. Lond. A308 (1983) 523-615. [2] T. Baird, L. Jeffrey, P. Selick, The space of commuting n-tuples in $SU(2)$, Illinois J. Math. 55 (2011), no. 3, 805–813. [3] M. Crabb, Spaces of commuting elements in $SU(2)$, Proc. Edin. Math. Soc. 54 (2011), no. 1, 67–75. [4] N. Ho, L. Jeffrey, K. Nguyen, E. Xia, The $SU(2)$-character variety of the closed surface of genus 2. Geom. Dedicata 192 (2018), 171–187. [5] N. Ho, L. Jeffrey, P. Selick, E. Xia, Flat connections and the commutator map for $SU(2)$, Oxford Quart. J. Math., to appear (in the Atiyah memorial special issue). [6] L. Jeffrey, A. Lindberg, S. Rayan, Explicit Poincar´e duality in the cohomology ring of the $SU(2)$ character variety of a surface. Expos. Math., to appear. [7] M.S. Narasimhan and C.S. Seshadri, Stable and unitary vector bundles on a compact Riemann surface. Ann. of Math. 82 (1965) 540–567. [8] P. Newstead, Topological properties of some spaces of stable bundles, Topology 6 (1967), 241–262. [9] C.T.C Wall, Classification problems in differential topology. V. On certain 6-manifolds. Invent. Math. 1 (1966), 355–374; corrigendum, ibid., 2 (1966) 306. [10] M. Thaddeus, Conformal field theory and the cohomology of the moduli space of stable bundles. J. Differential Geom. 35 (1992) 131–149. [11] E. Witten, Two dimensional gauge theories revisited, J. Geom. Phys. 9 (1992) 303-368. video ± Sept 24 Balázs Elek (Cornell University) Heaps, Crystals and Preprojective algebra modules abstract ± Kashiwara crystals are combinatorial gadgets associated to a representation of a reductive algebraic group that enable us to understand the structure of the representation in purely combinatorial terms. We will describe a type-independent construction of crystals of certain representations, using the heap associated to a fully commutative element in the Weyl group. Then we will discuss how these heaps also lead us to the construction of modules for the preprojective algebra of the Dynkin quiver. Using the rank-nullity theorem, we will see how the Kashiwara operators have a surprisingly nice description in terms of these preprojective algebra modules. This is work in progress joint with Anne Dranowski, Joel Kamnitzer, Tanny Libman and Calder Morton-Ferguson. video ± Oct 1 Andrey Smirnov (University of North Carolina at Chapel Hill) Quantum difference equations, monodromies and mirror symmetry abstract ± An important enumerative invariant of a symplectic variety $X$ is its vertex function. The vertex function is the analog of J-function in Gromov-Witten theory: it is the generating function for the numbers of rational curves in $X$. In representation theory these functions feature as solutions of various $q$-difference and differential equations associated with $X$, with examples including qKZ and quantum dynamical equations for quantum loop groups, Casimir connections for Yangians and other objects. In this talk I explain how these equations can be extracted from algebraic topology of symplectic dual variety $X^!$, also known as $3D$-mirror of $X$. This can be summarized as "identity" $$\text{Enumerative geometry of }X = \text{algebraic topology of }X^!$$ The talk is based on work in progress with Y.Kononov arXiv:2004.07862; arXiv:2008.06309. video ± Oct 8 Nicolle González (University of California at Los Angeles) A Skein theoretic Carlsson-Mellit algebra abstract ± The Carlsson-Mellit algebra arose for the first time in the proof of the shuffle conjecture, which gives an explicit combinatorial formula for the Frobenius character of the space of diagonal harmonics in terms of parking functions. Its polynomial representation, given by certain complicated plethystic operators over extensions of the ring of symmetric functions, plays a particularly important role as it encodes much of the underlying combinatorial theory. By various results of Gorsky, Mellit and Carlsson it was shown that this algebra can be used to construct generators of the Elliptic Hall algebra in addition to having deep connections to the homology of torus knots. Thus, a natural starting point in the search to categorify these structures is the categorification of the Carlsson-Mellit algebra and its polynomial representation. . In this talk I will explain joint work with Matt Hogancamp where we constructed a purely skein theoretic formulation of this algebra and realized its generators as certain braid diagrams on a thickened annulus. Consequently, we used this framework to categorify the polynomial representation of the Carlsson-Mellit algebra as a family of functors over the derived trace of the Soergel category. . Slides . video ± Oct 15 Soheyla Feyzbakhsh (Imperial) An application of Bogomolov-Gieseker type inequality to counting invariants abstract ± In this talk, I will work on a smooth projective threefold $X$ which satisfies the Bogomolov-Gieseker conjecture of Bayer-Macrì-Toda, such as the projective space $\mathbb{P}^3$ or the quintic threefold. I will show certain moduli spaces of 2-dimensional torsion sheaves on $X$ are smooth bundles over Hilbert schemes of ideal sheaves of curves and points in $X$. When $X$ is Calabi-Yau this gives a simple wall crossing formula expressing curve counts (and so ultimately Gromov-Witten invariants) in terms of counts of D4-D2-D0 branes. This is joint work with Richard Thomas. video ± No video available at this time. Please contact the organizers or check back later! Oct 22 Anton Alekseev (University of Geneva) Poisson-Lie groups, integrable systems and the Berenstein-Kazhdan potential abstract ± Integrable systems and Poisson-Lie groups are closely related topics. In this talk, we will explain how integrability helps in understanding Poisson geometry of the dual Poisson-Lie group $K^*$ of a compact Lie group $K$. One of our main tools will be the Berenstein-Kazhdan potential from the theory of canonical bases. . The talk is based on joint works with A. Berenstein, I. Davidenkova, B. Hoffman, J. Lane and Y. Li. video ± Oct 29 José Simental Rodríguez (University of California at Davis) Parabolic Hilbert schemes and representation theory abstract ± We explicitly construct an action of type A rational Cherednik algebras and, more generally, quantized Gieseker varieties, on the equivariant homology of the parabolic Hilbert scheme of points on the plane curve singularity $C = \{x^{m} = y^{n}\}$ where $m$ and $n$ are coprime positive integers. We show that the representation we get is a highest weight irreducible representation and explicitly identify its highest weight. We will also place these results in the recent context of Coulomb branches and BFN Springer theory. This is joint work with Eugene Gorsky and Monica Vazirani. video ± Nov 5 Noah White (Australian National University) Cactus group actions and cell modules abstract ± The cactus group associated to a Coxeter group can be thought of as an asymptotic version of the braid group. It has been observed by many authors that interesting cactus group actions can be constructed in many situations when one has a representation of the braid group. In this talk I will explain what the cactus group is, and what is meant by "asymptotic". I will also explain how to construct cactus group actions associated to cell modules of the Hecke algebra, a description of this action using Lusztig’s isomorphism between the Hecke algebra and group algebra and point to some interesting questions along the way. Much of this talk is work joint with Raphael Rouquier. video ± Nov 12 Ulrike Rieß (ETH Zurich) On the Kähler cone of irreducible symplectic orbifolds abstract ± In this talk I report on recent joint work with G. Menet: We generalize a series of classical results on irreducible symplectic manifolds to the orbifold setting. In particular we prove a characterization of the Kähler cone using wall divisors. This generalizes results of Mongardi for the smooth case. I will finish the talk by applying these results to study a concrete example. video ± Nov 19 Rekha Biswal (University of Edinburgh) Macdonald polynomials and level two Demazure modules for affine $\mathfrak{sl}_{n+1}$ abstract ± An important result due to Sanderson and Ion says that characters of level one Demazure modules are specialized Macdonald polynomials. In this talk, I will introduce a new class of symmetric polynomials indexed by a pair of dominant weights of $\mathfrak{sl}_{n+1}$ which is expressed as linear combination of specialized symmetric Macdonald polynomials with coefficients defined recursively. These polynomials arose in my own work while investigating the characters of higher level Demazure modules. Using representation theory, we will see that these new family of polynomials interpolate between characters of level one and level two Demazure modules for affine $\mathfrak{sl}_{n+1}$ and give rise to new results in the representation theory of current algebras as a corollary. This is based on joint work with Vyjayanthi Chari, Peri Shereen and Jeffrey Wand. video ± No video available at this time. Please contact the organizers or check back later! Nov 26 No Seminar. Thanksgiving Break. Dec 3 Li Yu (University of Chicago) Wonderful compactification of a Cartan subalgebra of a semisimple Lie algebra abstract ± Let $H$ be a Cartan subgroup of a semisimple algebraic group $G$ over the complex numbers. The wonderful compactification $\overline{H}$ of $H$ was introduced and studied by De Concini and Procesi. For the Lie algebra $\mathfrak{h}$ of $H$, we define an analogous compactification $\overline{\mathfrak{h}}$ of $\mathfrak{h}$, to be referred to as the wonderful compactification of $\mathfrak{h}$. The wonderful compactification of $\mathfrak{h}$ is an example of an "additive toric variety". We establish a bijection between the set of irreducible components of the boundary $\overline{\mathfrak{h}} - \mathfrak{h}$ of $\mathfrak{h}$ and the set of maximal closed root subsystems of the root system for $(G, H)$ of rank $r - 1$, where $r$ is the dimension of $\mathfrak{h}$. An algorithm based on Borel-de Siebenthal theory that constructs all such root subsystems is given. We prove that each irreducible component of $\overline{\mathfrak{h}}- \mathfrak{h}$ is isomorphic to the wonderful compactification of a Lie subalgebra of $\mathfrak{h}$ and is of dimension $r - 1$. In particular, the boundary $\overline{\mathfrak{h}} - \mathfrak{h}$ is equidimensional. We describe a large subset of the regular locus of $\overline{\mathfrak{h}}$. As a consequence, we prove that $\overline{\mathfrak{h}}$ is a normal variety. video ± End of Semester (Seminar resumes Spring 2021)

#### Previous Semesters

 Spring 2020 Fall 2019 Spring 2019 Fall 2018 Spring 2018 Fall 2017 Spring 2017 Fall 2016 Spring 2016 Fall 2015 Spring 2015