Palace of Catalan Music (S. Adams/GETTY)

## Geometry, Physics, and Representation Theory Seminar

### RTG Group Seminar

#### Spring 2018, Northeastern

 Meeting weekly on Thursdays 2:50-3:50pm in 509/511 Lake Hall at Northeastern. If you are not at Northeastern, but would like to recieve announcements, join the mailing list. If you have a question or would like to speak at seminar, email

### Schedule

 Date Speaker Title Jan 11 No Seminar Jan 18 No Seminar Wed, Jan 24 at 2:50 Martí Lahoz (U. de Barcelona / U. Paris Diderot) Bridgeland stability for semiorthogonal decompositions and cubic fourfolds We illustrate a new method to induce stability conditions on semiorthogonal decompositions and apply it to the Kuznetsov component of the derived category of cubic fourfolds. We use this to reprove the Torelli theorem for cubic fourfolds, to generalize the results of Addington-Thomas, and to study the rich hyperkähler geometry associated to these hypersurfaces. This is the content of joint works with Arend Bayer, Emanuele Macrì, Howard Nuer, Alex Perry, and Paolo Stellari. Wed, Jan 31 at 2:50 Kostya Tolmachov (MIT) Towards a categorification of a projection from an affine to a finite Hecke algebra in type $A$ Work of Bezrukavnikov on local geometric Langlands correspondence and works of Gorsky, Neguţ, Rasmussen and Oblomkov, Rozansky on knot homology and matrix factorizations suggest that there should be a categorical version of a certain natural homomorphism from an affine Hecke algebra to a finite Hecke algebra in type A, sending basis lattice elements on the affine side to Jucys-Murphy elements on the finite side. I will talk about recent progress towards a construction of such a categorification in the setting of Hecke categories. Feb 8 Seth Shelley-Abrahamson (MIT) The Dunkl Weight Function for Representations of Rational Cherednik Algebras 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 formulas for the invariant Hermitian forms on the "Verma modules" in the associated category $\mathcal{O}$. 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. Feb 15 Francois Greer (Stony Brook) Elliptic fibrations in the presence of singularities I will present recent progress on genus 0 curve counting for elliptic fibrations. The Gromov-Witten generating series for an elliptic fibration is expected to have modular properties by mirror symmetry. When the homology class in the base is irreducible and the total space is smooth, we obtain a classical modular form for the full modular group. If the base class is reducible, we expect the series to be quasi-modular. If the fibration does not admit a section, then the modular form has higher level. Both of these relaxations are related to the presence of singularities in the geometry. Feb 22 Zhixian Zhu (UC Riverside) Fujita's freeness conjecture for 5-fold. Let $X$ be a smooth projective variety of dimension $n$ and $A$ any ample line bundle. Fujita conjectured that the adjoint line bundle $\mathcal{O}(K_X+mA)$ is globally generated for any m greater or equal to $\mathrm{dim}(X)+1$. we prove Fujita's freeness conjecture for smooth 5-folds via a singularity approach. More explicitly, we constructed an effective $\mathbf{Q}$-divisor $D$, and study the singularity of the pair $(X, D)$, whose multiplier ideals providing the required vanishing result. Mar 1 Xiaolei Zhao (Northeastern) Twisted cubics on cubic fourfolds and stability conditions. It is a classical result of Beauville and Donagi that Fano varieties of lines on cubic fourfolds are hyper-Kahler. More recently, Lehn, Lehn, Sorger and van Straten constructed a hyper-Kahler eightfold out of twisted cubics on cubic fourfolds. In this talk, I will explain a new approach to these hyper-Kahler varieties via moduli of stable objects on the Kuznetsov components. Along the way, we will derive several properties of cubic fourfolds as consequences. This is based on a joint work with Chunyi Li and Laura Pertusi. Mar 8 No Seminar (Spring Break) Mar 15 Yuchen Liu (Yale) The normalized volume of a singularity is lower semi-continuous Motivated by work in differential geometry, Chi Li introduced the normalized volume of a klt singularity as the minimum normalized volume of all valuations centered at the singularity. This invariant carries some interesting geometric/topological information of the singularity. In this talk, we show that in a Q-Gorenstein flat family of klt singularities, normalized volumes are lower semicontinuous with respect to the Zariski topology. As an application, we show that K-semistability is a very generic or empty property in a Q-Gorenstein flat family of Q-Fano varieties. This is a joint work with Harold Blum. Mar 22 David Treumann (Boston College) Betti spectral curves and Betti spectral 3-manifolds. A Lagrangian $L$ in the cotangent bundle of $M$ can determine a family of local systems on $M$, by either Floer theory or microlocal sheaf theory. When $M$ is two-dimensional, there is a close analogy between that construction, and the family of Higgs bundles parametrized by a spectral curve in the cotangent of a Riemann surface. When $M$ is 3-dimensional you could do something similar, and the analogy is more compelling if you allow for the local system on M to have "irregular singularities" along a boundary -- but you have to make up a definition of irregular singularities, in 3d. I'll try to explain starting from Deligne's old irregular Riemann-Hilbert correspondence. The talk is partly based on joint work with Linhui Shen and Eric Zaslow, and partly based on joint work with Xin Jin. Mar 29 Vasily Dolgushev (Temple) What Drinfeld could have replied to Deligne My colleagues keep asking me if there is an explicit construction whose input is a Drinfeld associator and whose output is a formality quasi-isomorphism for the brace operad BR. In my talk, I will give a positive answer to this question by showing that there are no obstructions for constructing a quasi-isomorphism from the Cobar-Bar resolution of BR to the chain operad of parenthesized braids. In principle, Drinfeld could have sketched this construction as a possible response to Deligne's famous letter from 1993. Of course, this story would not be complete without Tamarkin's paper "Formality of chain operad of small squares" from 1998 and Fiedorowicz's recognition principle. Details of this construction will appear in a joint textbook (in preparation) with Boris Tsygan. Apr 5 Luca Schaffler (U Mass, Amherst) Equations for point configurations to lie on a rational normal curve Let $V_{d,n}\subseteq(\mathbb{P}^d)^n$ be the Zariski closure of the set of $n$-tuples of points lying on a rational normal curve. The variety $V_{d,n}$ was introduced because it provides interesting birational models of $\overline{M}_{0,n}$: namely, the GIT quotients $V_{d,n} /\!\!/ SL_{d+1}$. In this talk our goal is to find the defining equations of $V_{d,n}$. In the case $d=2$ we have a complete answer. For twisted cubics, we use the Gale transform to find equations defining $V_{3,n}$ union the locus of degenerate point configurations. We prove a similar result for $d\geq4$ and $n=d+4$. This is joint work with Alessio Caminata, Noah Giansiracusa, and Han-Bom Moon. Apr 12 Daniel Litt (Columbia) Canonical paths and monodromy Let $K$ be a local field (archimedean or non-archimedean) and $X$ a normal variety over K. Then, in several settings, there exist canonical linear combinations of paths between any two points in $X$. I'll explain several applications of this observation: (1) a monodromy-free theory of iterated integration on complex varieties, (2) some structural results about Galois actions on pro-$\ell$ geometric fundamental groups of varieties over p-adic local fields, for p different from $\ell$, and (3) results on the representation theory of arithmetic fundamental groups. Apr 19 1:35PM Frank Gounelas (Humboldt University of Berlin) Positivity of the cotangent bundle of a K3 surface In a program set out by Sakai, Kobayashi, Bogomolov and others, one aims to refine the Kodaira classification in terms of positivity of the cotangent sheaf instead of just that of its top exterior power the canonical divisor. In this talk I will discuss joint work with J.C. Ottem in which we study various notions of positivity for the cotangent bundle of a general K3 surface. Our results are applications of classical geometric constructions but also recent results of Bayer-Macri studying the cones of divisors on Hilbert schemes of points. Apr 19 2:50PM Mandy Cheung (Harvard) Quiver representations and theta functions. Scattering diagrams theta functions and broken lines were developed in order to describe toric degenerations of Calabi-Yau varieties and construct mirror pairs. Later, Gross-Hacking-Keel-Kontsevich unravel the relation of those objects with cluster algebras. In the talk, we will discuss how we can combine the representation theory with these objects. We will also see how the broken lines on scattering diagram give a stratification of quiver Grassmannians using this setting. End of Semester (Seminar Resumes Fall 2018)

#### Previous Semesters

 Fall 2017 Spring 2017 Fall 2016 Spring 2016 Fall 2015 Spring 2015