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# Algebra and Algebraic Geometry seminar at UBC

Except when noted otherwise, the seminar meets **Monday 3:00-4:00pm** on **Zoom** (password:the number of lines on a generic quintic threefold).

## Announcements

Talks and schedule changes are announced on the alggeom mailing list. Please email the organizer to be added to the list.

## Organizers

Kai Behrend (behrend@math.ubc.ca), Dylan G.L. Allegretti (dylan.allegretti@gmail.com), Ming Zhang (zhangming@math.ubc.ca)

## Schedule

### 2020/21 Winter Term 2

2021-01-18 | Dylan G.L. Allegretti (UBC) | Title: Wall-crossing and differential equations |

Abstract: The Kontsevich-Soibelman wall-crossing formula describes the wall-crossing behavior of BPS invariants in Donaldson-Thomas theory. It can be formulated as an identity between (possibly infinite) products of automorphisms of a formal power series ring. In this talk, I will explain how these same products also appear in the exact WKB analysis of Schrödinger's equation. In this context, they describe the Stokes phenomenon for objects known as Voros symbols. |

2021-01-25 | CANCELED | Title |

Abstract: |

2021-02-01 | Henry Liu (Columbia University) | Title: Quasimaps and stable pairs |

Recording (Passcode: M3=8E#SM) | Abstract: I will explain an equivalence between a flavor of Donaldson-Thomas theory (due to Bryan and Steinberg) on ADE surface fibrations and quasimaps to Hilbert schemes of ADE surfaces. The proof involves an explicit combinatorial description of vertices. The equivalence can be used to relate machinery from both sides, notably an equivariant K-theoretic DT crepant resolution conjecture and 3d mirror symmetry. |

2021-02-08 | Yu-Wei Fan (UC Berkeley) | Title: Stokes matrices, character varieties, and points on spheres |

Slides | Abstract: Moduli spaces of points on n-spheres carry natural actions of braid groups. For n=0,1, and 3, we prove that these symmetries extend to actions of mapping class groups of positive genus surfaces, through exceptional isomorphisms with certain character varieties. We also apply the exceptional isomorphisms to the study of Stokes matrices and exceptional collections of triangulated categories. Joint work with Junho Peter Whang. |

2021-02-22 | Mark Shoemaker (Colorado State University) | Title: A mirror theorem for gauged linear sigma models |

Abstract: Let G be a finite group acting on a smooth complex variety M. Let X —> M/G be a crepant resolution by a smooth variety X. The Crepant Resolution Conjecture predicts a complicated relationship between the Gromov—Witten invariants of X and the orbifold Gromov—Witten invariants of the stack [M/G].
In this talk I will describe an analogous conjecture involving Landau—Ginzburg (LG) models. An LG model is, roughly, a smooth complex variety Y together with a regular function w: Y—> \CC. LG models can be used to give alternate “resolutions” of hypersurface singularities in a certain sense and are related to so-called noncommutative resolutions. I will briefly discuss the gauged linear sigma model, which is used to define curve counting invariants for LG models, and describe a new technique for computing these invariants. |

2021-03-01 | Yefeng Shen (University of Oregon) | Title: Virasoro constraints in quantum singularity theory |

Abstract: In this talk, we introduce Virasoro operators in quantum singularity theories for nondegenerate quasi-homogeneous polynomials with nontrivial diagonal symmetries. Using Givental's quantization formula of quadratic Hamiltonians, these operators satisfy the Virasoro relations. Inspired by the famous Virasoro conjecture in Gromov-Witten theory, we conjecture that the genus g generating functions arise in quantum singularity theories are annihilated by the Virasoro operators. We verify the conjecture in various examples and discuss the connections to mirror symmetry of LG models and LG/CY correspondence. This talk is based on work joint with Weiqiang He. |

2021-03-08 | Davesh Maulik (MIT) | Title: Cohomology of the moduli of Higgs bundles and the Hausel-Thaddeus conjecture |

Abstract: In this talk, I will discuss some results on the structure of the cohomology of the moduli space of stable SL_n Higgs bundles on a curve. One consequence is a new proof of the Hausel-Thaddeus conjecture proven previously by Groechenig-Wyss-Ziegler via p-adic integration. We will also discuss connections to the P=W conjecture if time permits. Based on joint work with Junliang Shen. |

2021-03-15 | Jack Petok (Dartmouth College) | Title: Kodaira dimensions of some moduli spaces of hyperkähler fourfolds |

Abstract: We use modular forms to study the birational geometry of some moduli spaces of hyperkähler fourfolds. I'll review a bit of the algebraic geometry of these moduli spaces, and then I'll explain some methods, due to Gritsenko, Hulek, and Sankaran, for computing their Kodaira dimensions. These methods make use of special modular forms defined on high rank orthogonal groups. I'll also report on an ongoing project with Jen Berg applying related techniques to certain moduli spaces of Enriques surfaces. |

2021-03-22 | Qile Chen (Boston College) | Title: Counting curves in critical locus via logarithmic compactifications |

Abstract: I will introduce some recent developments and work in progress on studying Gauged Linear Sigma Models using logarithmic compactifications.
These logarithmic compactifications admit two types of virtual cycles --- the reduced virtual cycles that recover Gromov-Witten invariants of complete intersections, and the canonical virtual cycles that depend only on the geometry of ambient spaces. These two types of virtual cycles differ only by a third virtual cycle of the boundary of the logarithmic compactifications. Using the punctured logarithmic maps of Abramovich-Chen-Gross-Siebert, these virtual cycles can be computed via the tropical and equivariant geometry of the logarithmic compactifications. This leads to a new method for computing Gromov-Witten invariants of complete intersections. The talk consists of joint work with Felix Janda, Yongbin Ruan, Adrien Sauvaget and Rachel Webb. |

2021-03-29 | Arnav Tripathy (Harvard University) | Title: K3s as Hyperkahler Quotients |

Abstract: I'll explain how to construct K3 surfaces as hyperkahler quotients and, as time permits, our expected application to counting open GW invariants. This is all joint work with M. Zimet. |

2021-04-12 | Zijun Zhou (IPMU) | Title: 3d mirror symmetry, vertex function, and elliptic stable envelope |

Abstract: 3d mirror symmetry is a duality in physics, where Higgs and Coulomb branches of certain pairs of 3d N=4 SUSY gauge theories are exchanged with each other. Motivated from this, M. Aganagic and A. Okounkov introduced the enumerative geometric conjecture that the vertex functions of the mirror theories are related to each other. The two sets of q-difference equations satisfied by the vertex functions, in terms of the K\"ahler and equivariant parameters, are expected to exchange with each other. The conjecture therefore leads to a nontrivial relation between their monodromy matrices, the so-called elliptic stable envelopes. In this talk, I will discuss the proof in several cases of the conjecture for both vertex functions and elliptic stable envelopes. This is based on joint works with R. Rim\'anyi, A. Smirnov, and A. Varchenko. |

### 2020/21 Winter Term 1

2020-09-25 | Renzo Cavalieri (Colorado State University) | Title: Tropical Psi Classes |

Abstract: |

2020-10-02 | Pierrick Bousseau (ETH Zürich) | Title: The skein algebra of the 4-punctured sphere from curve counting |

Recording (Passcode: &&RAd62X) | Abstract: The Kauffman bracket skein algebra is a quantization of the algebra of regular functions on the SL_2 character variety of a topological surface. I will explain how to realize the skein algebra of the 4-punctured sphere as the output of a mirror symmetry construction based on higher genus Gromov-Witten invariants of a log Calabi-Yau cubic surface. This leads to a proof of a previously conjectured positivity property of the bracelets bases of the skein algebras of the 4-punctured sphere and of the 1-punctured torus. |

2020-10-09 | Tony Yue Yu (Université Paris-Sud, Paris-Saclay) | Title: Secondary fan, theta functions and moduli of Calabi-Yau pairs |

Recording (Passcode: xK1tqv?!) | Abstract: We conjecture that any connected component $Q$ of the moduli space of triples $(X,E=E_1+\dots+E_n,\Theta)$ where $X$ is a smooth projective variety, $E$ is a normal crossing anti-canonical divisor with a 0-stratum, every $E_i$ is smooth, and $\Theta$ is an ample divisor not containing any 0-stratum of $E$, is \emph{unirational}. More precisely: note that $Q$ has a natural embedding into the Kollár-Shepherd-Barron-Alexeev moduli space of stable pairs, we conjecture that its closure admits a finite cover by a complete toric variety. We construct the associated complete toric fan, generalizing the Gelfand-Kapranov-Zelevinski secondary fan for reflexive polytopes. Inspired by mirror symmetry, we speculate a synthetic construction of the universal family over this toric variety, as the Proj of a sheaf of graded algebras with a canonical basis, whose structure constants are given by counts of non-archimedean analytic disks. In the Fano case and under the assumption that the mirror contains a Zariski open torus, we construct the conjectural universal family, generalizing the families of Kapranov-Sturmfels-Zelevinski and Alexeev in the toric case. In the case of del Pezzo surfaces with an anti-canonical cycle of $(-1)$-curves, we prove the full conjecture. The reference is arXiv:2008.02299 joint with Hacking and Keel. |

2020-10-16 | Yan Soibelman (Kansas State University) | Title: Exponential integrals, Holomorphic Floer theory and resurgence |

Recording (Passcode: WcrGN=$\$$0) | Abstract: Holomorphic Floer theory is a joint project with Maxim Kontsevich, which is devoted to various aspects of the Floer theory in the framework of complex symplectic manifolds.
In my talk I will consider an important special case of the general story. Exponential integrals in finite and infinite dimension can be thought of generalization of the theory of periods (i.e variations of Hodge structure). In particular, there are comparison isomorphisms between Betti and de Rham cohomology in the exponential setting. These isomorphisms are corollaries of categorical equivalences which are incarnations of our generalized Riemann-Hilbert correspondence for complex symplectic manifolds. Furthermore, fomal series which appear e.g. in the stationary phase method or Feynman expansions (in infinite dimensions) are Borel re-summable (resurgent). If time permits I will explain the underlying mathematical structure which we call analytic wall-crossing structure. From the perspective of Holomorphic Floer theory it is related to the estimates for the number of pseudo-holomorphic discs with boundaries on two given complex Lagrangian submanifolds. |

2020-10-23 | Dragos Oprea (UCSD) | Title: Virtual invariants of Quot schemes of surfaces |

Recording (Passcode: Tx4+E5Yv) | Abstract: The Quot schemes of surfaces parametrizing quotients of dimension at most 1 of the trivial sheaf carry 2-term perfect obstruction theories. Several generating series of associated virtual invariants are conjectured to be given by rational functions. We show this is the case for several geometries including all smooth projective surfaces with p_g>0. This talk is based on joint work with Noah Arbesfeld, Drew Johnson, Woonam Lim and Rahul Pandharipande. |

2020-10-30 | Daniel Halpern-Leistner (Cornell University) | Title: Derived Theta-stratifications and the D-equivalence conjecture |

Recording (Passcode: 3N.pSg3y) | Abstract: The D-equivalence conjecture predicts that birationally equivalent projective Calabi-Yau manifolds have equivalent derived categories of coherent sheaves. It is motivated by homological mirror symmetry, and has inspired a lot of recent work on connections between birational geometry and derived categories. In dimension 3, the conjecture is settled, but little is known in higher dimensions. I will discuss a proof of this conjecture for the class of Calab-Yau manifolds that are birationally equivalent to some moduli space of stable sheaves on a K3 surface. This is the only class for which the conjecture is known in dimension >3. The proof uses a more general structure theory for the derived category of an algebraic stack equipped with a Theta-stratification, which we apply in this case to the Harder-Narasimhan stratification of the moduli of sheaves. |

2020-11-06 | Yifeng Huang (University of Michigan) | Title: Cohomology of configuration spaces of punctured varieties |

Recording (Passcode: 8o1$\$$FES=) | Abstract: Given a smooth complex variety X (not necessarily compact), consider the unordered configuration space Conf^n(X) defined as {(x_1,...,x_n)\in X^n: x_i \neq x_j for i\neq j} / S_n. The singular cohomology of Conf^n(X) has long been an active area of research. In this talk, we investigate the following phenomenon: "puncturing once more" seems to have a very predictable effect on the cohomology of configuration spaces when the variety we start with is noncompact. In specific, a formula of Napolitano determines the Betti numbers of Conf^n(X - {P}) from the Betti numbers of Conf^m(X) (m \leq n) if X is a smooth *noncompact* algebraic curve and P is a point. We present a new proof using an explicit algebraic method, which also upgrades this formula about Betti numbers into a formula about mixed Hodge numbers and generalizes this formula to certain cases where X is of higher dimension. |

2020-11-13 | Christopher Woodward (Rutgers University) | Title: Quantum K-theory of git quotients |

8:30-9:30 AM | Recording (Passcode: .Vj7Lquc) | Abstract: (w E. Gonzalez) I will discuss a generalization of the Kirwan map to quantum K-theory, a presentation of quantum K-theory of toric varieties, and some open questions. |

2020-11-13 | Ming Zhang (UBC) | Title: Verlinde/Grassmannian Correspondence |

9:45-10:45 AM | Recording (Passcode: .Vj7Lquc click $\rangle|$ button to see the second recording) | Abstract: In the 90s', Witten gave a physical derivation of an isomorphism between the Verlinde algebra of GL(n) of level l and the quantum cohomology ring of the Grassmannian Gr(n,n+l). In the joint work arXiv:1811.01377 with Yongbin Ruan, we proposed a K-theoretic generalization of Witten's work by relating the GL_n Verlinde numbers to the level l quantum K-invariants of the Grassmannian Gr(n,n+l), and refer to it as the Verlinde/Grassmannian correspondence. The correspondence was formulated precisely in the aforementioned paper, and we proved the rank 2 case (n=2) there.
In this talk, I will first explain the background of this correspondence and its interpretation in physics. Then I will discuss the main idea of the proof for arbitrary rank. A new technical ingredient is the virtual nonabelian localization formula developed by Daniel Halpern-Leistner. |

2020-11-20 | Emily Clader (San Francisco State University) | Title: Permutohedral Complexes and Curves With Cyclic Action |

Recording (Passcode: ?5H+jdKC) | Abstract: Although the moduli space of genus-zero curves is not a toric variety, it shares an intriguing amount of the combinatorial structure that a toric variety would enjoy. In fact, by adjusting the moduli problem slightly, one finds a moduli space that is indeed toric, known as Losev-Manin space. The associated polytope is the permutohedron, which also encodes the group-theoretic structure of the symmetric group. Batyrev and Blume generalized this story by constructing a "type-B" version of Losev-Manin space, whose associated polytope is a signed permutohedron that relates to the group of signed permutations. In joint work in progress with C. Damiolini, D. Huang, S. Li, and R. Ramadas, we carry out the next stage of generalization, defining a family of moduli space of curves with Z_r action encoded by an associated "permutohedral complex" for a more general complex reflection group, which specializes when r=2 to Batyrev and Blume's moduli space. |

2020-11-27 | Dustin Ross (San Francisco State University) | Title: Putting the "volume" back in volume polynomials |

Abstract: It is a strange and wonderful fact that Chow rings of matroids behave in many ways similarly to Chow rings of smooth projective varieties. Because of this, the Chow ring of a matroid is determined by a homogeneous polynomial called its volume polynomial, whose coefficients record the degrees of all possible top products of divisors. The use of the word "volume" is motivated by the fact that the volume polynomial of a smooth projective toric variety actually measures the volumes of certain polytopes associated to the variety. In the matroid setting, on the other hand, no such polytopes exist, and the goal of our work was to find more general polyhedral objects whose volume is measured by the volume polynomial of matroids. In this talk, I will motivate and describe these polyhedral objects. This is joint work with Anastasia Nathanson. |

2020-12-04 | Junliang Shen (MIT) | Title: Intersection cohomology of the moduli of of 1-dimensional sheaves and the moduli of Higgs bundles |

8:30-9:30 AM | Recording (Passcode: 39RHp$\$$v8) | Abstract: In general, the topology of the moduli space of semistable sheaves on an algebraic variety relies heavily on the choice of the Euler characteristic of the sheaves. We show a striking phenomenon that, for the moduli of 1-dimensional semistable sheaves on a toric del Pezzo surface (e.g. P^2) or the moduli of semistable Higgs bundles with respect to a divisor of degree > 2g-2 on a curve, the intersection cohomology of the moduli space is independent of the choice of the Euler characteristic. This confirms a conjecture of Bousseau for P^2, and proves a conjecture of Toda in the case of local toric Calabi-Yau 3-folds. In the proof, a generalized version of Ngô's support theorem plays a crucial role. Based on joint work in progress with Davesh Maulik. |