**Spring 2024 program.**For previous semesters including recordings, see the bottom of the page.

### February 12, 2024

## Felix Thimm : The 3-fold K-theoretic DT/PT vertex correspondence

** Abstract: **
Donaldson-Thomas (DT) and Pandharipande-Thomas (PT) invariants are
two curve counting invariants for 3-folds. In the Calabi-Yau case, a
correspondence between the numerical DT and PT invariants has been
conjectured by Pandharipande and Thomas and proven by Bridgeland and
Toda using wall-crossing. For equivariant K-theoretically refined
invariants, the DT/PT correspondence reduces to a DT/PT
correspondence of equivariant K-theoretic vertices. In this talk I
will explain our proof of the equivariant K-theoretic DT/PT vertex
correspondence using a K-theoretic version of Joyce's wall-crossing
setup. An important technical tool is the construction of a
symmetized pullback of a symmetric perfect obstruction theory on the
orginial DT and PT moduli stacks to a symmetric almost perfect
obstruction theory on auxiliary moduli stacks. This is joint work
with Henry Liu and Nick Kuhn.

### Monday,April 15th, 2024

## Miguel Moreira : The cohomology ring of moduli spaces of 1-dimensional sheaves on $\mathbb{P}^2$

** Abstract:**
The cohomology of moduli spaces of
1-dimensional sheaves, together with a special filtration called the
perverse filtration, can be used to give an intrinsic definition of
(refined) Gopakumar-Vafa invariants. While there are methods to
calculate the Betti numbers of these moduli spaces in low degree,
understanding the perverse filtration is more challenging. One way to
compute it is to fully determine the cohomology ring. In this talk I
will explain an approach to describing the cohomology rings of moduli
spaces and moduli stacks in terms of generators and relations, which
allowed us to determine them for curve classes up to degree 5 (for
moduli spaces) and 4 (for moduli stacks). I will explain the $P=C$
conjecture, which is an analogue of the $P=W$ conjecture in a Fano and
compact setting. This is joint work with Yakov Kononov, Woonam Lim
and Weite Pi.

### Past Schedule :

Links to video and notes below each talk when available.

### September 13th, 2021

## Eric Zaslow : The proper Landau-Ginzburg potential is the open mirror map

** Abstract:** I will
discuss toric Fano surfaces in the
complement of a smooth anticanonical
divisor and their mirror
Landau-Ginzburg theories. I will
focus on relations between open
Gromov-Witten invariants of fibers of
the Gross fibration, relative
invariants, scattering diagrams and
broken lines, tropical curves,
superpotentials and wall crossing.
This joint work with Tim Graefnitz and
Helge Ruddat builds on works of Chan,
Lau, Leung, Tseng; the wall crossing
takes its cue from the work Gross,
Pandharipande,
Siebert.
Lecture Notes.

### September 20th, 2021

## Renzo Cavalieri : The integral Chow ring of $M_{0,0}(\mathbb{P}^r,d)$

** Abstract:** We give
an efficient presentation of the
Chow ring with integral coefficients
of the open part of the moduli space
of rational maps of odd degree to
projective space. A less fancy
description of this space has its
closed points correspond to
equivalence classes of $(r+1)$-tuples
of degree $d$ polynomials in one
variable with no common positive
degree factor. We identify this
space as a $GL(2,\mathbb{C})$ quotient of an
open set in a projective space, and
then obtain a (highly redundant)
presentation by considering an
envelope of the complement. A
combinatorial analyis then leads us
to eliminating a large number of
relations, and to express the
remaining ones in generating
function form for all
dimensions. The upshot of this work
is to observe the rich combinatorial
structure contained in the Chow
rings of these moduli spaces as the
degree and the target dimension
vary. This is joint work with
Damiano Fulghesu.

Video.

### October 4th, 2021

## Richard Thomas : Higher rank DT theory from curve counting

** Abstract:** Fix a
Calabi-Yau 3-fold X. Its DT
invariants count stable bundles
and sheaves on X. The generalised
DT invariants of Joyce-Song count
semistable bundles and sheaves on
X. I will describe work with
Soheyla Feyzbakhsh showing these
generalised DT invariants in any
rank r can be written in terms of
rank 1 invariants.
By the MNOP conjecture the latter
are determined by the GW
invariants of X. Along the way we
also express rank r DT invariants
in terms of rank 0 invariants
counting sheaves supported on
surfaces in X. These invariants
are predicted by S-duality to be
governed by (vector-valued, mock)
modular forms.

Video.

### October 18th, 2021

## Ben Davison : The decomposition theorem for 2-Calabi-Yau categories

** Abstract:**
Examples of 2CY categories include the
category of coherent sheaves on a K3
surface, the category of Higgs
bundles, and the category of modules
over preprojective algebras or
fundamental group algebras of compact
Riemann surfaces. Let $p:M\rightarrow
N$ be the morphism from the stack of
semistable objects in a 2CY category
to the coarse moduli space. I'll
explain, using cohomological DT
theory, formality in 2CY categories,
and structure theorems for good moduli
stacks, how to prove a version of the
BBDG decomposition theorem for the
exceptional direct image of the
constant sheaf along $p$, even though
none of the usual conditions for the
decomposition theorem apply: $p$ isn't
projective or representable, $M$ isn't
smooth, the constant mixed Hodge
module complex $\mathbb{Q}_M$ isn't
pure... As applications, I'll explain
a proof of Halpern-Leistner's
conjecture on the purity of stacks of
coherent sheaves on K3 surfaces, and
if time permits, a (partly
conjectural) way to extend nonabelian
Hodge theory to Betti/Dolbeault
stacks.

Video.

Slides.

### October 25th, 2021

## Burt Totaro : Varieties of general type with doubly exponential asymptotics

** Abstract:** We
construct smooth projective
varieties of general type with the
smallest known volume and others
with the most known vanishing
plurigenera in high dimensions. The
optimal volume bound is expected to
decay doubly exponentially with
dimension, and our examples achieve
this decay rate. We also consider
the analogous questions for other
types of varieties. For example, in
every dimension we conjecture the
terminal Fano variety of minimal
volume, and the canonical Calabi-Yau
variety of minimal volume. In each
case, our examples exhibit doubly
exponential behavior. (Joint work
with Louis Esser and Chengxi
Wang.)

Video.

### November 1st, 2021

## Dori Bejleri : Wall crossing for moduli of stable log varieties

** Abstract:** Stable
log varieties or stable pairs $(X,D)$
are the higher dimensional
generalization of pointed stable
curves. They form proper moduli spaces
which compactify the moduli space of
normal crossings, or more generally
klt, pairs. These stable pairs
compactifications depend on a choice
of parameters, namely the coefficients
of the boundary divisor D. In this
talk, after introducing the theory of
stable log varieties, I will explain
the wall-crossing behavior that
governs how these compactifications
change as one varies the
coefficients. I will also discuss some
examples and applications. This is
joint work with Kenny Ascher, Giovanni
Inchiostro, and Zsolt Patakfalvi.

Video.

### November 8th, 2021

## Aaron Pixton : Formulas for double-double ramification cycles

** Abstract:** The
double-double ramification cycle
parametrizes curves of genus g
admitting two maps to the projective
line with specified ramification
profiles over two points. This cycle
is easy to define in families of
smooth curves, but much more subtle
for general stable curves. I will
discuss some special cases in which a
formula for the double-double
ramification cycle is now known. This
is joint work with David Holmes, Sam
Molcho, Rahul Pandharipande, and
Johannes
Schmitt.

Video.

### November 22nd, 2021

## Georg Oberdieck : From $K3$ surfaces to Hilbert schemes and back

** Abstract:** I will
discuss the relationship between three
different counting theories associated
to a $K3$ surface $S$: (i)
Gromov-Witten theory of the Hilbert
scheme of points of $S$ with complex
structure of the source curve fixed to
be an elliptic curve $E$ with some rational tails, (ii)
Donaldson-Thomas theory of $S\times E$, (iii)
Virtual Euler characteristics of Quot
schemes of stable sheaves on the $K3$
surfaces. This leads to new
evaluations of these virtual Euler
numbers, and to multiple cover
formulas for all three theories.

Lecture notes.

Preprint.

Link to UBC events page.

### December 13th, 2021

## Samir Canning : The Chow rings of moduli spaces of elliptic surfaces

** Abstract:** For
each nonnegative integer N, Miranda
constructed a coarse moduli space of
elliptic surfaces with section over
the projective line with fundamental
invariant N. I will explain how to
compute the Chow rings of these moduli
spaces when N is at least 2. The Chow
rings exhibit many properties
analogous to those expected for the
tautological ring of the moduli space
of curves: they satisfy analogues of
Faber's conjectures, and they exhibit
a stability property as N goes to
infinity. When N=2, these elliptic
surfaces are K3 surfaces polarized by
a hyperbolic lattice. I will explain
how the computation of the Chow ring
confirms a special case of a
conjecture of Oprea and Pandharipande
on the structure of the tautological
rings of moduli spaces of lattice
polarized K3 surfaces. This is joint
work with Bochao
Kong.

### January 10th, 2022

## Borislav Mladenov : Degeneration of the Ext spectral sequence and formality of RHom DG algebras associated to holomorphic Lagrangian subvarieties

** Abstract:** Let $L$
be a complex Lagrangian in a
holomorphic symplectic variety
$X$. Consider the DG algebra
$\operatorname{RHom}_X(\cal{L},\cal{L})$
a line bundle $\cal{L}$ on $L$. I
will introduce the local-to-global
spectral sequence computing its
cohomology and describe the
differential in terms of a
deformation-obstruction class of
Huybrechts-Thomas. I will then state
the simplest versions of the
degeneration and formality results and
briefly explain the motivation coming
from the work of Solomon-Verbitsky and
Kapustin’s Seiberg-Witten duality. To
conclude, I’ll give a quick reminder
of Deligne’s degeneration of the Leray
spectral sequence and explain how to
adapt it to our setting. If time
allows, I’d like to sketch a second
proof via deformation quantisation
without going into technicalities.

Link to UBC events page.

### January 17th, 2022

## Lothar Gottsche : Blowup formulas, strange duality and generating functions for Segre and Verlinde numbers of surfaces

** Abstract:**This is
based in part on joint work with
Martijn Kool, and is partially joint
work in progress with Anton
Mellit.

We formulate conjectural
blowup formulas for Segre and
Verlinde numbers of moduli spaces of
sheaves on projective surfaces with
$q=0$ and $p_g>0$. Combining these
with a virtual version of the
strange duality conjecture, we
obtain a lot of information about
the generating functions of these
invariants, which allows to
conjecturally determine them in many
cases. In work in progress we give a
complete conjectural determination
of the Verlinde and Segre series of
Hilbert schemes of points on
surfaces, as well as a proof of this
conjecture for surfaces with $K^2=0$.

Link to UBC events page.

### January 24th, 2022

## Dimitri Zvonkine : Gromov-Witten invariants of complete intersections

** Abstract:** We show
that there is an effective way to
compute all Gromov-Witten (GW)
invariants of all complete
intersections. The main tool is Jun
Li's degeneration formula: it allows
one to express GW invariants of a
complete intersection from GW
invariants of simpler complete
intersections. The main difficulty is
that, in general, the degeneration
formula does not apply to primitive
cohomology insertions. To circumvent
this difficulty we introduce simple
nodal GW invariants. These invariants
do not involve primitive cohomology
classes, but instead make use of
imposed nodal degenerations of the
source curve. Our work contains two
main statements: (i) simple nodal GW
invariants can be computed by the
degeneration formula, (ii) simple
nodal GW invariants determine all GW
invariants of a complete
intersection. The first statement is
geometric; the second uses the
invariance of GW invariants under
monodromy and some representation
theory. In this talk I will spend more
time on part (ii) to complete more
geometrically oriented talks by my
co-authors: Hulya Arguz, Pierrick
Bousseau and Rahul Pandharipande.

Link to UBC events page.

### February 7th, 2022

## Tudor Padurariu : Hall algebras in Donaldson-Thomas theory

** Abstract:**
Kontsevich-Soibelman defined the
cohomological Hall algebra (CoHA) of a
quiver with potential. By a result of
Davison-Meinhardt, CoHAs are
deformations of the universal
enveloping algebra of the BPS Lie
algebra of the quiver with potential.
My plan is to present the analogues
stories in the categorical and
K-theoretic contexts. I will
introduce the categorical and
K-theoretic Hall algebras of a
quiver with potential and explain
how to prove versions of the
Davison-Meinhardt theorem in these
contexts. These results have
applications in categorical
Donaldson-Thomas theory and in the
study of Hall algebras of surfaces.

Video.

### February 14th, 2022

## Kai Behrend : Donaldson-Thomas theory of the quantum Fermat quintic

** Abstract:**We study
non-commutative projective varieties
in the sense of Artin-Zhang, which are
given by non-commutative homogeneous
coordinate rings, which are finite
over their centre. We construct moduli
spaces of stable modules for these,
and construct a symmetric obstruction
theory in the CY3-case. This gives
deformation invariants of
Donaldson-Thomas type. The simplest
example is the Fermat quintic in
quantum projective space, where the
coordinates commute up to carefully
chosen 5th roots of unity. We explore
the moduli theory of finite length
modules, which mixes features of the
Hilbert scheme of commutative 3-folds,
and the representation theory of
quivers with potential. This is
mostly work of Yu-Hsiang Liu, with
contributions by myself and Atsushi
Kanazawa.

Lecture Notes

Video

Link to UBC events page.

### February 21st, 2022

## Anders Buch : A Vafa-Intriligator formula for Fano varieties

** Abstract:**I will
speak about a generalization of the
Vafa-Intriligator formula that
expresses the Gromov-Witten
invariants of a Fano variety, for
fixed markings and insertions of
even degrees, in terms of
eigenvalues of multiplication
operators on the small quantum
cohomology ring. This generalizes
earlier known formulas for
cominuscule flag varieties,
including Grassmannians of type A
and Grassmannians of maximal
isotropic subspaces. In the
Grassmannian cases, the relevant
eigenvalues are explicitly known
from the work of Rietsch and
Cheong. The eigenvalues can also be
described explicitly for Fano
complete intersections. Applications
include simple formulas for Tevelev
degrees. This is joint work with
Rahul Pandharipande.

Notes.

Video.

Link to UBC events page.

### February 28th, 2022 SPECIAL TIME: 8:45 am Pacific time / 5:45 pm Zurich time.

## Tamas Hausel : Fixed point scheme as spectrum of equivariant cohomology and Kirillov algebras

** Abstract:** As an
example of the multiplicity algebras
of the Arnold school we will look at
the equivariant cohomology of a
Grassmannian and notice that it is
inscribed in its regular fixed point
scheme generalising observations of
Brion-Carrell.
In turn, we will describe it
explicitely as the classical
Kirillov
algebra of a fundamental
representation of $SL_n$, originally
observed by Panyushev. These
observations are motivated by mirror
symmetry considerations originating
in recent work with Hitchin on the
mirror of very stable upward flows
in the Hitchin system, which we will
briefly indicate.

Video.

Link to UBC events page.

### March 7th, 2022

## Younghan Bae : Counting surfaces on Calabi-Yau fourfolds

** Abstract:** I will
describe a sheaf theoretic way to
enumerate surfaces on a smooth
quasi-projective Calabi-Yau
fourfold. By Borisov-Joyce/Oh-Thomas,
a moduli space of sheaves or complexes
on a Calabi-Yau fourfold has a virtual
class. When we count two dimensional
objects, this theory has to be
modified to get nontrivial
invariants. I will explain how to
reduce the theory and deformation
invariance along the Hodge loci. This
can be related to the variational
Hodge conjecture in some
examples. Next I will explain two ways
to compactly stable pairs, which we
call PT0, PT1 pairs. Several
conjectural correspondences between
DT, PT0 and PT1 invariants will be
discussed with evidences. This is a
joint work in progress with Martijn
Kool and Hyeonjun Park.

Notes.

Link to UBC events page.

### May 2nd, 2022

## Arend Bayer : Kuznetsov categories of Fano threefolds

** Abstract:** I will
give a survey of recent results on
Kuznetsov categories of Fano
threefolds of Picard rank one. These
results give additional structure to
their classification, and their moduli
spaces. The techniques involved
include moduli spaces of
Bridgeland-stable objects,
Brill-Noether statements, and
equivariant categories (spiced with a
pinch of derived algebraic geometry).

Link to UBC events page.

### May 9th, 2022

## Navid Nabijou : From orbifolds to logarithms via birational invariance

** Abstract:**
Logarithmic and orbifold structures
provide two different paths to the
enumeration of curves with fixed
tangencies to a normal crossings
divisor. Simple examples demonstrate
that the resulting systems of
invariants differ, but a more
structural explanation of this defect
has remained elusive.
I will discuss joint work with Luca
Battistella and Dhruv Ranganathan,
in which we identify birational
invariance as the key property
distinguishing the two theories. The
logarithmic theory is stable under
toroidal blowups of the target,
while the orbifold theory is not. By
identifying a suitable system of
“slope-sensitive” blowups, we define
a “limit orbifold theory” and prove
that it coincides with the
logarithmic theory. Our proof hinges
on a technique – rank reduction –
for reducing questions about normal
crossings divisors to questions
about smooth divisors, where the
situation is much-better understood.

Notes.

Link to UBC events page.

### June 20th, 2022

## Marc Levine : Equivariant localization in quadratic enumerative geometry

** Abstract:** We
explain how to adapt the classical
Atiyah-Bott torus localization methods
to the computation of invariants, such
as degrees of Euler classes, in the
Grothendieck-Witt ring of quadratic
forms, These quadratic enumerative
invariants lift the usual
integer-valued ones via the rank
function, and for fields contained in
the reals, the signature gives
invariants for problems over the
reals. Examples include quadratic
counts of twisted cubic curves on
hypersurfaces and complete
intersections, this last is joint work
with Sbarina Pauli.

Notes.

Link to UBC events page.

### September 26th, 2022

## Jim Bryan : A theory of Gopakumar-Vafa invariants for orbifold Calabi-Yau threefolds

** Abstract:**
We define integer valued invariants of
an orbifold Calabi-Yau threefold $X$
with transverse ADE orbifold
points. These invariants contain
equivalent information to the
Gromov-Witten invariants of $X$ and are
related by a Gopakumar-Vafa like
formula which may be regarded as a
universal multiple cover / degenerate
contribution formula for orbifold
Gromov-Witten invariants. We also give
sheaf theoretic definitions of our
invariants. As examples, we give
formulas for our invariants in the
case of a (local) orbifold K3
surface. These new formulas generalize
the classical Yau-Zaslow and
Katz-Klemm-Vafa formulas. This is
joint work with S. Pietromonaco.

### October 3rd, 2022

## Woonam Lim : Virasoro constraints in sheaf theory and vertex algebras

** Abstract:** In
enumerative geometry, Virasoro
constraints first appeared in the
context of moduli of stable curves and
maps. These constraints provide a rich
set of conjectural relations among
Gromov-Witten descendent
invariants. Recently, the analogous
constraints were formulated in several
sheaf theoretic contexts; stable pairs
on 3-folds, Hilbert scheme of points
on surfaces, higher rank sheaves on
surfaces with only
(p,p)-cohomology. In joint work with
A. Bojko, M. Moreira, we extend and
reinterpret Virasoro constraints in
sheaf theory using Joyce's vertex
algebra. This new interpretation
yields a proof of Virasoro constraints
for curves and surfaces with only
(p,p) cohomology by means of
wall-crossing formulas.

### October 24th, 2022

## Dhruv Ranganathan : Logarithmic enumerative geometry for curves and sheaves

** Abstract:**I will
discuss ongoing work with Davesh
Maulik in which we formulate a
generalization of the GW/DT
conjectures to the setting of simple
normal crossings pairs. When the
divisor in the pair is smooth, the
formulation of the conjecture
necessitates the study of the
cohomology of the Hilbert scheme of
points on a surface. The formulation
of the logarithmic GW/DT conjecture
requires new geometry coming from a
logarithmic Hilbert scheme of
points. We prove a strengthened
logarithmic degeneration formula on
both sides of the correspondence and
prove that the new conjectures are
compatible with the old ones via
degeneration. I’ll discuss this circle
of ideas, and explain which parts of
the conjectures are within reach.

### November 7th, 2022

## Davesh Maulik : The $P=W$ conjecture for $GL_n$

** Abstract:** The
$P=W$ conjecture, first proposed by de
Cataldo-Hausel-Migliorini in 2010,
gives a link between the topology of
the moduli space of Higgs bundles on a
curve and the Hodge theory of the
corresponding character variety, using
non-abelian Hodge theory. In this
talk, I will explain this circle of
ideas and discuss a recent proof of
the conjecture for $GL_n$ (joint with
Junliang Shen).

### November 21st, 2022

## Zhiyu Liu :Castelnuovo bound and Gromov-Witten invariants of the quintic 3-fold

** Abstract:** One of
the most challenging problems in
geometry and physics is to compute
higher genus Gromov-Witten invariants
of compact Calabi-Yau 3-folds, such as
the famous quintic 3-fold. I will
briefly describe how physicists
compute Gromov-Witten invariants of
the quintic 3-fold up to genus 53,
using five mathematical
conjectures. Three of them have been
already proved, and one of the
remaining two conjectures has been
solved in some genus. I will explain
how to prove the last open one, called
the Castelnuovo bound, which predicts
the vanishing of Gopakumar-Vafa
invariants for a given degree at
sufficiently high genus. This talk is
based on the joint work with Yongbin
Ruan.

### March 20th, 2023

Note 9:30 am Pacific Time, 5:30 pm Zurich time## Longting Wu :All-genus WDVV recursion, quivers, and BPS invariants.

** Abstract:**
Let $D$ be a smooth rational ample
divisor in a smooth projective surface $X$. In this talk, we will
present a simple uniform recursive formula for (primary) Gromov-Witten
invariants of $\mathcal{O}_X(-D)$. The recursive formula can be used to determine
such invariants for all genera once some initial data is known. The
proof relies on a correspondence between all-genus Gromov–Witten
invariants and refined Donaldson–Thomas invariants of acyclic
quivers. In particular, the corresponding BPS invariants are expressed
in terms of Betti numbers of moduli spaces of quiver
representations. This is a joint work with Pierrick Bousseau.

### May 1st, 2023

## Sam Payne :Cohomology groups of moduli spaces of curves

** Abstract:**
Algebraic geometry endows the
cohomology groups of moduli spaces of
curves with additional structures,
such as (mixed) Hodge structures and
Galois representations. Standard
conjectures from arithmetic, regarding
analytic continuations of L-functions
attached to these Galois
representations, lead to striking
predictions, by Chenevier and Lannes,
about which such structures can
appear. I will survey recent results
unconditionally confirming several of
these predictions and studying
patterns in the appearances of motives
of low weight. The latter are governed
by the operadic structures induced by
tautological morphisms and the
cohomology of graph complexes.

Based on joint work with Jonas Bergström and Carel Faber; with Sam
Canning and Hannah Larson; with Melody Chan and Søren Galatius; and
with Thomas Willwacher.

### September 11th, 2023

## John Pardon :Universally counting curves in Calabi--Yau threefolds

** Abstract:**
Enumerating curves in algebraic
varieties traditionally involves
choosing a compactification of the
space of smooth embedded curves in the
variety. There are many such
compactifications, hence many
different enumerative invariants. I
will propose a "universal" (very
tautological) enumerative invariant
which takes values in a certain
Grothendieck group of 1-cycles. It is
often the case with such "universal"
constructions that the resulting
Grothendieck group is essentially
uncomputable. But in this case, the
cluster formalism of Ionel and Parker
shows that, in the case of threefolds
with nef anticanonical bundle, this
Grothendieck group is freely generated
by local curves. This reduces the
MNOP conjecture (in the case of nef
anticanonical bundle and primary
insertions) to the case of local
curves, where it is already known due
to work of Bryan--Pandharipande and
Okounkov--Pandharipande.

### October 2nd, 2023

## Georg Oberdieck :Curve counting on the Enriques surface and the Klemm-Marino formula

** Abstract:**
An Enriques surface is the quotient of
a K3 surface by a fixed point-free
involution. Klemm and Marino
conjectured a formula expressing the
Gromov-Witten invariants of the local
Enriques surface in terms of
automorphic forms. In particular, the
generating series of elliptic curve
counts on the Enriques should be the
Fourier expansion of (a certain power
of) Borcherds automorphic form on the
moduli space of Enriques surfaces. In
this talk I will explain a proof of
this conjecture. The proof uses the
geometry of the Enriques Calabi-Yau
threefold in fiber classes. If time
permits, I will also discuss various
conjectures about non-fiber classes.