Algebraic Geometry Seminar
Spring 2019 — Tuesdays 3:30  4:30 PM, location LCB 222
Date  Speaker  Title — click for abstract (if available) 
January 8 


January 15 


January 22 


January 29 


February 5 


February 11 (note special day) in JWB 208 
Andreas Malmendier Utah State University 
Configurations of sixlines,
string dualities, and modular forms
A smooth K3 surface obtained as the blowup of the quotient of a fourtorus by the involution automorphism at all 16 fixed points is called a Kummer surface. Algebraic Kummer surfaces obtained from abelian varieties provide a fascinating arena for string compactification and string dualities as they are not trivial spaces but are sufficiently simple to analyze most of their properties in detail. However, their Picard rank is always bigger or equal 17, and they only provide a geometric description for heterotic string vacua with up to one Wilson line. In this talk, I give an explicit description of the family of K3 surfaces of Picard rank sixteen associated with the double cover of the projective plane branched along the union of six lines, and the family of its Van GeemenSarti partners, i.e., K3 surfaces with special Nikulin involutions, such that quotienting by the involution and blowing up recovers the former. The family of Van GeemenSarti partners is a fourparameter family of K3 surfaces with a H + E_{7}(1) + E_{7}(1) lattice polarization. We describe explicit Weierstrass models on both families using even modular forms on the bounded symmetric domain of type IV. We also show that our construction provides a geometric interpretation, called geometric twoisogeny, for the Ftheory/heterotic string duality in eight dimensions with two Wilson lines. If time allows, I will also describe special six line configurations (such as six lines tangent to a conic, three lines meeting in a point) that correspond to Kummer surfaces of Jacobians of genustwo curves with principal polarization and those associated to (1, 2)polarized abelian surfaces, as well as their applications in string theory and number theory. 
February 19 
Eduardo
Gonzalez University of Massachusetts Boston 
Stratifications in gauged GromovWitten theory
Let G be a reductive group and X be a smooth projective Gvariety.
In classical geometric invariant theory (GIT), there are
stratifications of X that can be used to understand the geometry of
the GIT quotients X//G and their dependence on choices. In this
talk, after introducing basic theory, I will discuss the moduli of
gauged maps, their relation to the GromovWitten theory of GIT
quotients X//G and work in progress regarding stratifications of the
moduli space of gauged maps as well as possible applications to
quantum Ktheory. This is joint work with D. HalpernLeistner, P.
Solis and C. Woodward.

February 20 at 3:00pm 
Felix
Janda University of Michigan 
Logarithmic GLSM moduli spaces
Understanding the structure of GromovWitten invariants of quintic
threefolds is an important problem in enumerative geometry which has
been studied since the early 90s. Together with Q. Chen, Y. Ruan and
A. Sauvaget, we construct new moduli spaces that we call
"logarithmic GLSM moduli spaces". One application is toward proving
conjectures from physics about higher genus GromovWitten invariants
of quintic threefolds, such as the holomorphic anomaly equations.
Another application, which also was the initial motivation to
develop logarithmic GLSM, is toward proving a conjecture of R.
Pandharipande, A. Pixton, D. Zvonkine and myself on loci of
holomorphic differentials with prescribed zeros. In this talk, I
will focus on the second application. Its main actor is the
socalled Witten's rspin class, the analog of the virtual class in
the FJRW theory of the A_{r1}singularity.

February 26 


March 5 
Linquan Ma Purdue University 
Homological conjectures, perfectoid spaces, and singularities in mixed characteristic
The homological conjectures have been a focus of research in commutative
algebra since 1960s. They concern a number of interrelated conjectures
concerning homological properties of commutative rings to their internal
ring structures. These conjectures had largely been resolved for rings
that contain a field, but several remained open in mixed
characteristicuntil recently Yves Andre proved Hochsterâ€™s direct
summand conjecture and the existence of big CohenMacaulay algebras,
which lie in the heart of these homological conjectures. The main new
ingredient in the solution is to systematically use the theory of
perfectoid spaces, which leads to further developments in the study of
mixed characteristic singularities. For example, using integral
perfectoid big CohenMacaulay algebras, one can define the mixed
characteristic analog of rational/Frational and log terminal/Fregular
singularities, and they have applications to study singularities when the
characteristic varies (based on recent joint work with Karl Schwede). In
this talk, we will give a survey on these results and methods.

March 12 
N/A Spring Break 

March 19 
Jonathan Campbell Vanderbilt University 
Bicategorical Duality Theory with Applications to Topology and
Algebra
In this talk I'll describe how a bicategorical gadget, called the
shadow, allows one to extract many interesting algebraic and
topological invariants. For example, in algebraic bicategories, one
easily recovers group characters, and in certain topological
categories, one recovers the Lefschetz number. I'll describe joint
work with Kate Ponto generalizing work of BenZviNadler which
allows us to simultaneously recover the Lefschetz theorem for
DGalgebras due to Lunts and the theory of 2characters due to
Ganter Kapranov, along with many other results. Prerequisites: An
appetite for category theory (but I will not assume knowledge of
bicategories!). There will be some motivation from stable homotopy
theory, but one need only believe in the stable homotopy category,
not have knowledge of it.

March 26 
Jihao
Liu University of Utah 
Boundedness of complements for
pairs with DCC coefficients
Shokurov introduced the theory of complements while he investigated
log flips of threefold. The theory is further developed by
ProkhorovShokurov and Birkar. As one of the key steps in the proof
of BAB conjecture, Birkar showed a conjecture of Shokurov, i.e., the
existence of ncomplements for Fano pairs with hyperstandard
coefficients. In this talk, I will show that the boundedness of
complements holds for pairs with DCC coefficients, and have some
additional properties: divisibility, rationality, approximation, and
anisotropic. If there's still enough time, I will show some of its
applications on the ACC for minimal log discrepancies. This is a
joint work with J.Han and V.V.Shokurov.

April 2 
HsianHua Tseng Ohio State University 
Relative and orbifold GromovWitten invariants
For a smooth projective variety X containing a smooth irreducible
divisor D, the question of counting curves in X with prescribed
contact conditions along D is a classical one in enumerative
geometry. In more modern approaches to this question, there are two
ways to define these counts: as GromovWitten invariants of X
relative to D, or as GromovWitten invariants of the stack X_{D,r}
of rth roots of X along D (for r large). In genus 0, explicit
calculations in examples suggested that these two sets of
GromovWitten invariants are always the same, whether they actually
count curves or not. This is proven in full generality by
AbramovichCadmanWise. The situation in genus > 0 is not so simple,
as an example of D. Maulik showed that the two sets of GromovWitten
invariants are not equal, even in genus 1. In this talk, we'll
explain how these two sets of GromovWitten invariants are related
in all genera, in full generality. This is based on a joint work
with Fenglong You.

April 9 


April 16 


April 23 


April 30 
N/A Finals Week 

This web page is maintained by Aaron Bertram, Christopher Hacon, and Bronson Lim.