| Date |
Speaker |
Title |
|
January 9, LCB 215
|
| Andrew Snowden |
| (Princeton University) |
|
The equations of the GIT quotient (P1)n/PGL(2)
Let n >= 4 be an even integer and let Mn be the moduli
space of
n points on P1 modulo the action of PGL(2), thought of as the
GIT quotient (P1)n/PGL(2). The space Mn has a natural
projective embedding; let Rn be its projective coordinate ring.
The ring Rn was studied classically in the context of invariant
theory. In the late nineteenth centruy, Kempe proved that Rn
is generated by its degree one piece. Since that time however,
generators for the ideal of relations have not been determined.
I will talk about my recent work with Howard, Millson and Vakil
towards understanding generators of this ideal. Two of our
results: 1) the ideal of relations is always generated in
degrees <= 3; and 2) for all n not equal to 6 the image of Mn
in projective space is cut out by quadrics.
|
|
|
| Dave Anderson |
| (University of Michigan) |
|
Chern class formulas for G2 degeneracy loci
Let V be a vector bundle of rank n on a variety X, with subbundles E and F
of respective ranks e and f. The locus of points of X where the fibers of
E and F intersect in dimension more than e+f-n is a basic example of a
degeneracy locus, and it is useful to have formulas for the cohomology
classes of such loci in terms of the Chern classes of E, F, and V. Many
variations are possible: there should be one for each Lie type, and for
each element of the corresponding Weyl group. For classical types,
formulas were given by Giambelli-Thom-Porteous, Kempf-Laksov, Harris-Tu,
and Fulton. In this talk, I will give formulas corresponding to
exceptional type G2. Along the way, I'll discuss octonion bundles,
and describe the G2 flag variety in concrete, linear-algebraic
terms.
|
|
January 22
|
Tommaso de Fernex
|
A vanishing theorem for log canonical pairs
|
|
January 29
|
|
The orbifold cohomology of abelian and non-abelian quotients
Let G be a semisimple Lie group and which is acting on X.
Then we consider X//G and X//T where T is a maximal
torus of G. We will explain a comparison theory of orbifold
cohomologies of X//G and X//T when both spaces are orbifolds.
|
| January 31, 2.45 pm |
| (LCB 323) |
|
| Sam Payne |
| (Clay Institute/Stanford) |
|
Frobenius splitting of toric varieties
|
|
|
Mihnea Popa (UIC) |
|
February 5
|
|
The Witten equation, mirror symmetry and quantum singularity theory
I will describe recent joint work with Huijun Fan and Yongbin Ruan in which
we construct, for every non-degenerate quasi-homogeneous singularity, a moduli
space of decorated stable curves and a virtual cycle on that space. In the
special case of the Ar-1 singularity, our constructions give a
refinement of the theory of r-spin curves.
For simple singularities the resulting Frobenius algebra is "mirror dual"
to the Milnor ring of the singularity. For other singularities, we
conjecture a more complicated mirror symmetry relation.
Finally, I will describe some work in progress, in which we generalize the
Faber-Shadrin-Zvonkine proof of the Witten conjecture for r-spin curves
(An) to prove an analogous theorem about integral hierarchies
associated to the simple singularities D2n and E6,
E7 and E8
|
|
|
Fumitoshi Sato (KIAS) |
|
February 12
|
| Mircea Mustaţă |
| (University of Michigan) |
|
Towards an inductive approach to singularities of pairs
|
|
|
| Mircea Mustaţă |
| (University of Michigan) |
|
|
|
| Pramod Achar |
| (Louisiana State) |
|
| February 19 |
| 3 pm, LCB 222 |
|
| Matthew Ballard |
| (University of Washington) |
|
Graded reconstruction for Gorenstein varieties
Singularity of a projective variety manifests itself
categorically in the loss of essential surjectivity of the natural
inclusion of the perfect derived category into the bounded derived category
of coherent sheaves. We shall see, that when working over a perfect field,
these two categories determine each other through a type of duality. Using
this knowledge, we can then extend Bondal and Orlov's result on
reconstruction of a smooth projective variety from the graded structure on
the bounded derived category of coherent sheaves to the case of Gorenstein
projective varieties.
|
|
February 26
|
Milena Hering (IMA)
|
The moduli space of points in P1 and the Koszul property
|
|
March 4
|
Christopher Hacon
|
Deformation of canonical pairs
|
|
March 11
|
Gueorgui Todorov
|
The Gromov-Witten potential of the local P(1,2)
|
|
March 18
|
|
No Seminar | | (Spring break) |
|
|
|
March 25
|
| Emanuele Macri |
| (University of Bonn, Germany) |
|
Derived equivalences of K3 surfaces and orientation
We will consider the problem of describing equivalences between the
derived categories of coherent sheaves of smooth projective K3
surfaces. After recalling the `classical' Derived Torelli Theorem, we
will prove a conjecture by Szendroi which improves the result and
involves the preservation of
the orientation of some 4-dimensional space in the real cohomology of
the K3 surfaces. Our approach relies on the proof of the same
conjecture for generic (non-projective) K3 surfaces. This is joint
work with D. Huybrechts and P. Stellari (arXiv:0710.1645).
|
|
April 1
|
| Alfred Chen |
| (National Taiwan University/Utah) |
|
Some further remarks on birational geometry of threefolds
We notice that our previous study on the basket of singularities also
applicable to weak Q-Fano threefolds. More precisely, we obtained the
following results:
1. non-vanishing of antiplurigenera
2. lower bound for anti-canonical volume for weak Q-Fano threefolds.
This bound is sharp.
We will also describe the relation between flips and "unpacking", which
provide another viewpoint to see the termination of flips.
|
TBA
|
April 8
|
Tommaso de Fernex
|
Deformation of canonical pairs, part 2
|
|
April 15
|
|
Relating relative and orbifold Gromov-Witten invariants
|
|
April 22
|
???
|
TBA
|