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Valley Geometry SeminarFridays, LGRT 1634
Tea: 3:45 PM
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23 January | Business Meeting | |
| 3:00-4:00 | |||
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31 January | Frank Sottile, University of Massachusetts | |
| 4:00-5:00 | Enumerative Real Algebraic Geometry | ||
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7 February | David Cox, Amherst College | |
| 4:00-5:00 | Vanishing and Codimension Theorems for Toric Varieties | ||
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21 February | John Fogarty, UMass | |
| 4:00-5:00 | Quotients of Algebraic Surfaces | ||
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28 February | Julianna Tymoczko, Princeton | |
| 4:00-5:00 | An Introduction to Hessenberg varieties | ||
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7 March | Misha Kogan, Northeastern U | |
| 4:00-5:00 | Schubert varieties and Gelfand-Cetlin polytopes | ||
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21 march | No Seminar: Spring Break | |
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28 March | Charlie Epstein, U Penn | |
| 4:00-5:00 | Adventures in Magnetic Resonance | ||
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4 April | Eyal Markman, University of Massachusetts | |
| 4:00-5:00 | Stability conditions on triangulated categories and Mirror symmetry, after Douglas and Brideland. | ||
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11 April | Mircea Mustata, Harvard University | |
| 4:00-5:00 | Asymptotic invariants of line bundles | ||
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18 April | Alex Yong, University of Michigan | |
| 4:00-5:00 | Formulas for Schubert classes via Grobner bases and degeneracy loci | ||
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25 April | Mark Goresky, IAS | |
| 4:00-5:00 | The topological trace formula | ||
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2 May | Carlos D'Andrea, UC Berkeley | |
| 4:00-5:00 | Inversion of birational maps | ||
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9 May | Emma Carberry, MIT | |
| 4:00-5:00 | Special Lagrangian cones in C3 over tori | ||
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16 May | Laura Matusevich, Harvard University | |
| 4:00-5:00 | Title | ||
Abstract:
Consider the following hard problem: Give meaningful information
about the number r of real solutions to a system of multivariate
polynomial equations. Quite often much more is known than merely that
r is between 0 and the number d of complex solutions. This stronger
information is typically available when the system has some geometric
structure. Enumerative real algebraic geometry treats this hard problem
for systems coming from geometry.
In this talk, I will survey some of what is known. In particular,
I will discuss recent better understanding of the bounds of 0 and d.
A picture is emerging: for systems from geometry, the upper bound
of d can always be obtained, and in many cases there are non-trivial
lower bounds on the number of real solutions, with certain gaps that
have been discovered experimentally.
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Hessenberg varieties form a family of subvarieties of the flag
variety with important relations to representation theory, numerical
analysis, quantum cohomology, and other areas. Significant examples
include the Springer fiber (whose cohomology carries information about
representations of the Weyl group) and the Peterson variety (which can
be stratified so that the open stratum's coordinate ring gives the quantum
cohomology of the flag variety).
I will discuss the geometric structure of Hessenberg varieties and will
describe how several major classes of Hessenberg varieties can be paved
by affines. I will show how these affines are indexed by filled Young
tableaux and how their dimensions can be computed by simple
combinatorial
rules.
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Schubert polynomials S_w represent cohomology classes of
Schubert varieties X_w in the flag manifold. Billey-Jockusch-Stanley and
Fomin-Kirillov expressed Schubert polynomials as positive sums of
monomials indexed by combinatorial objects called rc-graphs. We discuss
the geometry underlying these monomial expressions. In particular, we
construct a flat degeneration of the flag manifold to the toric variety
Y associated to the Gelfand-Cetlin polytope. Every Schubert variety
X_w degenerates to a reduced union of toric subvarieties of Y. The
components of the degeneration of X_w correspond to certain faces of the
Gelfand-Cetlin polytope. Moreover, these components are in bijection with
rc-graphs which enter the monomial expression for S_w.
This is joint
work with Ezra Miller.
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Mirror symmetry relates the symplectic geometry of Calabi-Yau
varieties to the complex geometry of the mirror Calabi-Yau.
Geometric objects in the symplectic side, such as (special)
Lagrangian subvarieties, should correspond, roughly, to
equivalence classes holomorphic vector bundles,
which satisfy a stability condition.
The concept of stability of vector bundles is well understood,
but it is not preserved under the equivalence relation,
defining objects in the derived category. The work of Douglas
and Brideland resolves this difficulty. There are many
different ways to embed the relevant (stable) symplectic category
in the holomorphic derived category.
The space of all such embeddings is itself
a complex manifold with interesting geometry.
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I will describe certain invariants
associated to big line bundles on smooth projective varieties,
which measure the asymptotic behaviour of the base locus.
Similar invariants can be studied in a pure
algebraic setting. This is based on joint work
with L. Ein, R. Lazarsfeld, M. Nakamaye and M. Popa.
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I will describe two approaches for obtaining formulas for
Schubert classes in a generalized flag variety X=G/B. Both analogize
known results in type A.
The first approach generalizes to arbitrary Lie types an interesting linear basis (due to Lascoux and Schützenberger) for the cohomology ring of X=SL(n,C)/B. In general, it is formed by products of special classes h(i,d), which turn out to be short, positive integer sums of Schubert classes. Remarkably, all of these classes satisfy the same recurrence:
h(i,d)=h(i,d-1)[h(i,1)-h(i-1,1)]+h(i-1,d)
(here i labels a node of the Dynkin diagram). We will explain some questions related to this basis. This approach uses a small bit of Grobner basis theory.
The second approach gives an analogue of a formula from type A
(previously obtained in joint work with A. Buch, A. Kresch and
H. Tamvakis) to types BCD. The type A formula expresses a
Schubert class as a positive integer sum of products of Schur polynomials.
The coefficients involved are the quiver coefficients
of Buch and Fulton. We obtain similar formulas in types BCD, in terms of
new collections of positive integer coefficients that we can describe
combinatorially, but for which we seek a degeneracy loci interpretation.
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In 1981, J. Arthur described a formula for the
trace of a Hecke correspondence on the L2 cohomology of a
locally symmetric space. At that time it was suggested by
Arthur, Casselman, and Langlands, that there should be a
topological interpretation of this formula as a Lefschetz
fixed point formula. In this talk I will discuss the
mathematics which resulted from attempts to interpret this
formula geometrically.
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Properness of rational parametrizations of hypersurfaces can
be stated algorithmically. In this talk I will present different
algorithmic criteria to decide whether a given rational parametrization
is proper. Furthermore, if the parametrization is proper, I will also
show how to compute the inverse of the parametrization.
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Special Lagrangian 3-folds are of interest in mirror symmetry, and in particular play an
important role in the SYZ conjecture. One wishes to understand the singularities that can
develop in families of these 3-folds; the relevant local model is provided by special
Lagrangian cones in complex 3-space. When the link of the cone is a torus, there is a
natural invariant
g associated to the cone, namely the genus of its spectral curve. We show that for each
g there are countably many real (g-2)-dimensional families of such special Lagrangian
cones.
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