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F. R. A. GME. N. T.
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This is a seminar series intended to involve people in the
For more information, or to provide a speaker, contact Renzo, renzo AT math.colostate.edu, or Yano, casa AT math.colorado.edu
Schedule of Talks:
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Date |
Speaker |
At |
Title (click on title to view summary) |
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Aug 27th |
Brian Osserman (UC Davis) |
CSU |
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Sep 4th (note unusual day and room) Eng E105 |
Matt Ballard |
CSU |
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Sep 10th |
Graeme Wilkin (CU Boulder) |
CU |
Introductory
talk: Finite-dimensional GIT Second talk: Hyperkahler Kirwan surjectivity for rank 2 Higgs bundles |
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Sep 17th |
Thomas Markwig ( Hannah Markwig (Goettingen) |
CSU |
Counting tropical elliptic plane curves with fixed j-invariant |
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Sep 24th |
Josh Thompson (CSU) |
CSU |
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Oct 1st |
Sam Payne (Clay) |
CSU |
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Oct 8th |
Michael Rose (UC Berkeley) |
CSU |
Differential
graded structure on the moduli of stable sheaves |
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Oct 12th |
Francois-Xavier Machu (U. |
CSU |
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2:15-3:15pm
Room 350
Oct 15th
4-5pm
Room 350
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Eric Stade (CU) Taku Ishii ( |
CU |
Automorphic forms, archimedean L factors,
Whittaker functions, and Barnes integrals |
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Oct 22nd |
Sarah Kitchen (U.Utah) Luke Oeding ( |
CSU |
Representation Theory and the Geometry
of Flag Varieties Variety of
principal minors and the set theoretic defining equations |
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Oct 29th |
Steffen Marcus (Brown) |
CSU |
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Nov 5th |
Jesse Kass (U |
CU |
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Nov 12th |
Yusuf Mustopa (U.Michigan) Philipp Rostalski ( |
CSU |
Subordinate Loci on Symmetric Products and Syzygies of Points |
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Nov 19th |
Aaron Bertram ( |
CSU |
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Dec 3rd |
Marc Krawitz (U.Michigan) |
CSU |
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Dec 4th WEBER 202 3-5pm |
Emanuele Macri (U.Utah) |
CSU |
Introduction to derived categories and
semi-orthgonal decompositions. Fano varieties of
lines of cubic fourfolds containing a plane. |
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Dec 10th |
Ahmet Emin Tatar ( |
CSU |
Logistics:
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CU Boulder Campus
map
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CSU Campus map
Abstracts:
Hurwitz numbers, which count branched covers of the projective line, are a very classical topic, with equivalent formulations in algebraic geometry, group theory, complex geometry, and topology, as well as connections to number theory and string theory. After reviewing the basic definitions, we discuss some of the difficulties of generalizing Hurwitz numbers to positive characteristic, and present some preliminary progress in this direction.
Sep 4: Matthew Ballard
In the first hour, we will start from coherent sheaves and sheaf cohomology and build to Beilinson's resolution of the diagonal.
In 2003, Rouquier defined the notion of dimension of a triangulated category. Roughly, it measures how quickly the category can be built from a single object. Orlov conjectured that the dimension of the bounded derived category of coherent sheaves on a smooth variety is equal to the dimension of the variety. In this talk, we discuss joint work with D. Favero which allows one to verify some cases of the conjecture.
In the first talk I will introduce the notion of a
representation of a quiver, and describe how one can use Geometric Invariant
Theory (GIT) to construct a moduli space of stable representations. It is a
“well-known fact” that these spaces also have a description as a
sym- plectic quotient, and I will explain what this is and sketch how to prove
this correspondence. The first examples will be familiar moduli spaces
(pro jective spaces and Grassmannians), and there will be plenty of pictures
describing how GIT works in this case. If time permits, I will also describe an
“unstable version” of this correspondence.
In the second talk, I will define Higgs bundles and show how the methods
of the previous talk apply to the space of Higgs bundles. A major question in
the field is that of hype kahler Kirwan surjectivity, and I will explain
how in joint work with George Daskalopoulos, Jonathan Weitsman and Richard
Wentworth, we proved this using an approach outlined by Kirwan involving Morse
theory on singular spaces. I plan to emphasise the algebraic aspects of our
construction, rather than the analytic aspects.
It turns out that many interesting properties of algebraic varieties are preserved under the process of tropicalisation. We will explain during this talk what tropicalisation is, and we will give one example for the above statement. To be more precise, if we start on the algebraic side with a smooth elliptic curve, i.e. a curve of genus one, then under good circumstances the tropicalisation will be a piecewise linear with precisely one loop, i.e. the graph will have genus one. It turns out that the length of this loop is related to the j-invariant of the elliptic curve.
(joint work with Michael Kerber)
In tropical geometry, algebraic curves are replaced by piece-wise linear
degenerations called tropical curves. Even though we "lose a lot of
information" with this degeneration, many properties of the algebraic
curve can be read off the tropical curve, and many theorems that hold for
algebraic curves can be shown on the tropical side.
One of the fields in which tropical geometry has had a lot of success recently
is enumerative geometry. In this talk, we present an enumerative
invariant - namely the number of plane elliptic curves of a given degree d with
fixed j-invariant through 3d-1 points in general position - that can
be read off the tropical side. We present a tropical way to determine these
numbers.
This talk follows a talk given by Thomas Markwig who shows how the j-invariant
of a curve is reflected in the tropical world.
Sep 24: Josh Thompson
A Complex Projective Structure can be thought of as a
Riemann surface with the added notion of circles. These structures are
defined by charts to the Riemann sphere whose transition maps are Mobius
transformations. The transition maps are then "globalized" into
a (holonomy) representation of the fundamental group of S into PSL(2,C).
In fact, almost every representation of the fundamental group of S into
PSL(2,C) is the holonomy group of SOME projective structure. This
raises the question: How well does a holonomy representation characterize
the underlying Complex Projective Structure? Goldman answered this
question when the representation is faithful onto a discrete subgroup of
PSL(2,R), showing that all such structures come from a unique
"hyperbolic parent" by a surgery process called grafting.
In the first hour we'll discuss the definitions, introduce grafting and review
some hyperbolic geometry. Along the way we'll show that Complex
Projective Structures produce a unique signature in the form of a homotopy
class of curves on the surface. .
In the second hour, we'll use this signature to first form a new proof of the
Goldman result and later to address the problem when the holonomy is a Schottky
subgroup of PSL(2,R). We'll sketch why, in this case, a generic Complex
Projective Structure has infinitely many "hyperbolic
parents".
Oct 1: Sam Payne
Deligne's mixed Hodge theory gives
a canonical increasing filtration on the singular cohomology of an algebraic
variety, called the weight
filtration, that is strictly respected by natural operations such as cup
product and pullbacks under algebraic morphisms. I will discuss
how this weight filtration for noncompact varieties is related to the
combinatorics of boundary strata in algebraic compactifications.
I will give a new, simple,
explicit GIT construction of the moduli of stable sheaves on a projective
scheme, which avoids the Quot scheme. Then I will outline several
applications that serve to demonstrate the advantages of this construction.
This is joint work with Kai Behrend and Ionut Ciocan-Fontinane.
Oct 12: Francois-Xavier
Machu
We provide a handy example of a
family of logarithmic rank-2
connections over an elliptic curve, for which the question on the
stability of the underlying vector bundles is completely determined.
These connections are obtained as direct images of regular connections on
line bundles over genus-2 double covers of the elliptic curve, such curves
are called bielliptic curves. We give an explicit parameterization of all
such connections, determine their monodromy and differential Galois group.
The underlying rank-2 vector bundle is described in terms of elementary
transforms and birational maps of ruled surfaces. We finally determine the
local structure of these moduli spaces of connections and give an example of
a moduli space of connections having a singularity, and saying a few words
on this singularity (as its minimal resolution via geometry toric).
Oct 15: Eric Stade
Mellin transforms of class one
Whittaker functions arise as archimedean L factors of automorphic L functions.
More-or-less recent work of the speaker and of Taku Ishii of
Oct 15: Taku Ishii
I will talk about direct computation of archimedean zeta integrals for certain automorphic L-functions. By using Mellin-Barnes type integral representations of Whittaker functions, we can show the coincidence of archimedean zeta integrals and archimedean L-factors on SO(2n+1)?GL(m) (m=n,n+1). This is a joint work with Eric Stade.
Oct 22: Sarah Kitchen
The Borel-Weil-Bott theorem identifies finite dimensional representations for a complex semi-simple algebraic group with line bundles on a projective variety associated to that group. I will discuss generalizations of this principle of associating representations to sheaves. In particular, the structure of a representation will give the associated sheaf the structure of a D-module. We aim to understand representations by relating D-modules on various partial flag varieties associated to our group. I will begin with a brief overview of D-modules and equivariant sheaves, highlighting some advantageous differences from O-modules on projective varieties, then explain how to extend known results relating representations and sheaves on the full flag variety to improve our understanding of the geometric picture.
A principal minor of a matrix is
the determinant of a submatrix centered about the main diagonal. A basic linear
algebra question asks
to what extent is it possible to prescribe the principal minors of a matrix. Holtz and Sturmfels studied the
algebraic variety of principal
minors of symmetric matrices. The equations of this variety tell when it is
possible for a given vector of length 2^n to be the principal
minors of a symmetric n by n matrix. Holtz and Sturmfels found these equations
in the cases n=3,4 and discovered an interesting connection
to Cayley's hyperdeterminant. They conjectured that the hyperdeterminantal
module generates the equations of the variety.
I will describe the applications that motivate the original question. Then I
will give several examples indicating some of the beautiful
symmetry and geometry of this variety. Finally I will sketch a proof of the set
theoretic version of the Holtz and Sturmfels conjecture.
Oct 29: Steffen Marcus
The Kontsevich spaces of stable
maps (and their
generalizations) are the key moduli spaces used to study the
Gromov-Witten theory of smooth projective varieties and stacks. The
first hour of this talk will be a basic introduction to Stable Maps
and Gromov-Witten Theory of the Projective Plane. We will do at least
one simple calculation.
The second hour will begin with a discussion of the basics of
Gromov-Witten theory for smooth projective varieties and
Deligne-Mumford stacks. After this we will show how the moduli stack
of twisted stable maps can be extended to allow generic stabilizers on
the source curves of the twisted stable maps.
Nov 5: Jesse Kass
In algebraic geometry, one way to
study a variety is to put that variety into a family of varieties and then
study the family's degenerate elements.
An obstacle to applying this technique is that one must first construct an
appropriate family. For Abelian varieties, this construction problem
was first studied by N\'{e}ron and Lang.
One question posed by Lang was ``do N\'{e}ron models admit good
completions." I will discuss my work in affirmatively answering
Lang's question for certain Jacobian varieties. Of particular interest is
that I can construct extensions for certain families with additive reduction.]
Much of the first hour will be spend surveying the general topic of extending
families of varieties.
Nov 12:Yusuf Mustopa
The dth symmetric product C_d of a smooth projective curve C is a smooth projective variety which encodes the "degree-d aspect" of the geometry of C. The subordinate loci on C_d associated to linear series on C encode the degree-d aspect of maps from C to projective space. In this talk, I will discuss how these loci govern the cone of effective divisors of C_d, how some natural divisors on C_d may be characterized as subordinate loci associated to higher-rank vector bundles, and also a conjectural description of the effective cone of C_d when C is a general curve of genus g and d is at least (g/2)+1.
Nov 12: Philipp Rostalski
Polynomial equations play an
important role in mathematics,
engineering and science. Many practical problems can be reduced to the
task of computing all real roots of a system of polynomial equations
or a certain distinguished basis for the corresponding vanishing
ideal. While for the task of computing all complex roots a plethora of
algebraic tools is readily available, real root solving is still in
its infancy.
In this talk we discuss the relation between Hankel operators and real
algebraic geometry, more precisely real radical ideals. A new tool for
characterizing and computing the real radical ideal is proposed. Based
on this characterization, we devise an algorithm using numerical
linear algebra and semidefinite optimization to approximately compute
the real variety (assuming it is finite) of an ideal as well as a
border basis.
This talk is based on joined work with Jean Lasserre (LAAS-CNRS) and
Monique Laurent (CWI).
First hour (Woes).
Triangulated categories are more challenging to work with than abelian
categories, but Fourier-Mukai transforms, Beilinson's diagonal resolution for
projective space and good old functoriality should convince you of their necessity
in the context of coherent sheaves.
Second hour (Joys). Every moduli theorist loves
a new invariant, and the manifold of (Bridgeland) stability conditions is a
marvelous one, with beautiful wall and chamber structures, producing all
sorts of new insights into the moduli spaces of coherent sheaves on surfaces.
We'll look closely at the projective plane, following some recent work of Arend
Bayer and Emanuele Macri.
Dec 3: Marc Krawitz
In physics, a Landau-Ginzburg theory is
specified by a weighted homogeneous polynomial satisfying a mild non-degeneracy
condition. The "chiral ring" associated to the theory is the
orbifolded A-model constructed mathematically by Fan, Jarvis and Ruan
(arXiv:0712.4021). The unorbifolded B-model of the theory is given by the local
algebra (Milnor ring) of the defining polynomial. I will show how a natural
transposition operation suggested by Berglund-Huebsch implements mirror
symmetry for Landau-Ginzburg theories having `invertible potentials'. This
includes those specified by the simple and unimodal singularities of Arnol'd's
classification.
I will describe a duality between symmetry groups for transposed potentials and
an orbifolding construction on the B-model, leading to a generalization of the
main result, and suggesting a relationship with Arnol'd's Strange Duality.
Dec 4: Emanuele
Macri
First hour: We give a brief introduction to derived categories through Bondal-Orlov and Kuznetsov work.
Sedcond hour: We report on joint work with P. Stelari on realizing Fano varieties of lines of cubic fourfolds containing a plane as moduli spaces of twisted complexes in derived categories of K3 surfaces.
In previous semesters the seminar page was maintained by Rachel Pries and Jeff Achter.