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Valley Geometry SeminarFridays, LGRT 1634
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 | 1 February | Cristian Lenart, SUNY-Albany | |
| 4:00-5:00 | Multiplication formulas in the K-theory of flag varieties | ||
| 8 February | David Cox, Amherst College | |
| 4:00-5:00 | Three Stories about Algebraic Geometry and its Applications | ||
| 15 February | Nick Schmitt, U Mass Amherst | |
| 4:00-5:00 | Constant mean curvature surfaces via analytic methods: theory, numerics and visualization | ||
| 22 February | Manfred Minimair, Seton Hall | |
| 3:30-4:15 & 4:30-5:15 | Resultants of Composed Polynomials | ||
| 1 March | Eyal Markman, U Mass Amherst | |
| 4:00-5:00 | Orbifold cohomology for global quotients, after B. Fantechi and L. Gottsche | ||
| 8 March | Tevian Dray, Mount Holyoke College | |
| 4:00-5:00 | The Geometry of the Octonions | ||
| 13 March | Amit Khetan, University of California, Berkeley | |
| 4:00-5:00 | Special Lecture | ||
| 15 & 22 March | No Seminar (Spring Break) | |
| 29 March | Anders Buch, MIT | |
| 4:00-5:00 | Quantum cohomology of Grassmannians | ||
| 5 April | Katrin Leschke, TU Berlin and UMass | |
| 4:00-5:00 | Willmore Spheres in Quaternionic Projective Space | ||
| 12 April | Paul Seidel, IAS | |
| 4:00-5:00 | Fukaya Categories and Deformations | ||
| 19 April | Victoria Powers, Emory University | |
| 4:00-5:00 | Representation Theorems in Real Algebraic Geometry and Applications to Optimization on Semialgebraic Sets. | ||
| 24 April | Robert MacPherson, Institute for Advanced Study | |
| Wednesday! 4:30-5:30 | A functor from topology to geometry. | ||
| 3 May | Eugene Xia, U Mass, Amherst | |
| 4:00-5:00 | Higgs bundles on representation varieties | ||
| 10 May | Andrei Caldararu, U Mass Amherst | |
| 4:00-5:00 | On an example of Vafa-Witten or how to get rid of discrete torsion | ||
Grothendieck and Schubert polynomials are representatives for
Schubert classes in the K-theory and cohomology of complex flag
varieties, respectively. The main object of the talk is an explicit
formula for expanding in the basis of Grothendieck polynomials the
product of two such polynomials, one of which is indexed by an arbitrary
permutation, and the other by a simple transposition. This is a K-theory
generalization of Monk's formula for Schubert polynomials, which, in
turn, is a special case of Chevalley's formula for multiplying Schubert
classes in the cohomology of flag varieties corresponding to Lie groups.
Our formula is in terms of increasing chains in a certain suborder of
the Bruhat order on the symmetric group with certain labels on its
covers. Some related results and a Hopf algebra perspective will also be
presented, if time permits.
TOP
This talk will discuss three separate stories which illustrate some
of the unexpected ways that algebraic geometry and commutative algebra can
arise in applied situations. The first story deals with the regularity of
graded module over a polynomial ring, which is computed using a free
resolution. I will explain how regularity relates to implicitization
questions asked by computer scientists. The second story concerns a formula
useful in interpolation and numerical analysis which is related to results
about local complete intersections. The final story begins with a paper in
Nature describing a synthetic oscillatory network of transcriptional
regulators. The authors make some unsupported claims about the number of
steady-state solutions of their system. Proving their claims leads to a
question in computational real algebraic geometry which reveals that
cleverness is more useful than a computer.
TOP
We talk on efficiently computing resultants of composed polynomials. By the resultant of several polynomials in several variables (one fewer variables than polynomials) we mean an irreducible polynomial in the coefficients of the polynomials that vanishes if they have a common zero. By a composed polynomial we mean the polynomial obtained from a given polynomial by replacing each variable by a polynomial.
In the first part of the talk we present the main results and
in the second part we give detailed proofs.
TOP
The octonions are the last in the sequence of 4 division algebras generalizing
the real and complex numbers. The octonions play a somewhat surprising role
in a wide range of geometric phenomena, some of which will be described in
this talk. In particular, the octonions provide a natural description of the
rotation groups SO(7), SO(8), and G2, 2x2 octonionic Hermitian matrices lead
to a natural description of the Lorentz group SO(9,1) (and the last Hopf
fibration), and 3x3 octonionic Hermitian matrices lead to a natural
description of SO(27), SO(26,1), F4, and E6. Furthermore, this geometric
structure is important for physics, from quantum mechanics to supersymmetry to
particle physics.
TOP
The (small) quantum cohomology ring of a Grassmann variety encodes the
enumerative geometry of rational curves in this variety. By using
degeneracy loci formulas on quot schemes, Bertram has proved quantum
Pieri and Giambelli formulas which give a complete description of the
quantum cohomology ring. In this talk I will present elementary new
proofs of these results which rely only on the definition of
Gromov-Witten invariants and standard facts about the usual cohomology
of Grassmannians. I will also report on work in progress with Andrew
Kresch and Harry Tamvakis towards obtaining a quantum
Littlewood-Richardson rule for the genus zero Gromov-Witten invariants
on Grassmannians.
This work is part of a project where a ``quaternionified'' complex
analysis is used to give new results in surface theory. The important aspect
in the quaternionic set up is that conformal maps from a Riemann surface into
S^4 = HP^1 play the role of the meromorphic functions in complex analysis.
Basic constructions of complex Riemann surface theory, such as holomorphic
line bundles, holomorphic curves in projective space, the Kodaira embedding,
the Pl"ucker relations and the Riemann-Roch estimate, carry over to the
quaternionic setting. Additionally, an important new invariant of the
quaternionic holomorphic theory is the Willmore energy of a holomorphic
curve. We consider critical points of the Willmore energy under variations
by holomorphic curves, the so-called Willmore surfaces, and explain the
classification result for Willmore spheres in HP^n: every Willmore sphere has
integer Willmore energy, and arises from complex holomorphic data.
"Open string theory" is the physicists' name for
a theory of maps from Riemann surfaces with
boundary to a Kaehler manifolds. I will explain
what algebraic structures this gives rise to,
and how one might want to use algebraic
deformation theory (of a slightly nonstandard
kind) to understand this. The talk is largely
speculative and conjectural.
Back
A real representation theorem is a theorem characterising real polynomials
which are positive on a given semialgebraic set, i.e., a set defined by finitely
many polynomial inequalities. Usually the characterization is given in
terms of the polynomials defining the semialgebraic set and sums of squares.
A classic example of such a theorem (in one variable) is a result of Fekete
which says that any polynomial which is non-negative on [-1,1] can be written
in the form f + (1 - x2)g, where
f and g are globally non-negative and hence
sums of squares. Recently, such theorems have been used to solve problems
involving optimization on semialgebraic sets. The link between the abstract
theorems and algorithms for solving optimization problems is provided
by semidefinite programming. We will briefly survey real representation
theorems, explain the connection with semidefinite programming, and describe
some applications in optimization.
Back
Let X be an appropriate topological space and let T be a torus acting on it. Then Spec H*T(X) is an affine algebraic variety associated to X and the torus action. Suppose X satisfies the hypotheses that H*T(X) is generated by H2T(X) and T has only isolated fixed points. Then Spec H*T(X) is the union of all the linear spaces of a particular arrangement of linear subspaces in a vector space. Many of our favorite spaces satisfy these hypotheses: certain flag varieties, Schubert varieties, Springer fibers, affine analogues of these things, toric varieties, complete symmetric varieties, ... As might be expected, nice spaces lead to nice arrangements of linear subspaces.
Tomorrow's lecture will show an easy way to calculate the arrangement
of linear subspaces from the the geometry of the torus action, in
certain cases.
Back
This talk gives an overview of a theorem that relates flat
connections and Higgs bundles introduced by Hitchin et al. Applications
include the study of complex and real representation varieties.
Back
In 1994 Vafa and Witten examined an example of a smooth string theory, whose central feature was that it appeared that it had to be compactified on a singular Calabi-Yau 3-fold (a number of ordinary double points seemed unavoidable, and they could not be removed in the physical context either by deformation or by blowing up). Later, Aspinwall, Morrison and Gross suggested that these singularities were intricately related to the existence of a non-trivial element of the Brauer group of this singular space.
We'll explain how all these ideas come together in a very clear picture, and present a way of removing the singularities by moving to an entirely new space. We'll also discuss how this could in principle be used in general to get rid of discrete torsion, by way of the SYZ conjecture.
Most of the talk shall be at a relatively elementary level, requiring no
knowledge of string theory.
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