All talks, unless otherwise mentioned, are on Thursday afternoon at 3:30 pm in Room 304 of
the Boyd Graduate Studies Building.
The colloquium talk is pitched at a general mathematical audience, including graduate students. All are warmly encouraged to attend. Typically, there is a post-colloquium dinner (time and place TBA at the colloquium), for which a subsidy is available for graduate students.
Abstract: A quiver is a finite directed graph, that is, a finite set of vertices some of which are joined by arrows. A quiver representation assigns a finite-dimensional vector space to each vertex, and a linear map between the corresponding spaces to each arrow. The theory of quiver representations is a beautiful and well-developed field. A fundamental role in this theory is played by Bernstein-Gelfand-Ponomarev reflection functors associated to every source or sink of a quiver. In an ongoing joint work with Harm Derksen and Jerzy Weyman we extend these functors to arbitrary vertices. This construction is based on a framework of quivers with potentials; their representations are quiver representations satisfying relations of a special kind between the linear maps attached to arrows. The motivations for this work come from several sources: superpotentials in physics, Calabi-Yau algebras, cluster algebras. However no special knowledge will be assumed in any of these subjects, and the exposition should be accessible to graduate students.
September 18, 2008: Xiao-Li Meng, Harvard University
Self-consistency: A General Recipe for Semi-parametric and Non-parametric Estimation
with Incomplete and Irregularly Spaced Data
Abstract: A common frustration in statistical estimation and, more generally, information recovery is when data are missing, incomplete or irregularly spaced (e.g., as with wavelets). Self-consistency is a general principle for handling such “bad” data problems in semi-parametric and non-parametric estimation and under an arbitrary loss function. It also provides a theoretical criterion to regulate and improve estimation procedures even when there is no missing data. Indeed, efficient estimation procedures, such as maximum likelihood estimation, are automatically self-consistent (asymptotically under square-loss). Conceptually, self-consistency is extremely appealing; it is essentially a mathematical formalization of the iteration of common-sense "trial-and-error" methods until no more improvement is possible.
Mathematically it is elegant, with one fixed-point equation to solve and a general projection theorem to establish its optimality. Practically, it is straightforward to program because it directly uses the regular/complete-data method for each iteration, much like the EM algorithm, which can be viewed as a version of self-consistency for maximum likelihood estimation. Its major disadvantage is that it can be computationally intensive. However, increasingly efficient (approximate) implementations are being discovered, such as for wavelet de-noising with hard and soft thresholding. This talk summarizes those these findings, based on joint work with Thomas
Lee on wavelet applications and with Zhan Li on the theoretical foundation of the self-consistency principle.
October 23, 2008:
Monika Ludwig, Polytechnic Institute of NYU
Affine surface areas
Abstract: For smooth convex sets in Rn, there are two classical SLn invariant notions of surface area: equi-affine surface area and centro-affine surface area. In important recent papers by Leichtweiss , Lutwak, Schuett and Werner, these definitions have been extended to general convex sets.
Using valuations on convex sets, equi-affine surface area and centro-affine surface area can be characterized by their transformation properties. Moreover, a much richer family of affine surface areas can be obtained. These characterization theorems (joint work with Matthias Reitzner) will be described and applications of affine surface areas will be discussed.
Low-Dimensional Characterization of Neuronal Network Activity
in a Large-Scale Model of the Visual Cortex
Abstract: A major theoretical challenge in systems neuroscience
modeling is to summarize the dynamics of complex neuronal networks in
low dimensional models. While most approaches have focused either on
developing reduced descriptions of single neurons or on mean-field,
population density models of networks, here, we describe our progress
in developing low-dimensional dynamical systems models of large-scale
cortical networks using a data-driven approach. Taking a model visual
cortical network to be our experimental system, we use empirical
principal components analysis (PCA) of simulation data as a dimension
reduction tool to generate low-dimensional dynamical systems which
allow us to predict (and postdict) the simulation data in an
approximate, but mathematically consistent, fashion. Furthermore, we
use this empirical, data-driven PCA on a small subset of model
neurons; our results suggest that it may be possible to generate such
target dynamical systems from simultaneous electro-physiological
measurements of network neurons "in vivo".
January 29, 2009:
Gerard Besson, Directeur de Recherches C.N.R.S., Institut Fourier Université de Grenoble, France.
A simple proof of the Mostow rigidity theorem
Abstract: We shall present a natural and simple construction of maps between negatively curved simply connected manifolds with nice properties with respect to the volume element. We use a family of measures on the boundary at infinity. The description will be done in the simplest case, namely unit discs in R^n endowed with the hyperbolic metric. As an application we shall prove Mostow's rigidity theorem for compact hyperbolic manifolds. This amounts to showing that the above construction yields, in the right set up, an isometry. This is a joint work with G. Courtois and S. Gallot.
February 19, 2009:
Julia Hartmann, RWTH Aachen University.
Galois Groups of Linear Differential Equations
Abstract: Differential Galois theory is the algebraic theory of
linear homogeneous differential equations. It mimics and
generalizes the usual Galois theory for polynomials. The talk gives an introduction to differential Galois theory and then discusses the inverse problem, i.e. which linear
algebraic groups occur as differential Galois groups over a given
field. We focus on situations where this problem can be attacked
using patching methods.
March 19, 2009:
Herbert Lange, Friedrich-Alexander-Universität Erlangen-Nürnberg.
Brill-Noether theory of vector bundles on curves
Abstract: After recalling classical Brill-Noether theory of line bundles on curves, I will introduce and explain Brill-Noether theory of vector bundles with an emphasis on the differences between the cases of rank one and higher rank. At the end of the talk I will present a recent joint result on the subject with Peter Newstead.
A new look at the moduli space of hyperelliptic curves
Abstract: There are two natural ways to compactify the moduli
space of smooth hyperelliptic curves of genus g: one is to take the closure
in the moduli space of stable curves of genus g, and the other is to construct
the GIT quotient of semistable binary forms of degree 2g + 2. Avritzer and
Lange showed that there is a projective birational morphism f from the
former to the latter extending the natural isomorphism between the loci of
smooth curves. By carrying out a log minimal model program (MMP), we
prove that f decomposes into a series of divisorial contractions
collapsing the boundary divisors in natural order. We also obtain a conjectural
formula for critical values in the log MMP for the moduli of stable
hyperelliptic curves which are also expected to be critical values for
the log MMP for the moduli of stable curves. They are then identified
with the values from a formula obtained by considering the GIT stability of
curves. This suggests that log canonical models can be constructed
as GIT quotients of Hilbert scheme of curves that are 'rationally' semistable.