Fri, 17 Jun 2016

16:00 - 17:00
L1

Conjugacy classes and group representations

David Vogan
(MIT)
Abstract

One of the big ideas in linear algebra is {\em eigenvalues}. Most matrices become in some basis {\em diagonal} matrices; so a lot of information about the matrix (which is specified by $n^2$ matrix entries) is encoded by by just $n$ eigenvalues. The fact that lots of different matrices can have the same eigenvalues reflects the fact that matrix multiplication is not commutative.

I'll look at how to make these vague statements (``lots of different matrices...") more precise; how to extend them from matrices to abstract symmetry groups; and how to relate abstract symmetry groups to matrices.

Tue, 16 Jun 2015

14:30 - 15:00
L3

Are resultant methods numerically unstable for multidimensional rootfinding

Alex Townsend
(MIT)
Abstract
A popular class of algorithms for global multidimensional rootfinding are hidden-variable resultant methods. In two dimensions, when significant care is taken, 
they are competitive practical rootfinders.  However, in higher dimensions they are known to be notoriously difficult, if not impossible, to make numerically robust.  We will show that the most popular variant based on the Cayley resultant is inherently and spectacularly numerically unstable by a factor that grows exponentially with the dimension. Disastrous. Yet, perhaps, it can be circumnavigated. 
Fri, 19 Jun 2015
17:30
L2

Social Capital and Microfinance

Esther Duflo
(MIT)
Abstract
This talk will review the literature on the interaction between social capital and microfinance: how microfinance adoption diffuses through the social network, how its functioning leverages existing links and strengthen some links while weakening others
Tue, 21 Jan 2014

14:30 - 15:30
L6

Sparse graph limits and scale-free networks

Yufei Zhao
(MIT)
Abstract

We introduce and develop a theory of limits for sequences of sparse graphs based on $L^p$ graphons, which generalizes both the existing $L^\infty$ theory of dense graph limits and its extension by Bollob\'as and Riordan to sparse graphs without dense spots. In doing so, we replace the no dense spots hypothesis with weaker assumptions, which allow us to analyze graphs with power law degree distributions. This gives the first broadly applicable limit theory for sparse graphs with unbounded average degrees.

Joint work with Christian Borgs, Jennifer T. Chayes, and Henry Cohn.

Mon, 13 Jan 2014

17:20 - 18:10
L4

Null singularities in general relativity

Jonathan Luk
(MIT)
Abstract

We consider spacetimes arising from perturbations of the interior of Kerr

black holes. These spacetimes have a null boundary in the future such that

the metric extends continuously beyond. However, the Christoffel symbols

may fail to be square integrable in a neighborhood of any point on the

boundary. This is joint work with M. Dafermos

Tue, 10 Jan 2012

15:45 - 16:45
SR1

Clone of (HoRSE seminar) Real variation of stabilities and equivariant quantum cohomology II

Roman Bezrukavnikov
(MIT)
Abstract

I will describe a version of the definition of stability conditions on a triangulated category to which we were led by the study of quantization of symplectic resolutions of singularities over fields of positive characteristic. Partly motivated by ideas of Tom Bridgeland, we conjectured a relation of this structure to equivariant quantum cohomology; this conjecture has been verified in some classes of examples. The talk is based on joint projects with Anno, Mirkovic, Okounkov and others

Tue, 10 Jan 2012

14:00 - 15:00
SR1

(HoRSE seminar) Real variation of stabilities and equivariant quantum cohomology I

Roman Bezrukavnikov
(MIT)
Abstract

I will describe a version of the definition of stability conditions on a triangulated category to which we were led by the study of quantization of symplectic resolutions of singularities over fields of positive characteristic. Partly motivated by ideas of Tom Bridgeland, we conjectured a relation of this structure to equivariant quantum cohomology; this conjecture has been verified in some classes of examples. The talk is based on joint projects with Anno, Mirkovic, Okounkov and others

Fri, 21 Oct 2011

14:30 - 15:30
DH 3rd floor SR

The Timescales of The Ocean Circulation and Climate

Prof. Carl Wunsch
(MIT)
Abstract

Studies of the ocean circulation and climate have come to be dominated by the results of complex numerical models encompassing hundreds of thousands of lines of computer code and whose physics may be more difficult to penetrate than the real system. Some insight into the large-scale ocean circulation can perhaps be gained by taking a step back and considering the gross time scales governing oceanic changes. These can derived from a wide variety of simple considerations such as energy flux rates, signal velocities, tracer equilibrium times, and others. At any given time, observed changes are likely a summation of shifts taking place over all of these time scales.

Thu, 21 Jun 2001

14:00 - 15:00
Comlab

Tridiagonal matrices and trees

Prof Gilbert Strang
(MIT)
Abstract

Tridiagonal matrices and three term recurrences and second order equations appear amazingly often, throughout all of mathematics. We won't try to review this subject. Instead we look in two less familiar directions.

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Here is a tridiagonal matrix problem that waited surprisingly long for a solution. Forward elimination factors T into LDU, with the pivots in D as usual. Backward elimination, from row n to row 1, factors T into U_D_L_. Parlett asked for a proof that diag(D + D_) = diag(T) + diag(T^-1).^-1. In an excellent paper (Lin Alg Appl 1997) Dhillon and Parlett extended this four-diagonal identity to block tridiagonal matrices, and also applied it to their "Holy Grail" algorithm for the eigenproblem. I would like to make a different connection, to the Kalman filter.

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The second topic is a generalization of tridiagonal to "tree-diagonal". Unlike the interval, the tree can branch. In the matrix T, each vertex is connected only to its neighbors (but a branch point has more than two neighbors). The continuous analogue is a second order differential equation on a tree. The "non-jump" conditions at a meeting of N edges are continuity of the potential (N-1 equations) and Kirchhoff's Current Law (1 equation). Several important properties of tridiagonal matrices, including O(N) algorithms, survive on trees.

Thu, 06 Jun 2002

14:00 - 15:00
Comlab

Filtering & signal processing

Prof Gilbert Strang and Per-Olof Persson
(MIT)
Abstract

We discuss two filters that are frequently used to smooth data.

One is the (nonlinear) median filter, that chooses the median

of the sample values in the sliding window. This deals effectively

with "outliers" that are beyond the correct sample range, and will

never be chosen as the median. A straightforward implementation of

the filter is expensive for large windows, particularly in two dimensions

(for images).

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The second filter is linear, and known as "Savitzky-Golay". It is

frequently used in spectroscopy, to locate positions and peaks and

widths of spectral lines. This filter is based on a least-squares fit

of the samples in the sliding window to a polynomial of relatively

low degree. The filter coefficients are unlike the equiripple filter

that is optimal in the maximum norm, and the "maxflat" filters that

are central in wavelet constructions. Should they be better known....?

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We will discuss the analysis and the implementation of both filters.

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