Spring 2011
Colloquium Schedule
Department of Mathematics
Spring 2011
Colloquia
(and other events) Schedule:
Except as noted, all talks are on Mondays from
4:00 to 4:50pm, in Avery Hall 115, preceded by refreshments at 3:30 pm in Avery
Hall 348.
Here is a discussion of
what to expect at Colloquium talks and here
is a discussion of what might make a good colloquium talk.
Go here
for information on scheduling a future colloquium
and eventually
here
for
the tentative Fall 2011
Colloquia schedule.
Mon Jan 10:
First Day of classes
Jan 10: No colloquium
Mon Jan 17:
Martin Luther King Day/ No colloquium
Jan 24:
Speaker: Dana
Williams
Affiliation: Dartmouth College
Local host: Allan Donsig
Title: The Equivariant Brauer Group
Abstract:
In algebraic topology, we learn to associate groups Hn(T) to
locally compact spaces which "count the
n-dimensional holes in T''.
In this talk, I want to describe how to realize H3(T) as a set
Br(T) of
equivalence classes of certain well-behaved
C*-algebras. The group structure imposed on Br(T)
via
its identification with H3(T) is very natural in its
C*-setting. With this group structure, Br(T) is
called
the "Brauer group" of T. Depending on your point of
view, this result can be viewed either as a
concrete realization of
H3(T) or as a classification result for a class of
C*-algebras. In the last part of
the talk, I want to describe an
equivariant version of Br(T) developed jointly with
David Crocker,
Alex Kumjian and Iain Raeburn. No prior knowledge of
C*-algebras or operator algebras will be
assumed.
This colloquium is funded by the
UNL Research Council.
January 28-30
Nebraska Conference for Undergraduate Women in Mathematics
Invited addresses: Friday January 28 Linda Petzold
Saturday January 29 Fan Chung)
Jan 31: No colloquium
(Because of the activities and talks at NCUWM)
Feb 7:
Speaker: Paul Wenger
Affiliation: University of Colorado Denver
Local host: Stephen Hartke
Title: Unique Saturation and Eigenvalue
Methods for Graphs
Abstract:
There is a long history of using tools from linear algebra on problems from
graph theory. In this talk
we present various results that are proved by
analyzing the eigenvalues of the adjacency matrix of a
graph. Two classic
results in this area are the Hoffman-Singleton proof that there are at most
five
Moore graphs (graphs with diameter d and girth 2d+1), and the
Friendship Theorem of Erdös, Rènyi,
and Sös. We demonstrate a
surprising connection between Moore graphs and friendship graphs.
In particular, we discuss uniquely H-saturated graphs. Given a graph
H, a graph G is uniquely
H-saturated if G does not contain a copy of
H, but adding any edge to G completes exactly one copy
of H. This is a
structural variant of a traditional extremal graph problem first explored by Turàn, and
subsequently by Erdös, Hajnal, and Moon. We discuss uniquely
H-saturated graphs for various H and
show that the Moore graphs and
friendship graphs classify the uniquely H-saturated graphs when H is
a
cycle with three or five vertices, respectively.
Feb 14: No colloquium
Feb 21:
Speaker: Vladimir Itskov
Affiliation:UNL
Local host:
Title: Algebraic approaches
to the network encoding problem
Abstract:
The brain stores memories by modifying synaptic weights in networks of neurons. A long-standing
problem is: how one can design a neural network that encodes a list of desired memory patterns? One
can also interpret the structure of the set of memory patterns encoded by a network in terms of an
underlying "stimulus space" represented by the neurons. What kinds of stimulus spaces can be
represented by a recurrent network?
We approach these problems in the context of a simplified network model that allows us to convert
the original questions into questions about the combinatorial structure of eigenvalues for the defining
matrices of the network and its subnetworks. We prove that, for any homotopy type of a finite
simplicial complex, there exist networks that represent a stimulus space with this homotopy type. In
other words, one can realize stimulus spaces with prescribed topological properties. In order to
explore more detailed properties of network encoding, we investigate networks that are perturbations
of “maximally flexible” networks. Here we employ a variety of algebraic tools, and find some
surprising connections to discrete geometry.
Feb 28:
Speaker: Dr. Matthias Eller
Affiliation: Georgetown University
Local host: Daniel Toundykov
Title: The boundary value
problem for hyperbolic systems of partial differential equations
Abstract:
A survey of the classical theory of the boundary value problem for hyperbolic systems will be given.
Friedrichs's Theory originated with symmetric hyperbolic systems whereas Kreiss's theory is
concerned with strictly hyperbolic systems. Both theories establish well-posedness for the boundary
value problem provided the boundary condition satisfies certain criteria and the coefficients are
sufficiently regular.
Newer results concerning rough coefficients and more general boundary conditions due to Metivier
and Coulombel will be presented. Finally, conservative boundary conditions which are
of particular interest in applied problems will be discussed.
This colloquium is funded by the
National Science Foundation.
March 7:
Speaker: Ryan Karr
Affiliation:University of Central Florida
Local host: Roger Wiegand
Title: Cubic orders and their lattices
Abstract:
The ring of integers in a number field K contains a certain
collection of subrings, the orders
of K. When K is a quadratic
number field and R is an order in K, the structure of torsion-free
R-modules, a.k.a. R-lattices, is easy to describe. However, when K is a
cubic number field, much less
is known about the structure of
R-lattices. In this colloquium talk, we’ll examine the structure of
these lattices in the closely-related contexts of direct-sum
cancellation and representation type.
March 14:
Speaker: Andrei Zelevinsky
Affiliation: Northeastern University
Local host: Luchezar Avramov
Title: Cluster algebras via quivers with potentials
Abstract:
Cluster algebras are commutative rings of a special kind making a
surprising appearance in
a
variety of settings, including tilting
theory, Poisson geometry, Teichmuller theory, representations of
semisimple groups, etc. Their structure is governed by several
piecewise-polynomial and rational
recurrences on a regular tree.
Although these recurrences are quite explicit and elementary, a direct
proof of
their conjectural properties seems to be hard to find. Jointly with
Harm Derksen and Jerzy
Weyman we find their representation-theoretic
interpretation in terms of quivers with
potentials,
allowing us to prove most of the conjectures in question.
I will try to provide a reasonably self-contained introduction to
cluster algebras and quivers with potentials.
March 20-27
Spring break/no colloq
March 28:
Speaker: John Neuberger
Affiliation: University of North Texas
Local host:Lynn Erbe and Allan Peterson
Title:
Semidynamical systems and differential equations
Abstract:
We discuss generators for certain kinds of functions, called semidynamical
systems, on a
complete separable metric space X. A "semidynamical
system" T is actually a set of transformations
on X, one for each natural
number n, such that T(0) is the identity transformation and, for n,m
natural numbers, the composition T(n)T(m) is the same as T(n+m).
A complete characterization of such generators is given. This solves an old problem
on the relationship
between semidynamical
systems and time-dependent differential
equations.
An analogous generator for local (in "time") semigroups is given. A possible application
to the open
local-global existence problem for Navier-Stokes equations
is given, together with an indication of a
numerical attack on this problem.
April 1 2011
Rowlee Lecture Avery 115;
Reception, 3:15 p.m. in Avery 348
Speaker: David Eisenbud
Affiliation: University of California, Berkeley
Local host: John Meakin
Title:
Plato's Cave: Some things we know and some things we don't know about shadows on
the wall
Abstract:
What can you tell about an object from its shadow? There are many versions of this
question, important not only in geometry but in data analysis and elsewhere. I'll describe some of
these, and then focus on classical geometric constructions having to do with curves and surfaces and
things with more dimensions, and some of the mysteries about them that remain.
April 1-3
KUMUNU (Kansas-Missouri-Nebraska) Algebra conference
April 4: No colloquium (we have talks March 28, April 1 already)
April 11:
Speaker: Gene Abrams
Affiliation:University of Colorado at Colorado Springs
Local host: Roger Wiegand
Title:
Leavitt path algebras: a five year update
Abstract:
Most of the rings one encounters as basic examples have what's known as the ``Invariant Basis
Number"
property,
namely, for every pair of positive
integers m and n, if the free left R-modules RRm
and
RRn are isomorphic,
then m=n.
(For instance,
the IBN property of fields is used to show that the
dimension of a vector space over a field is well
defined.) In seminal work completed the early 1960's,
Bill Leavitt produced a specific, universal
collection of algebras which fail to have IBN. While it's fair
to say that these algebras were
initially viewed as mere pathologies, it's just as fair to say that these
now-so-called Leavitt
algebras currently play a central, fundamental role in numerous lines of
research in both algebra
and analysis.
In a Spring 2006 talk presented in the University of Nebraska Colloquium Series (in honor of Bill
Leavitt's 90th birthday), I described one such line of research. Specifically, for any directed
graph E and
field K one can define the Leavitt path algebra LK(E).
When E is a
graph with one vertex and at least
two loops, the Leavitt path algebra construction yields precisely
the Leavitt algebras of Bill's previous
work. In Spring 2006 the subject of Leavitt path algebras
was barely out of infancy, the initial results
in this area having been developed in late 2004.
As of Spring 2011 the subject has matured well into
adolescence, currently enjoying a seemingly
constant opening of new lines of investigation, and the
significant advancement of existing lines.
I'll give an overview of some of the work on Leavitt path
algebras which has occurred in the
intervening five years, as well as an overview of some of the future
directions in which this
topic may lead.
There should be something for everyone in this presentation, including and especially
algebraists,
analysts, graph theorists, and folks working in symbolic dynamics. The talk
will be aimed at a general
audience; for most of the presentation, a basic course in rings
and modules will provide more-than-
adequate background.
April 18:
Speaker: Carly Klivans
Affiliation: University of Chicago
Local host: Carina Curto
Title: Spanning Trees and Generalized
Matrix Tree Theorems
Abstract:
The classical matrix-tree theorem, first observed by Kirchoff in his work on electrical networks,
expresses the number of spanning trees of a graph. It is a fundamental theorem in combinatorics with a
wide range of applications. This talk concerns generalizations of spanning trees and the matrix-tree
theorem from graphs to more general cell complexes of arbitrary dimension. Central to this
investigation is the study of a combinatorial Laplacian which is a discrete version of the usual
Laplace
operator on differential forms on a Riemannian manifold. I will give a brief history of
the combinatorial
Laplacian and more recent considerations from topological combinatorics. In
particular, we present
cellular matrix-tree theorems which enumerate cellular spanning trees in
terms of the spectra and
determinant of a combinatorial Laplacian.
This talk reflects joint work with Art Duval and Jeremy Martin.
April 25:
Speaker: Zoran Grujic
Affiliation: University of Virginia
Local host: Daniel Toundykov and Lorena Bociu
Title: Energy cascade
in physical scales of 3D incompressible flows
Abstract:
Classical turbulence phenomenology (Kolmogorov, Onsager; 1940's) is based primarily
on the
empirical evidence, and it has been a great challenge for both physicists and
mathematicians to derive
the key features of the turbulence phenomenon directly from
the basic continuum model for 3D
incompressible viscous flows -- the 3D Navier-Stokes
equations.
The lecture will review mathematical results pertaining to hallmarks of the turbulence
phenomenology:
existence of the energy cascade (a nearly-constant, nonlinear transfer
of the averaged energy across a
range of scales -- the so-called inertial range), and
locality (in scale) of the averaged flux, including a
new setting for the study of
cascades in actual physical scales of the flow.
This colloquium is funded by the
UNL Research Council.
Mon May 2:
First Day of Spring Semester Final Exams
May 2 : No Colloquium
Current
Schedule of Open Dates for Fall 2011
(will be here eventually).