This seminar features expository talks on topics in analysis that
reflect the participants' interests, but often don't get into courses.
In previous years we've heard talks on functional, complex, and
harmonic analysis, probability, and dynamical systems. The atmosphere
is relaxed and supportive; everyone is welcome to participate: give a
talk, or just be part of the audience.
Spring Term Schedule 2017:
Friday, June 2 & 9, 2:00--3:00 PM in NH 373
Gary Sandine, PSU
will speak on:
"Radiography for Optimists"
The idea to present this work was inspired by the recent talk, "Whatever happened to lp for p < 1?" We will take a brief tour of computerized tomography (CT) and tomographic reconstruction methods that work very well when one has views from hundreds of angles around the object of interest, such as in medical CT scans. In some non-medical CT applications I had the good fortune of collaborating on, limitations of the imaging systems and other factors meant we almost always had less than ten views, so existing reconstruction methods were inapplicable. We had some success framing it as an optimization problem using total variation regularization with a non-convex p-norm (p<1). I will show what we did and hope to make a case for why using an lp norm with p<1 was a good idea.
Slides for Gary's talk are here
Friday, May 19, 2:00--3:00 PM in NH 373 (please note corrected date!)
Bala Krishnamoorthy, Washington State University
will speak on:
We introduce new ideas for the average of a set of general shapes, which we
represent as currents from geometric measure theory. Using the flat norm to
measure the distance between currents, we present a mean and a median shape. We
present existence and regularity results for the median shape in codimensions 1
and 2. In the discrete setting, we model shapes as chains on a finite simplicial
complex. We demonstrate that the median shape of chains can be found efficiently
by solving a linear program (joint work with Yunfeng Hu, Matthew Hudelson, Altaa Tumurbaatar, and
Friday, April 28, 2:00--3:00 PM in NH 373
will speak on:
"Whatever happened to lp for p < 1?"
The sequence spaces lp are defined for all indices p > 0. For p ≥ 1 they are Banach spaces familiar to every student of functional analysis, but for p < 1 they’re almost never mentioned. Why? What are they hiding? We’ll investigate.
Notes for this talk are here
Friday, April 7, 14, & 21, 2:00--3:00 PM in NH 373
will speak on:
"Hamilton's mechanics and Legendre's transformation"
Second-order differential equations often ``look better'' as systems of
first-order equations. One of the systems equivalent to Newton's laws
of mechanics, Hamilton's mechanics, is quite beautiful.
This talk provides a self-contained derivation of Hamiltonian
mechanics from Newtonian mechanics. The derivation requires only the
simplest tools of analysis: the product and chain rules, a little
linear algebra, and the equality of mixed partial derivatives. The crux
of the computation --- the Legendre transformation --- arises naturally
in this setting.
Notes for Jim's previous talk on "Kinetic Energy as Potential" are here
Winter Term Schedule 2017:
Friday, February 24, March 3 & March 10, 2:00--3:00 PM in NH 373
Robert Lyons, Digimarc & PSU, will speak on:
"PDE's, Jet Bundles, and the Cartan Distribution"
Jets are a generalization of tangent vectors that provide a convenient
language for expressing the derivatives of maps. In a jet bundle we
associate derivatives with new independent variables and so turn a PDE
into a functional equation. To recover the PDE solution the functional
solutions must be integral manifolds of the Cartan distribution.
In these talks we'll define jet bundles and compute the Cartan
distribution. We'll then illustrate how a PDE system is re-written in
the jet manifold, and illustrate the constraints imposed by the Cartan
distribution. As a prelude, we’ll review the basics of differentiable
Friday, February 10 & 17, 2:00--3:00 PM in NH 373
Prof. Mau Nam Nguyen, PSU will speak on:
"The Theorems of Helly, Radon, and Carathéodory"
Helly's theorem, an important result from convex geometry, gives
sufficient conditions for a family of convex sets to have a nonempty
intersection; it has many proofs and many applications. Helly's theorem
has close connections to two other well-known theorems: Radon's theorem
and Carathéodory's theorem. In this talk, we’ll give the proof of
Helly's theorem and, using tools of convex analysis and optimization,
will study its relations to the theorems of Radon and Carathéodory.
Friday, January 27, 2:00--3:00 PM in NH 373
Joel Shapiro will speak on:
"Who proved the Hahn-Banach Theorem?"
The Hahn-Banach Theorem, often called ``the crown jewel of functional
analysis,’’ states that every bounded linear functional on a subspace
of a normed linear space can be extended linearly to the whole space
without increasing the norm. The result is attributed to Hans Hahn
(Germany) and Stefan Banach (Poland) who, in the late 1920’s,
independently published the proof for real scalars taught today in
every functional analysis course. For complex scalars the result is
usually attributed to Bohnenblust and Sobczyk who based their proof on
the real case in the late 1930’s.
In fact, the proof given by Hahn and Banach was discovered much earlier
in the setting of C([0,1]). In this talk I’ll sketch the origins of the
Hahn-Banach theorem, and tell you who first proved it. I’ll also tell
you who first proved the complex case.
Friday, January 20, 2:00--3:00 PM in NH 346
Ilya Spitkovsky, New York University, Abu Dhabi will speak on:
"Factorization of AP Matrix Functions"
The set AP of Bohr almost-periodic functions is the closed subalgebra
generated by pure imaginary exponentials. An AP factorization of a
square matrix function G is a representation of the form G = G+ D G-, where D is an exponential-valued diagonal matrix and G+, G-
are AP-valued matrix functions which, along with their inverses, have
non-negative Bohr-Fourier coefficients. This generalizes the
Wiener-Hopf factorization of continuous matrix-valued functions on the
unit circle. Time permitting, some open problems will be described.
Fall Term Schedule 2016:
Friday, December 2, 2:00--3:00 PM in NH 373
Wondi Geremew, Stockton University School of General Studies will speak on:
"NEW DCA BASED ALGORITHMS FOR SOLVING BILEVEL HIERARCHICAL CLUSTERING PROBLEMS"
A bilevel hierarchical clustering model is commonly used in designing
an optimal multicast networks. In this presentation we will consider
three different problem formulations of the bilevel hierarchical
clustering problem – a discrete optimization problem which can be shown
to be NP-hard.
Our approach is to reformulate the problem as a continuous
optimization problem by making some relaxations on the discreteness
conditions. This approach was considered by other researchers earlier,
but their proposed method depends on the square of the Euclidian norm –
because of its differentiability. Instead, our approach is to replace
the Euclidean norm by the Minkoski gauge function. Then we will apply
the Nesterov’s Smoothing Approximation technique to approximate the
Minkoski gauge function by a smooth function.
With this approach, we are able to propose three new DCA
based algorithms to solve the problems, and our preliminary numerical
results are promising.
(This talk is based on the joint work with N.M. Nam, S. Reynolds, and T. Tran)
Friday, November 4 & 18, 2:00--3:00 PM in NH 373
Tuyen Tran will speak on:
"Fenchel conjugate and its calculus"
Abstract. These talks focus on a central
concept of convex analysis called the Fenchel conjugate. We begin with
basic concepts and properties of the Fenchel conjugate. After that, we
develop a geometric approach to derive major calculus results for
Fenchel conjugates of extended-real-valued convex functions in infinite
Friday, October 21 & 28, 2:00--3:00 PM in NH 373
Jim Rulla will speak on:
"Kinetic Energy as Potential"
Prerequisite: Enough linear algebra to know that xTAx
is a Newton's laws of motion relate vector forces and momentum. The
vec-tors are easy to visualize and measure, and Newton's laws appeal to
the ``British'' school of experimental mechanics. The ``Continental''
school of theoretical mechanics prefers Lagrange's formulation using
(scalar) kinetic and potential energies. Kinetic energy is not
intuitive. Mechanics texts seldom motivate how kinetic energy arises
from Newton's laws, so it feels like a ``lucky guess''. This talk gives
a simple, logical derivation of the Lagrangian formulation from
Newton's laws, showing how kinetic energy is not merely a lucky guess,
but a logical consequence of Newton's laws.
Notes for this talk are here.
Friday, September 30 & October 7, 2:00--3:00 PM in NH 346
Mau Nam Nguyen will speak on:
"Convex Analysis and Optimization: from Convexity to Nonconvexity"
Convex analysis has been well recognized as an important area of
mathematics providing mathematical foundation for convex op-timization,
a field with many applications to different fields such as ma-chine
learning, facility location, and compressed sensing. At the same time,
many recent applications in these fields require optimization
tech-niques beyond convexity. In this talk, we discuss some results of
convex analysis used in optimization of nonconvex functions, especially
func-tions that are representable as differences of convex functions.
Then we present applications of nonconvex optimization to solving a
number of problems in multi-facility location and clustering.
Seminar schedule and lecture notes 2015-2016
Seminar schedule and lecture notes 2014-2015
Seminar schedule and lecture notes 2013-2014
Seminar schedule and lecture notes 2012-2013