Portland State University

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.

Friday, June 2 & 9, 2:00--3:00 PM in NH 373

"Radiography for Optimists"

Slides for Gary's talk are here.Abstract.The idea to present this work was inspired by the recent talk, "Whatever happened to l^{p}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 l^{p}norm with p<1 was a good idea.

Friday, May 19, 2:00--3:00 PM in NH 373 (please note corrected date!)

"Median Shapes"

Abstract.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 Kevin Vixie).

Friday, April 28, 2:00--3:00 PM in NH 373

"Whatever happened to l^{p}for p < 1?"

Notes for this talk are here.Abstract.The sequence spaces l^{p}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.

Friday, April 7, 14, & 21, 2:00--3:00 PM in NH 373

"Hamilton's mechanics and Legendre's transformation"

Notes for Jim's previous talk on "Kinetic Energy as Potential" are here.Abstract.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.

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"Abstract.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 manifolds.

Friday, February 10 & 17, 2:00--3:00 PM in NH 373

Prof. Mau Nam Nguyen, PSUwill speak on:"The Theorems of Helly, Radon, and Carathéodory"Abstract.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 Shapirowill speak on:"Who proved the Hahn-Banach Theorem?"Abstract.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"Abstract.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 formG = G, where_{+}D G_{-}Dis an exponential-valued diagonal matrix andGare 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._{+}, G_{-}

Fall Term Schedule 2016:

Friday, December 2, 2:00--3:00 PM in NH 373

Wondi Geremew, Stockton University School of General Studieswill speak on:"NEW DCA BASED ALGORITHMS FOR SOLVING BILEVEL HIERARCHICAL CLUSTERING PROBLEMS"Abstract.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 Tranwill 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 dimensions.

Friday, October 21 & 28, 2:00--3:00 PM in NH 373

Jim Rullawill speak on:"Kinetic Energy as Potential"Notes for this talk are here.Abstract.Prerequisite: Enough linear algebra to know that x^{T}Ax 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.

Friday, September 30 & October 7, 2:00--3:00 PM in NH 346

Mau Nam Nguyenwill speak on:"Convex Analysis and Optimization: from Convexity to Nonconvexity"Abstract.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