Pasquale Porcelli Lecture Series
Special Lecture Series

Posted March 30, 2007

Last modified April 7, 2007

Richard Kadison, University of Pennsylvania
Member of the National Academy of Sciences

The Pythagorean Theorem: A Closer Look

Extensions and variants of the Pythagorean theorem are presented, first from the point of view of finite-dimensional, linear algebra and, later, in the framework of infinite-dimensional Hilbert space. The results discussed make contact with the work of Kostant, Atiyah, and Guillemin-Sternberg in the convex geometry of symmetric spaces, the work of Horn and Schur on spectral theory, matrix inequalities, majorization, and convex polytopes, and semi-commutative, metric geometry from the point of view of conditional expectations. The first of the two Pythagoras lectures will be relatively elementary, the second will be slightly more advanced, relying somewhat on the operator-algebra, survey lecture that follows the first lecture. There will be coffee and cookies in the Atrium, Howe-Russell E, at 3:00.

Pasquale Porcelli Lecture Series
Special Lecture Series

Posted March 30, 2007

Last modified April 7, 2007

Richard Kadison, University of Pennsylvania
Member of the National Academy of Sciences

Operator Algebras: A Sampler

There will be coffee and cookies in the Atrium, Howe-Russell E, at 3:00.

Pasquale Porcelli Lecture Series
Special Lecture Series

Posted March 30, 2007

Last modified April 7, 2007

Richard Kadison, University of Pennsylvania
Member of the National Academy of Sciences

The Pythagorean Theorem: An Advanced View

There will be coffee and cookies in the Atrium, Howe-Russell E, at 3:00.

Pasquale Porcelli Lecture Series
Special Lecture Series

Posted January 25, 2008

Last modified February 21, 2008

Don Zagier, Max Planck Institut, Bonn and College de France
Recipient of the Frank Nelson Cole Prize in Number theory, the Prix Elie Cartan of the Académie des Sciences and the Chauvenet Prize of the Mathematical Association of America

The "q" in "q-series"

There will be refreshments before the lecture at 3pm in the Howe-Russell Atrium.

Pasquale Porcelli Lecture Series
Special Lecture Series

Posted January 25, 2008

Last modified February 21, 2008

Don Zagier, Max Planck Institut, Bonn and College de France
Recipient of the Frank Nelson Cole Prize in Number theory, the Prix Elie Cartan of the Académie des Sciences and the Chauvenet Prize of the Mathematical Association of America

The "q" in "quadratic"

There will be refreshments before the lecture at 3pm in the Howe-Russell Atrium.

Pasquale Porcelli Lecture Series
Special Lecture Series

Posted January 25, 2008

Last modified February 21, 2008

Don Zagier, Max Planck Institut, Bonn and College de France
Recipient of the Frank Nelson Cole Prize in Number theory, the Prix Elie Cartan of the Académie des Sciences and the Chauvenet Prize of the Mathematical Association of America

The "q" in "quantum"

There will be refreshments before the lecture at 3pm in the Howe-Russell Atrium.

Pasquale Porcelli Lecture Series
Special Lecture Series

Posted October 14, 2010

Last modified February 17, 2011

Craig Evans, University of California, Berkeley

Linearity and linearization

Abstract: In this expository lecture aimed at a general audience, I will first discuss the profound advantages of linear structure in mathematical problems and then survey several interesting ways to "linearize" nonlinear problems, primarily differential equations. Examples and applications will include perturbation and implicit function procedures, blow-up techniques, kinetic formulations, and adjoint methods based upon formal linearization.

Pasquale Porcelli Lecture Series
Special Lecture Series

Posted February 17, 2011

3:40 pm - 4:30 pm 103 Design Auditorium
Craig Evans, University of California, Berkeley

Convexity as one-sided linearity

Abstract: I will continue the themes of the previous talk, surveying for differential equations various convexity methods that can be interpreted as "one-sided linearity" tricks. These are especially useful since, as I will show, several important and highly nonlinear problems possess "hidden" convex structures of various sorts.

Pasquale Porcelli Lecture Series
Special Lecture Series

Posted February 17, 2011

3:40 pm - 4:30 pm 103 Design Auditorium
Craig Evans, University of California, Berkeley

Linear adjoint methods for sup-norm variational problems

Abstract: This final lecture will present some technical details about a recent application of linearization and adjoint methods for proving differentiability for weak solutions of the so-called "infinity Laplacian" PDE. This highly degenerate and nonlinear equation is fundamental in the emerging field of sup-norm variational problems and their applications.

Pasquale Porcelli Lecture Series
Special Lecture Series

Posted October 2, 2012

Last modified April 3, 2013

S. R. S. Varadhan, Courant Institute
National Medal of Science (2010), Abel Prize (2007)

What is large deviations?

Pasquale Porcelli Lecture Series
Special Lecture Series

Posted October 2, 2012

Last modified April 3, 2013

S. R. S. Varadhan, Courant Institute
National Medal of Science (2010), Abel Prize (2007)

Scaling limits of large systems

Pasquale Porcelli Lecture Series
Special Lecture Series

Posted October 2, 2012

Last modified April 3, 2013

S. R. S. Varadhan, Courant Institute
National Medal of Science (2010), Abel Prize (2007)

Counting Graphs

Pasquale Porcelli Lecture Series
Special Lecture Series

Posted March 12, 2014

Last modified April 25, 2014

Susan Murphy, University of Michigan H.E. Robbins Professor of Statistics & Professor of Psychiatry, Research Professor, Institute for Social Research
2013 MacArthur Fellow

L 1: Getting SMART about Adapting Interventions L2: Adaptive Confidence Intervals for Non-smooth Parameters

Refreshments at 3pm in foyer by Digital Media Center Lecture Hall

Lecture 1 for General Audience (3:30-4:20)

Getting SMART about Adapting Interventions

Abstract: Imagine you are a child with ADHD. Wouldn't you like your

doctors to periodically adapt your treatment to your unique--and

ever-changing--condition? And wouldn't you be excited to learn that an

algorithm used to analyze your medical data was originally developed for

applications in robotics and artificial intelligence? This lecture will

explain how a randomized clinical trial design (Sequential Multiple

Assignment Randomized Trial or SMART) is being used to develop adaptive

interventions--protocols that systematize sequential decision-making that

is key to effective treatment of health problems. Examples include a

study of children with ADHD and an ongoing study to improve care at

mental health clinics.

Lecture 2 for more specialized audience (4:30-5:20)

Adaptive Confidence Intervals for Non-smooth Parameters

Abstract: Non-regular, aka "non-smooth" parameters are of scientific

interest occur frequently in modern day inference. In particular when

scientific

interest centers on a non-smooth function of regular parameters such as in

the assessment of a machine learning classifier's performance, in the

estimation of multistage decision making policies and in the use of

methods that use assumptions of sparsity to threshold estimators. If

confidence intervals are

considered at all, most research assumes potentially implausible

"margin-like" conditions in order to justify the proposed confidence

interval method. We describe a different approach based on

constructing smooth upper and lower bounds on the parameter and then

basing the confidence interval on the smooth upper and lower bounds.

In particular two settings will be discussed and contrasted, that of a

confidence interval for the mis-classification rate and a confidence

interval for a parameter in multistage decision making policies.

Pasquale Porcelli Lecture Series
Special Lecture Series

Posted June 26, 2015

Last modified March 17, 2016

Maria Chudnovsky, Princeton University
MacArthur Foundation Fellowship recipient 2012.

Perfection and Beyond

About 10 years ago one of the central open problems in graph theory at the time, the Strong Perfect Graph Conjecture, was solved. The proof used structural graph theory methods, and spanned 155 journal pages. The speaker was part of the team of authors of this mathematical beast. In this talk we will explain the problem, describe some of the ideas of the proof (that has since been shortened somewhat), and discuss related problems that have been a subject of more recent research.

Pasquale Porcelli Lecture Series
Special Lecture Series

Posted January 22, 2016

Last modified March 17, 2016

Maria Chudnovsky, Princeton University
MacArthur Foundation Fellowship recipient 2012.

Coloring some perfect graphs

Perfect graphs are a class of graphs that behave particularly

well with respect to coloring. In the 1960's Claude Berge made two

conjectures about this class of graphs, that motivated a great deal of

research, and by now they have both been solved.

The following remained open however: design a combinatorial algorithm that

produces an optimal coloring of a perfect graph. Recently, we were able to

make progress on this question, and we will discuss it in this talk. Last

year, in joint work with Lo, Maffray, Trotignon and Vuskovic we were able

to construct such an algorithm under the additional assumption that the

input graph is square-free (contains no induced four-cycle). More

recently, together with Lagoutte, Seymour and Spirkl, we solved another

case of the problem, when the clique number of the input graph is fixed

(and not part of the input).

Pasquale Porcelli Lecture Series
Special Lecture Series

Posted January 22, 2016

Last modified March 17, 2016

Maria Chudnovsky, Princeton University
MacArthur Foundation Fellowship recipient 2012.

Induced cycles and coloring

The Strong Perfect Graph Theorem states that graphs with no no induced odd cycle of length at least five, and no complements of one behave very well with respect to coloring. But what happens if only some induced cycles (and no complements) are excluded? Gyarfas made a number of conjectures on this topic, asserting that in many cases the chromatic number is bounded by a function of the clique number. In this talk we discuss recent progress on some of these conjectures. This is joint work with Alex Scott and Paul Seymour.

Pasquale Porcelli Lecture Series
Special Lecture Series

Posted December 12, 2016

Last modified February 3, 2017

Ken Ono, Emory University

Gems of Ramanujan and their Lasting Impact on Mathematics

Abstract: Ramanujan's work has has a truly transformative effect on modern mathematics, and continues to do so as we understand further lines from his letters and notebooks. In this lecture, some of the studies of Ramanujan that are most accessible to the general public will be presented and how Ramanujan's findings fundamentally changed modern mathematics, and also influenced the lecturer's work, will be discussed. The speaker is an Associate Producer of the film *The Man Who Knew Infinity* (starring Dev Patel and Jeremy Irons) about Ramanujan. He will share several clips from the film in the lecture.

Pasquale Porcelli Lecture Series
Special Lecture Series

Posted December 12, 2016

Last modified February 3, 2017

Ken Ono, Emory University

Cool Theorems Proved by Undergraduates

Abstract. The speaker has been organizing summer research programs for undergraduate students for many years. This lecture will give a sample of their accomplishments. The speaker will talk about partitioning integers, prime numbers, number fields, and generalizations of classical theorems of Euler, Gauss, and Jacobi.

Pasquale Porcelli Lecture Series
Special Lecture Series

Posted December 12, 2016

Last modified February 4, 2017

Ken Ono, Emory University

Can't you just feel the Moonshine?

Borcherds won the Fields medal in 1998 for his proof of the Monstrous Moonshine Conjecture. Loosely speaking, the conjecture asserts that the representation theory of the Monster, the largest sporadic finite simple group, is dictated by the Fourier expansions of a distinguished set of modular functions. This conjecture arose from astonishing coincidences noticed by finite group theorists and arithmetic geometers in the 1970s. Recently, mathematical physicists have revisited moonshine, and they discovered evidence of undiscovered moonshine which some believe will have applications to string theory and 3d quantum gravity. The speaker and his collaborators have been developing the mathematical facets of this theory, and have proved the conjectures which have been formulated. These results include a proof of the Umbral Moonshine Conjecture, and Moonshine for the first sporadic finite simple group which does not occur as a subgroup or subquotient of the Monster. The most recent Moonshine (announced here) yields unexpected applications to the arithmetic elliptic curves thanks to theorems related to the Birch and Swinnerton-Dyer Conjecture and the Main Conjectures of Iwasawa theory for modular forms. This is joint work with John Duncan, Michael Griffin and Michael Mertens.

Pasquale Porcelli Lecture Series
Special Lecture Series

Posted November 7, 2017

Last modified March 13, 2018

Irene Fonseca, Carnegie Mellon University

Porcelli Lecture 1: Mathematics and Imaging Science

(The talk is intended to be accessible to High School Students.) In this talk, we will address the mathematical treatment of image processing, including inpainting, recolorization, denoising, and machine learning schemes.

Pasquale Porcelli Lecture Series
Special Lecture Series

Posted November 7, 2017

Last modified March 13, 2018

Irene Fonseca, Carnegie Mellon University

Porcelli Lecture 2: Mathematics and Materials Science

(Intended to be accessible to Undergraduate Students.) Abstract: Quantum dots are man-made nanocrystals of semiconducting materials. Their formation and assembly patterns play a central role in nanotechnology, and in particular in the optoelectronic properties of semiconductors. Changing the dots' size and shape gives rise to many applications that permeate our daily lives, such as the new Samsung QLED TV monitor that uses quantum dots to turn "light into perfect color"! Quantum dots are obtained via the deposition of a crystalline overlayer (epitaxial film) on a crystalline substrate. When the thickness of the film reaches a critical value, the profile of the film becomes corrugated and islands (quantum dots) form. As the creation of quantum dots evolves with time, materials defects appear. Their modeling is of great interest in materials science since material properties, including rigidity and conductivity, can be strongly influenced by the presence of defects such as dislocations. In this talk, we will use methods from the calculus of variations and partial differential equations to model and mathematically analyze the onset of quantum dots, the regularity and evolution of their shapes, and the nucleation and motion of dislocations.

Pasquale Porcelli Lecture Series
Special Lecture Series

Posted November 7, 2017

Last modified March 13, 2018

Irene Fonseca, Carnegie Mellon University

Porcelli Lecture 3: Homogenization of Integral Energies Under Periodically Oscillating Differential Constraints

(The talk is intended to be accessible to Graduate Students.) Abstract: A homogenization result for a family of integral energies is presented, where the fields are subjected to periodic first order oscillating differential constraints in divergence form. We will give an example that illustrates that, in general, when the operators differential operators have non constant coefficients then the homogenized functional maybe be nonlocal, even when the energy density is convex. This work is based on the theory of A-quasiconvexity with variable coefficients and on two-scale convergence techniques.

Pasquale Porcelli Lecture Series
Special Lecture Series

Posted September 30, 2018

Last modified February 10, 2019

Robert Bryant, Duke University
Fellow, American Academy of Arts and Sciences (2002), Member, National Academy of Sciences (2007), AMS Fellow (2013), AMS President (2015-2017)

Porcelli Lecture 1 (High-School Level): Mathematical Mysteries of the Ellipse

Abstract: After lines and circles, the simplest curves are the so-called conic sections, hyperbolas, parabolas, and ellipses. Not only are they the next simplest curves, but they have many applications in the physical world and have been studied for more than two thousand years.

However, these curves have many surprising properties that were not discovered until fairly recently.

For example, it has been known for a long time that light emitted from one focus of an ellipse collects at the other focus, and a similar property for the parabola is used in designing headlights. However, this turns out to be a special case of a much more interesting and surprising special property discovered in the 19th century and that has given rise to problems that we still don't know how to solve today.

In this talk, which will use nothing beyond high school algebra (and lots of pictures), I'll explain some of these mysteries and why we study them.

Pasquale Porcelli Lecture Series
Special Lecture Series

Posted September 30, 2018

Last modified February 10, 2019

Robert Bryant, Duke University
Fellow, American Academy of Arts and Sciences (2002), Member, National Academy of Sciences (2007), AMS Fellow (2013), AMS President (2015-2017)

Porcelli Lecture 2 (Undergraduate Level): Geometry Old and New: From Euclid to String Theory

Abstract: Classical geometry is based on notions of symmetry and congruence, and these ideas, while very old, have deeply influenced our understanding of the physical world. The idea of modeling the world through principles of least action or least energy are tied to symmetry in deep ways. In this talk, I will survey the history of how this relationship was uncovered by mathematicians such as Euler, Gauss, Lie, and Noether and is still developing in our modern understanding of the world, from Einstein's theory of relativity even to contemporary versions of string theory.

Pasquale Porcelli Lecture Series
Special Lecture Series

Posted September 30, 2018

Last modified February 10, 2019

Robert Bryant, Duke University
Fellow, American Academy of Arts and Sciences (2002), Member, National Academy of Sciences (2007), AMS Fellow (2013), AMS President (2015-2017)

Porcelli Lecture 3 (Graduate Student Level): The Best Possible Shapes of Surfaces

Abstract: Much of classical mathematics involves finding a configuration or shape that provides an optimum solution of a problem. For example, it has long been known (though a rigorous proof took quite a while to find) that the surface of least area enclosing a given volume is a round sphere. There are many other ways to measure surfaces, though, and finding 'the' surface that optimizes a given 'measurement' (subject to some given constraints) remains a challenging problem that has motivated some of the deepest recent work in the mathematics of geometric shapes.

In this talk, I will explain some of the classic ways to measure shapes of surfaces and relate this to classical problems involving surface area (soap films and bubbles) and total curvature as well to as recent progress by myself and others on these important optimization problems.