Past Seminars

Clemson Analysis Seminar

Fridays, 3:30--4:30pm

Martin M-102

Fall 2017

September 8 Walton Green (Clemson University)

September 15 Jeong-rock Yoon (Clemson University)

September 22 Oleg Yordanov (Clemson University & Bulgarian Academy of Sciences)

September 26 (Tuesday) James Melbourne (University of Minnesota)

October 6

October 13

October 20

October 27

November 3

November 10

November 17

December 1

December 8

Title & Abstract

September 26 James Melbourne

Title: A R\'enyi Entropy Trilogy

Abstract: In part of an effort to properly axiomatically characterize Shannon entropy, Alfred R\'enyi put forth a family of "information measures", parameterized $r \in [0,\infty]$. The Shannon entropy corresponded to $r=1$, and his famed entropy power inequality (EPI), fully proved by Stam some years later, could be written $N_1(X+Y) \geq N_1(X)+N_1(Y)$ for independent random variables $X,Y$. This provides an archetype for exploring further convolution inequalities under other Renyi entropy parameters. In particular, when $r=0$ one can interpret the Brunn-Minkowski inequality of Convex Geometry as a Renyi EPI of a nearly identical form, while setting r=\infty allows one to cleanly formulate some new projection inequalities important in Random Matrix Theory. We will properly define the terminology and notation used above in order to discuss this background and motivation before describing some recent progress in the understanding of $r=\infty$ case. Time permitting some general superadditivity properties.will be explained as well.

September 22 Oleg Yordanov

Title: Approximate, Saturated and Blurred Scaling of Random Fields: Applications

Abstract: Scaling (homogeneous, power-law) functions are empirically identified in a variety of natural phenomena and structures. An important class of irregular structures and processes, modeled as random fields, exhibit scaling of their second order, two-point correlation functions. Among these, referred to also as random fractals, are the morphology of rough surfaces, the fully developed turbulence, star and galaxy clusters, and many others. In all these cases, the scaling is accounted for by using power-law functions, which are singular and have limited range of validity. In this talk, I present examples of random fields whose correlation functions are defined over the entire real line and are analytic: yet they exhibit scaling properties albeit not exact. The fields are constructed over a finite band of wavenumbers/frequencies in the Fourier space. The scaling arises as an asymptotic behavior and therefore is only approximate. I also present applications of the above fields and discuss certain technical subtleties involved in these applications.

September 15 Jong-rock Yoon

Title: Various models of viscoelasticity including fractional derivative model

September 8 Walton Green

Title: Brownian Rotation of Magnetized Particles in Magnetic Particle Imaging

Abstract: Magnetic Particle Imaging (MPI) is a medical imaging technique which is implemented by measuring the voltage emitted by magnetized particles in a domain. I will derive the current model (equilibrium model) and propose a more sophisticated one (relaxation model) and compare the two in both simulation and reconstruction.

Spring 2017

March 3 Pei Pei (Otterbein University)

April 24 (Monday) Christian Wolf (City College of New York)

May 5 Jim Brannan (Clemson University)

Title & Abstract

May 5 Jim Brannan

Title: Unstable Behavior in Capitalistic Economic Systems

Abstract: We present a simple extension of the well-known Solow model of economic growth that describes the statistical behavior of business cycle fluctuations observed in the U.S. Gross Domestic Product (GDP). The extension shows how the network structure of economic supply chains induces a positive feedback loop in macrodynamics that causes the GDP to be very sensitive to small random disturbances in consumption and investment. The model suggests that recessions are inevitable in any well developed capitalistic economic system and that increasing globalization of the world economy will result in economic expansions and contractions that are less likely to be locally confined and more likely to affect economies on a global scale.

April 24 Christian Wolf

Title: On computability of rotation sets and their entropy

Abstract: Given a continuous dynamical system f:X→ X on a compact metric space X and a m-dimensional continuous potential

Φ = (φ_1, ..., φ_m):X→ R, the generalized rotation set Rot(Φ) is defined as the set of all μ-integrals of Φ, where μ runs

over all invariant probability measures. Analogously to the classical entropy of f, one can associate to each w ∈ Rot(Φ)

the localized entropy H(w) at w. In this talk, we consider the question concerning the computability of rotation sets and

localized entropy. We present positive results for shift maps and interior points of the rotation set. We also show that the

situation is more complicated when dealing with points at the boundary of the rotation set.

This is a joint work with Michael Burr and Martin Schmoll.

March 3 Pei Pei

Title: Well-posedness and stability of two nonlinear damped systems

Abstract: This talk contains theoretical analysis for two nonlinear damped systems: nonlinear Mindlin plate systems and wave equations of p-Laplacian type.

The first part studies well- posedness and long-time behavior of (Reissner) Mindlin-Timoshenko plate equations. The main feature of the considered model is the interplay between nonlinear viscous damping and nonlinear source terms. The results include Hadamard local well-posedness, global existence, blow-up in finite time, as well as estimates of the uniform energy decay rates.

The second part investigates a quasilinear wave equation of p-Laplacian type with Kelvin-Voigt damping and nonlinear source term. The source feedback is allowed to have a supercritical exponent, in the sense that the associated Nemytskii operator is not locally Lipschitz. We prove the existence of local weak solutions, which can be extended globally provided the damping term dominates the source in an appropriate sense. Moreover, a blow-up result is proved for solutions with negative initial total energy.

Fall 2016

Oct.28 Todd Wittman (The Citadel) (Joint Analysis/Computation Seminar)

Nov.1 (Tuesday) Chris Johnson (Wake Forest University)

Nov.11

Nov.18

Title & Abstract

November 1 Chris Johnson

Title: From billiards in the T-fractal to non-singular transformations

Abstract: The motion of an ideal point-mass in a self-similar polygonal region in the plane, called the T-fractal, naturally leads to the study of an infinite interval exchange. The self-similarity of this infinite interval exchange yields a finite affine interval exchange. Though such maps do not preserve Lebesgue measure, they are non-singular (i.e., preserve null sets) and many ergodic theoretic questions can still be asked about these maps.

In this talk I will describe a family of maps generalizing the affine interval exchange associated with the T-fractal, mention some simple ergodic properties of these maps, and describe a construction which associates a measure-preserving transformation on a space of infinite measure to each such map.

October 28 Todd Wittman

Title: Wavelet-based Sharpening of Remote Sensing Images

Abstract: Image processing is an interdisciplinary field that draws on various branches of mathematics including optimization, differential equations, and numerical analysis. I will discuss a mathematical approach to enhancing satellite imagery based on the calculus of variations. Satellite spectral images give more information about the objects in the scene, but this comes at the cost of reduced spatial resolution. To address this issue, we can fuse the spectral image with a high-resolution panchromatic image. This process is called pan-sharpening. Traditional pan-sharpening methods work well for low-dimensional multispectral datasets (4-6 bands), but do not extend to high-dimensional hyperspectral datasets (100-200 bands). We present a variational method that incorporates wavelets and Total Variation to sharpen hyperspectral images. Time permitting, we will discuss applications to density estimation. This is a joint work with Michael Moeller, Andrea Bertozzi, and Martin Burger.

Spring 2016

Feb.22 (Monday) Stefan Siegmund (Dresden University of Technology)

Mar.4

Mar.11

Mar.18 Spring Break

Mar.25 Loc Nguyen (UNC Charlotte)

Apr.1

Apr.7 (Thursday) Adrian Tudorascu (West Virginia University) (Joint Analysis/Computation Seminar)

Apr.15

Apr.22

Title & Abstract

April 7 Adrian Tudorascu

Title: Lagrangian solutions for the Semi-Geostrophic Shallow Water system in physical space

Abstract: SGSW is a third level specialization of Navier-Stokes (via Boussinesq, then Semi-Geostrophic), and it accurately describes large-scale, rotation-dominated atmospheric flow under the extra-assumption that the horizontal velocity of the fluid is independent of the vertical coordinate. The Cullen-Purser stability condition establishes a connection between SGSW and Optimal Transport by imposing semi-convexity on the pressure; this has led to results of existence of solutions in dual space (i.e., where the problem is transformed under a non-smooth change of variables). In this talk I will present recent results on existence and weak stability of solutions in physical space (i.e., in the original variables) for general initial data, the very first of their kind. This is based on joint work with M. Feldman (UW-Madison).

March 25 Loc Nguyen

Title: Invisibility cloak in finite frequency regime

Abstract: In the talk, I will describe a cloaking device, making an object invisible, in the finite frequency regime. The main part of the device is made of negative index materials (NIMs). NIMs are artificial and characterized by their negative electromagnetic parameters (permittivity and permeability). There are two main difficulties in the study of this problem. Due to the presence of NIMs, the governing equation has sign-changing coefficients and, hence, its ellipticity is lost. Secondly, the localized resonance near the inter-surface of NIMs and the object, which we want to hide, might occur. These difficulties can be handled by the reflection technique, a new three-sphere inequality and the technique of removing singularity. Our new three-sphere inequality is inspired from the unique continuation principle for elliptic equations. Thanks to it, the size of the cloaked region is independent on the frequency. This is a joint work with Hoai-Minh Nguyen.

February 22 Stefan Sigmund

Title: Dynamical Systems on Graphs

Abstract: Motivated by neural networks and the evolution of cooperation in biology, we study dynamical systems on arbitrary networks modeled by directed graphs. We present a new method to find phase-locked solutions which might conceptually contribute to understand how our brain uses functional connectivity to learn categories.

Fall 2015

Oct.1 Stefan Mueller (Georgia Southern University)

Oct.2 Amy Novick-Cohen (Technion IIT, Israel)

Oct.9

Oct.16

Oct.23

Oct.30 Irina Holmes (Georgia Tech)

Nov.6

Nov.13 Junshan Lin (Auburn University)

Nov.19 Kevin McGoff (UNC Charlotte)

Title & Abstract

November 19 Kevin McGoff

Title: Optimal Tracking for Dynamical Systems

Abstract: In this talk, I will discuss preliminary results from ongoing work with Andrew Nobel. We begin with the following tracking problem for dynamical systems. Fix two systems, S mapping X to itself and T mapping Y to itself. Which segment of trajectory from the (X,S) system best ``tracks” an observed segment of trajectory from the (Y,T) system? To make this question precise, we introduce cost functions for pairs of trajectories from the two systems. Then we analyze the asymptotic behavior of the associated optimization problems as the length of the observed trajectory tends to infinity. After presenting some basic results, I will discuss implications of this work for statistical inference, as well as connections to ergodic optimization.

November 13 Junshan Lin

Title: A Rigorous Mathematical Theory for Electromagnetic Field Enhancement in Metallic Nanogaps

Abstract: Subwavelength apertures and gaps on surfaces of noble metals (e.g., gold or silver) induce strong electric field and extraordinary optical transmission. This remarkable phenomenon can lead to novel applications in biological and chemical sensing, spectroscopy, and THz semiconductor devices. In this talk, I will present a quantitative analysis for the field enhancement when an electromagnetic wave passes through tiny metallic gaps. Based upon a rigorous study of the perfect electrical conductor model, we show that enormous electric field enhancement occurs inside the nanogap when there is extreme scale difference between the wavelength of radiation and the thickness of metal films. The analysis also leads to efficient asymptotic numerical method for calculating the wave field in the nanostructure. The ongoing work along this research direction will also be highlighted.

October 30 Irina Holmes

Title: Commutators in the two-weight setting

Abstract: In a foundational paper, Coifman, Rochberg and Weiss relate the norm of the commutator [b, T], where T is a Calderon-Zygmund operator, with the BMO norm of b. In this talk we explore a recent weighted version of this result. Specifically, we study the case when the commutator acts between two different weighted L^p spaces. A first result in this direction was first given by Bloom in 1985, for the Hilbert transform. We discuss an extension of Bloom's result to all Calderon-Zygmund operators, using dyadic methods.

October 1 Stefan Mueller

Title: C^0-symplectic topology and topological Hamiltonian dynamics

Abstract: C^0-symplectic topology is a fairly recently established subfield of symplectic topology that encompasses the study of C^0-phenomena and of C^0-analogs of smooth objects in symplectic topology. In particular, topological Hamiltonian dynamics can best be described as a continuous version of classical (i.e. smooth) Hamiltonian dynamics. A similar story can (and very well may) be told for contact topology/dynamics.

I will first introduce symplectic topology from its origins in classical mechanics. At least this part of the talk should be accessible to an audience that is not familiar with the basic notions of symplectic geometry, and may in fact serve as a motivation for the theory. We will naturally encounter some of the questions that motivate the development of C^0-symplectic topology, and then discuss a number of applications (to classical symplectic topology, topological dynamics, hydrodynamics, and Riemannian geometry).

October 2 Amy Novick-Cohen

Title: Coupling surface diffusion with mean curvature motion

Abstract: Mean curvature motion as well as surface diffusion both constitute geometric interfacial motions which have received considerable attention. In many applications, however, a more complex situation occurs in which surfaces evolving by these two different motions are dynamically coupled. In my lecture, we outline some of these applications, and survey a variety of relevant analytic results and conjectures.

Spring 2015

Feb.20 Brett Wick (Georgia Institute of Technology)

Feb.26 Ben Duncan (North Dakota State University) (Joint with ADM seminar)

Mar.6

Mar.13

Mar.20 Spring Break

Mar.27

Apr.3 Rachel Grotheer (Clemson University)

Apr.10

Apr.17 Robert Rahm (Georgia Institute of Technology)

Apr.24

Title & Abstract

April 17 Robert Rahm

Title: Entropy Bump Conditions for Fractional Maximal and Fractional Integral Operators

Abstract: We investigate weighted inequalities for fractional maximal and fractional integral operators. We approach the problem from a "dyadic" point of view and dyadic and sparse operators play a central role. Using the innovative framework of "entropy bounds", introduced by Treil-Volberg, and the techniques developed by Lacey-Spencer, we are able to deduce the weighted inequalities. In particular, we give conditions on two weights to ensure that the fractional maximal operator or the fractional integral operator is bounded between the weighted Lebesgue spaces. This is joint work with Scott Spencer.

April 3 Rachel Grotheer

Title: Introduction to the Reduced Basis Method and its Application to Hyperspectral Diffuse Optical Tomography

Abstract: In this talk, we develop a reduced basis method approach to solve the forward problem in hyperspectral diffuse optical tomography (hyDOT), a relatively new medical imaging modality. Our work is motivated by the computationally expensive image reconstruction problem in hyDOT which requires solving the forward problem hundreds, if not thousands, of times. We show initial results on a simple model of hyDOT and demonstrate the benefits of and difficulties in applying this method to hyDOT.

February 26 Ben Duncan

Title: Operator Algebras from Commutative Algebra

Abstract: The study of algebras of operator on Hilbert spaces can be fairly technical and intuition can be lacking. As a result, one way that operator algebraists have used to develop examples and motivate theorems is to consider operator algebras associated to familiar objects (groups, directed graphs, dynamical system, etc.) In this talk, after a very brief overview of operator algebras, I will discuss how one assigns an operator algebra to a module over a commutative ring. I will then discuss some questions and ongoing work concerning the algebras so constructed.

February 20 Brett Wick

Title: Linear Bound for Matrix A_2 Weights

Abstract: In this talk we will discuss recent results related to the norm of Calderon-Zygmund operators acting on vector-valued spaces normed by a matrix A_2 weight.

Fall 2014

Sept.19 Organizational Meeting

Sept.26

Oct.3

Oct.10

Oct.17

Oct.24 Thilo Strauss (Clemson University)

Oct31.

Nov.7

Nov.13 Maciej Wojtkowski (University of Warmia i Mazury)

Nov.21

Dec.5 Yang Yang (Purdue University)

Title & Abstract

October 24 Thilo Strauss

Title: Statistical Inversion in Electrical Impedance Tomography.

Abstract: Electrical impedance tomography (EIT) is a well known technique to estimate the resistivity distribution γ of a body Ω with unknown electromagnetic properties. EIT is a severely ill-posed inverse problem. In this presentation, we formulate the EIT problem in the Bayesian framework using mixed total variation (TV) and non-convex lp regularization prior. We use the Markov Chain Monte Carlo (MCMC) method to solve the ill-posed inverse problem and present simulations to estimate the distribution for each pixel for the image reconstruction of the resistivity.

November 13 Maciej Wojtkowski

Title: Bi-partitions of the 2-d torus, 1-dimensional tilings, hyperbolic automorphisms and their Markov partitions

Abstract: Bi-partitions are partitions of the 2-dim torus by two parallelograms. They give rise to 2-periodic tilings of the plane, and further to 1-dim tilings which have a host of well known combinatorial properties, e.g. these are Sturmian sequences. When a bi-partition is a Markov partition for a hyperbolic toral automorphism (= Berg partition), the tilings are substitution tilings. Substitutions preserving Sturmian sequences have the remarkable ``3-palindrome property''. The number of different substitutions was determined by Seebold '98, and the number of nonequivalent Berg partitions by Siemaszko and Wojtkowski '11. The two formulas coincide. Using tilings we explain the formula by the 3 palindrome property. The coincidence then shows that every combinatorial substitution preserving a Sturmian sequence is realized geometrically as a Berg partition.

December 5 Yang Yang

Title: Quantitative photo-acoustic tomography with partial data

Abstract: Hybrid imaging methods are a class of newly developed medical imaging methods. In a hybrid imaging method one attempts to combine the high resolution of one type of wave with the high contrast of another type of wave through a physical coupling. One typical example is the photo-acoustic tomography (PAT), where electromagnetic waves are coupled with acoustic waves. Mathematically implementing the quantitative PAT amounts to the inverse problem of recovering the diffusion and absorption coefficients from internal measurements. In this talk we will introduce the mechanism of PAT, and show that such recovery can be uniquely determined from the knowledge of well-chosen partial boundary conditions. This is joint work with Dr. Jie Chen.

Spring 2014

Jan.24 Organizational Meeting

Jan.31 Taufiquar Khan (Clemson University)

Feb.7 Job Candidate Interview

Feb.13 Cancelled due to Snow

Feb.21 Job Candidate Interview

Feb.27 (Thursday) Julia Barnes (Western Carolina University)

Mar.7 SEAM 2014

Mar.14

Mar.21 Spring Break

Mar.28

Apr.4

Apr.11

Apr.18

Apr.23 Fioralba Cakoni (University of Delaware) (Department Colloquium)

Title & Abstract

February 27 Julia Barnes

Title: Connections between bounded Julia sets and the level curves generated from related functions.

Abstract: Julia sets for the family of complex functions z^2 + c have been well studied for years, and the images of these Julia sets are commonplace. In this talk, we consider viewing this family of functions from a different perspective, beginning with the question of what the graphs of these functions and their iterates look like. Since the graph of any function from the complex plane to the complex plane is four dimensional, we analyze the level curves of the real and imaginary parts of the iterates of given functions and find some interesting connections between these images and the Julia sets of the corresponding functions. We will end with some known extensions into rational maps.

April 23 Fioralba Cakoni

Title: A Qualitative Approach to Inverse Scattering for Inhomogeneous Media: The Transmission Eigenvalue Problem

Abstract: Since the introduction of the linear sampling method in 1996 followed by the factorization method in 1998 and later the proof of the existence and monotonicity properties of real transmission eigenvalues in 2010, qualitative methods have become a popular method for solving inverse scattering problems. Interest in this area has exploded and the vast amount of literature currently available is an indication of the myriad directions that this research has taken. In this talk we consider the inverse scattering problem for an inhomogeneous (possibly anisotropic) media and show how to obtain information about the support as well as the physical properties of the media based on an investigation of the corresponding far field operator. At the foundation of this investigation is the so-called the transmission eigenvalue problem, which is a non-linear and non-selfadjoint eigenvalue problem. In particular, we will discuss the relevance and state of the art of the transmission eigenvalue problem and present what type of information transmission eigenvalues provide about the inhomogeneity.

Fall 2013

Sept.19 Organizational Meeting

Sept.27 Shitao Liu (Clemson University)

Oct.3 Sung Ha Kang (Georgia Institute of Technology)

Oct.11 No seminar scheduled

Oct.18 Karl Petersen (UNC Chapel Hill) (Department Colloquium)

Oct.25 Suzanne Lenhart (University of Tennessee) (Department Colloquium)

Nov.1 Hongyu Liu (UNC Charlotte)

Nov.8 Nándor Simányi (University of Alabama) (Department Colloquium)

Nov.15 Jeong-Rock Yoon (Clemson University)

Nov.22 No seminar scheduled

Dec.5 Martin Schmoll (Clemson University) (4:30-5:30pm at M203)

Title & Abstract

September 27 Shitao Liu

Title: Determining a magnetic Schrödinger operator from partial data in an infinite slab

Abstract: In this talk we study an inverse boundary value problem with partial data in an infinite slab in R^n, n>2, for the magnetic Schrödinger operator with bounded magnetic potential and electric potential. We show that the magnetic field and the electric potential can be uniquely determined, when the Dirichlet and Neumann data are given on either different boundary hyperplanes or on the same boundary hyperplanes of the slab.

October 3 Sung Ha Kang

Title: Mathematical approaches to Image deblurring and some aspects of Segmentation

Abstract: This talk is an introduction to mathematical approaches to image processing. Starting from Total Variation minimizing denoising, a few variational/PDE-based models in image restoration such as deblurring, inpainting and colorization will be presented. Each of the models will be validated with computational results to show the effect. Image segmentation separates the image into different regions to simplify the image and identify the objects easily. Mumford-Shah and Chan-Vese model are one of the most well-known variational models in the field. Starting from these models, this talk will include unsupervised model and its analytical properties of the minimizer.

October 18 Karl Petersen

Title: Concepts of entropy and complexity in dynamical systems

Abstract: There are many ways to compare dynamical systems with respect to the complexity of their behavior. We will look at a few of the standard ones and how they apply to some examples and then describe a new one that more closely takes into account internal influences within a system.

October 25 Suzanne Lenhart

Title: Using optimal control of PDEs to investigate population questions

Abstract: We use optimal control of partial differential equations to investigate conservation questions in population models. One example will address a question about resource allocation to increase population abundance with limited resources; the control represents the availability of resources. A second example is motivated by the question: Does movement toward a better resource environment benefit a population? The control is the advective coefficient in a parabolic PDE with nonlinear growth.

November 1 Hongyu Liu

Title: Approximate Cloaking of Acoustic and Electromagnetic Waves

Abstract: In this talk, I will describe the recent theoretical and computational progress on our work on regularized transformation-optics cloaking. Ideal cloak makes use of singular metamaterials, posing much challenge for practical realization. Regularization is incorporated into the construction in order to avoid the singular structures.

November 8 Nándor Simányi

Title: Brief History of the Boltzmann-Sinai Hypothesis

Abstract: The Boltzmann-Sinai Hypothesis dates back to 1963 as Sinai's modern formulation of Ludwig Boltzmann's statistical hypothesis in physics, actually as a conjecture: Every hard ball system on a flat torus is (completely hyperbolic and) ergodic (i. e. "chaotic", by using a nowadays fashionable, but a bit profane language) after fixing the values of the obviously invariant kinetic quantities. In the half century since its inception quite a few people have worked on this conjecture, made substantial steps in the proof, created useful concepts and technical tools, or proved the conjecture in some special cases, sometimes under natural assumptions. Quite recently I was able to complete this project by putting the last, missing piece of the puzzle to its place, getting the result in full generality. In the talk I plan to present the brief history of the proof by sketching the most important concepts and technical tools that the proof required.

November 15 Jeong-Rock Yoon

Title: Uniqueness in viscoelasticity reconstruction in Elastography

Abstract: We aim to prove the uniqueness of identifying viscoelasticity in a transient elasticity experiment, which enables us to acquire interior displacement data in the form of a movie. Viscoelasticity may be used to characterize pathological tissue. The main technique is to incorporate the unique continuation principle (as a solution of elliptic equation) and the finite propagation speed (as a solution of hyperbolic equation), which is called a shrink-and-spread argument. The finite propagation speed part is complete and the unique continuation part is being investigated as a joint work with Shitao Liu.

December 5 Martin Schmoll

Title: A short introduction into optimal transportation

Abstract: Optimal Transportation is a topic which is highly popular because of its many applications to mathematics and physics. The question asked is simple: How can we allocate a pile of dirt from A to B most efficiently? A straightforward conversion of this problem into mathematics leads to Monge's formulation of optimal transportation. Unfortunately Monge's problem is highly nonlinear and even the existence of solutions is unclear. Kantorovich proposed a measure based approach in the 1940's. Roughly speaking we are now allowed to spread out a piece of mass picked up in the source in the target, i.e. points can be mapped to larger sets. This turns out to be a linear problem. Moreover there is a duality principle, now called Kantorovich duality, which identifies the minimum we want to find as a maximum of a different problem. This formulation opens the door to apply analytic techniques and solve the problem in many interesting cases. Besides relating the optimal transportation problem a la Kantorovich to linear programming, we try to include a few consequences, like Brenier's polar factorization theorem and eventually some of the geometric ideas of Lott and Villani used in a study of concepts introduced by Perelman to prove the Poincar\'e conjecture.