Spring 2027
Facundo Mémoli
Subtitle:
Mathematics of Data Science
Course Description:
This course will explore ideas from geometry and topology that arise naturally in data science. Topics will include multidimensional scaling, hierarchical clustering, shape comparison via the Gromov--Hausdorff distance, persistent homology of metric spaces, optimal transport through the Wasserstein distance between probability measures and the Gromov--Wasserstein distance between metric-measure spaces, and spectral methods such as those used in diffusion geometry. The final selection of topics will be partly determined by the interests of the course participants, and related research directions will be discussed.
Textbook:
References: there’s no single textbook for this course. We will follow several sources including:
A course in Metric Geometry. Burago, Burago and Ivanov. AMS.
Cox, Trevor F., and Michael AA Cox. Multidimensional scaling. CRC press, 2000.
Mémoli, Facundo. "Gromov–Wasserstein distances and the metric approach to object matching." Foundations of computational mathematics 11 (2011): 417-487.
Villani, Cédric. Topics in optimal transportation. Vol. 58. American Mathematical Soc., 2021.
Mémoli, Facundo. "Some properties of Gromov–Hausdorff distances." Discrete & Computational
Geometry 48 (2012): 416-440.
Edelsbrunner, Herbert, and John L. Harer. Computational topology: an introduction. American Mathematical Society, 2022.
Chazal, Frédéric, Vin De Silva, and Steve Oudot. "Persistence stability for geometric complexes." Geometriae Dedicata 173.1 (2014): 193-214.
Bérard, Pierre, Gérard Besson, and Sylvain Gallot. "Embedding Riemannian manifolds by their heat kernel." Geometric & Functional Analysis GAFA 4 (1994): 373-398.
Coifman, Ronald R., and Stéphane Lafon. "Diffusion maps." Applied and computational harmonic analysis 21.1 (2006): 5-30.
Kim, W. and Memoli, F. Persistence over posets. AMS Notices https://www.ams.org/notices/202308/rnoti-p1214.pdf.
Phylogenetics. Semple, Charles, and Mike Steel. Phylogenetics. Vol. 24. Oxford University Press on Demand, 2003.
Dress, A., 2012. Basic phylogenetic combinatorics. Cambridge University Press.
Villani, C., 2009. Optimal transport: old and new (Vol. 338, p. 23). Berlin: springer.
Sturm, K.T., 2023. The space of spaces: curvature bounds and gradient flows on the space of metric measure spaces (Vol. 290, No. 1443). American Mathematical Society.
Mémoli, F., Wan, Z. and Wang, Y., 2022. Persistent Laplacians: Properties, algorithms and implications. SIAM Journal on Mathematics of Data Science, 4(2), pp.858-884.
Lim, L.H., 2020. Hodge Laplacians on graphs. Siam Review, 62(3), pp.685-715.
Prerequisites:
undergraduate courses in topology, algebra and analysis.
Fioralba Cakoni
Subtitle:
Mathematical foundations and computational aspects of inverse scattering theory
Course description:
Scattering theory is concerned with the effect an inhomogeneous medium has on an incident particles or wave. Although the basic mathematical model of scattering theory (in terms of partial differential equations) is deceptively simple, scattering phenomena continues to attract, perplex and challenge mathematicians from diverse disciplines.
Inverse scattering theory is concerned with extracting geometrical and physical properties, i.e. the coefficients in the PDEs, of the scattering inhomogeneity from
the effects that is has on probing waves measured far away. In the past thirty years the field of inverse scattering theory has become a major theme of the analysis and PDEs, as well as applied mathematics since it has applications to such diverse areas as medical imaging, geophysical exploration, and nondestructive testing. The growth of this field has been characterized by the realization that the inverse scattering problem is both nonlinear and ill-posed, thus presenting particular problems in the development of efficient inversion algorithms. The investigation of uniqueness lies at the foundation of the mathematical theory of inverse problems for PDEs. It has led to the development of beautiful mathematical analysis and has provided insight into designing reconstruction methods. The need to deal with ill-posed problems, i.e. solving problems that have no solution with noisy data, has led to the regularization theory which guaranties that the reconstruction algorithms work.
This course is a self-contained discussion on the mathematical foundations and computational aspects of inverse scattering theory.
The files of the textbook will be provided. Lecture notes will also be available.
Text:
D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer, 4rth Ed. 2019 and F. Cakoni, D. Colton and H. Haddar, Inverse Scattering Theory and Transmission Eigenvalues, CBMS-NSF, SIAM Publication, 2nd Ed. 2023.
Prerequisites:
Real analysis (equivalent to 501 and 502), Introductory Partial Differential Equations
*************************************
Fioralba Cakoni
Subtitle:
Mathematical Foundation of Inverse Problems
Course description:
This course is designed as a self-contained and up-to-date discussion of the mathematical methods in inverse problems and imaging which is a growing area of applied mathematics due to many applications in science and engineering. Such problems are typically non-linear and ill-posed. Topics to be covered include regularization theory for ill-posed problems, inverse spectral problems, impedance tomography, inverse scattering and other examples of imaging.
Text:
TBA
Prerequisites:
No formal prerequisites, but the course assumes a solid foundation in real analysis ( preferably 640:501-502) and a basic knowledge of partial differential equations. Introductory functional analysis is valuable but not requires.
Schedule of Sections: