Course Descriptions

Fall 2025

Konstantin Mischaikow

Subtitle:

Computation Topology Nonlinear Dynamics

Course Description:

This course provides an introduction to a combinatorial/algebraic topological approach to understanding nonlinear dynamics. While an introductory course in algebraic topology would be ideal, it is not absolutely necessary as the basic ideas of homology will be discussed as needed.

Topics to be discussed include:

   1. Methods for the efficient computation of homology groups, induced maps on homology, persistent homology, and Conley complexes. This is essential knowledge for working with large data sets.

   2. Order theoretic representation of global dynamics and algorithms for computing these representation.

  3. Homological invariants for nonlinear dynamics

  4. Applications of these techniques to

    a) nonlinear dynamics generated by interaction of continuous maps

    b) nonlinear dynamics of ordinary differential equations

    c) move from data to the identification/characterization of dynamics with probabilistic guarantees, and

   Concrete applications of these ideas will be given using problems arising in robotic control, systems biology, and ecology.

Text:

None

Prerequisites:

Math 311, Math 350 or equivalent

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Spring 2025

Facundo Memoli

Subtitle:

Mathematics of Data Science

Course Description:

This course will explore various ideas from geometry and topology that have manifestations in Data Science. Examples include: Multidimensional Scaling, Hierarchical Clustering, Shape comparison through the Gromov-Hausdorff distance, Persistent Homology of metric spaces, Optimal Transport through the notions of the Wassertsein distance between probability measures and the Gromov-Wasserstein distance between metric measure spaces, as well as spectral methods as those used in Diffusion Geometry.  The choice of topics will be partly determined by the interest of participants in the course.

Text:

References: there’s no single textbook for this course. We will follow several sources including:

Prerequisites:

Undergraduate courses in topology, algebra and analysis.

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Spring 2023

Konstantin Mischaikow

Subtitle:

Applied and Computational Topology

Course Description:

This course is an introduction to Topological Data Analysis (TDA). Though still relatively new TDA has blossomed over the past two decades, thus we will only have time to touch on a few subjects and these topics will be decided in part by the interests of students in the course. While an introductory course in algebraic topology would be ideal, it is not absolutely necessary as the basic ideas of homology will be discussed as needed.

We will start the course by considering:

1. Methods for the efficient computation of homology groups, induced maps on homology, persistent homology, and Conley complexes. This is essential knowledge for working with large data sets.

The following topics will be covered (the depth of coverage will depend on the class)

A. Data driven nonlinear dynamics. We will discuss topological and statistical tools for extracting specific dynamical structures from time series data.

2. Persistent homology with a focus on persistence diagrams. We will view persistence as providing nonlinear method of reduction of data that preserves topological structures measured by homology.
We will discuss applications to a variety of topics.

4. Machine learning. We will explore the dynamics of machine learning via topological dynamics and/or the use of machine learning to reduce dimension of data for topological analysis.

The choice of topics to which these techniques will be applied will be determined in part by the interest of participants in the course, but will may include the problems arising from image processing,
protein structures, materials science, gene regulatory networks, fluid flow, social choice, and dense granular media.

Text:

will provide lecture notes

Prerequisites:

Undergraduate courses in topology and algebra.

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Spring 2023

Fioralba Cakoni

Subtitle:

Mathematical Aspects of Inverse and Imaging Problems

Course Description:

Inverse and imaging problems arise in many applications from life sciences and engineering. These are problems where causes for a desired or an observed effect are to be determined.

Through some fundamental and contemporary examples of inverse and imaging problems, this course aims to introduce students to the rich mathematical and computational aspects of this area. Some of the examples to be discussed in this course include inverse spectral problems, inverse integral geometry, inverse problems for partial differential equations such as the Calderon problem and inverse scattering.

Text:

Lecture notes and reading material will be provided during the course

Prerequisites:

Real Analysis (equivalent to Math 501), introductory Functional Analysis (at least what is covered in Math 502), introductory Partial Differential Equations (equivalent to math 517), basis Fourier Analysis (at least what is covered in Math 502).

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Spring 2022

Konstantin Mischaikow

Subtitle:

Topological Data Analysis

Course Description:

This course is an introduction to Topological Data Analysis (TDA). Though still relatively new TDA has blossomed over the past two decades, thus we will only have time to touch on a few subjects and these topics will be decided in part by the interests of students in the course. While an introductory course in algebraic topology would be ideal, it is not necessary as the basic ideas of homology will be discussed as needed.

We will start the course by considering:

1. Methods for the efficient computation of homology groups, Conley complexes, persistent homology and induced maps on homology. This is essential knowledge for working with large data sets.

2. Persistent homology with a focus on persistence diagrams. We will view persistence as providing nonlinear method of reduction of data that preserves topological structures measured by homology. We will discuss applications to a variety of topics

3. Topological approach to nonlinear dynamics. We will discuss topological tools for extracting specific dynamical structures from time series data.

4. Exploring the dynamics of machine learning via topological dynamics.

The choice of topics to which these techniques will be applied will be determined in part by the interest of participants in the course, but will may include the problems arising from image processing, protein structures, materials science, gene regulatory networks, fluid flow, social choice, and dense granular media.

Text:

None

Prerequisites:

Undergraduate courses in topology and algebra

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Fioralba Cakoni

Subtitle:

Mathematics of Inverse Problems and Imaging

Course Description:

This course will discuss fundamental mathematical issues behind many problems in tomography and imaging, including medical imaging, geometrical inverse problems, inverse scattering, and Bayesian inversion and machine learning.

Text:

Lecture notes and reading material will be provided throught the course

Prerequisites:

No formal prerequisites, but the course assumes a solid foundation in real analysis (at least 640:411, 640:412 and preferably 640:501), and basic knowledge of partial differential equations. Introductory functional analysis is valuable but not required.

Schedule of Sections:

Previous Semesters:

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