Thomas Pietraho

Research Interests

• Representation theory and related combinatorics.

About the image: Machine learning models can sometimes reproduce results from pure mathematics; a challenge is interpreting the mechanism by which they do this. Above is a visualization of a hidden layer from a neural network that has learned how to multiply permutations in the symmetric group S₅ . To compute the product c= ab, the network learns the left coset of c with respect to each of the six dihedral subgroups of order 10. Since these subgroups intersect trivially, c is identified uniquely. Each image in the sequence represents a left coset partition of  S₅  with respect to a different dihedral subgroup; each petal a different permutation.

Courses

	Math 1800: Multivariate calculus. Multivariate calculus in two and three dimensions. Vectors and curves in two and three dimensions; partial and directional derivatives; the gradient; the chain rule in higher dimensions; double and triple integration; polar, cylindrical, and spherical coordinates; line integration; conservative vector fields; and Green’s theorem. Course laboratories are focused on the applications of multivariate calculus to data science and machine learning.
	Math 2603: Introduction to analysis. Building on the theoretical underpinnings of calculus, develops the rudiments of mathematical analysis. Concepts such as limits and convergence from calculus are made rigorous and extended to other contexts, such as spaces of functions. Specific topics include metric spaces, point-set topology, sequences and series, continuity, differentiability, the theory of Riemann integration, and functional approximation and convergence.
	Math 2805: Mathematical principles of machine learning. An introduction to the mathematical theory and practice of machine learning. Supervised and unsupervised learning problems, including regression, classification, clustering, and component analysis, focusing on techniques most relevant to the study and applications of neural networks. Additional topics may include dimension reduction, data visualization, denoising, norms and loss functions, optimization, universal approximation theorems, and algorithmic fairness. Class will include computer lab and projects, but no formal programming experience is necessary.
	Math 3603: Advanced analysis. Measure theory and integration with applications to probability and mathematical finance. Topics include Lebesgue measure and integral, measurable functions and random variables, convergence theorems, analysis of random processes including random walks and Brownian motion, and the Ito integral.

Please see the courses website for links to additional courses.

Publications

Drafts of and links to the following articles are available on my personal web page.

• Orbital Varieties and Unipotent Representations of Classical Semisimple Lie Groups, PhD Thesis, MIT.
• Components of the Springer Fiber and Domino Tableaux, Journal of Algebra, 272 (2):711-729, 2004. arXiv:math.RT/0210416
• A Relation for Domino Robinson-Schensted Algorithms,  Annals of Combinatorics, 13 (4):519-532, 2010. arXiv:math.CO/0603654.
• Orbital Varieties and Unipotent Representations Attached to Spherical Orbits, arXiv:math.RT/0603685
• Cells in the Weyl Groups of type B(n), Journal of Algebraic Combinatorics, 27(2):247-262, 2008. arXiv:math/0607231
• Cells and Constructible Representations in Type B, New York Journal of Mathematics 14:411-430, 2008. arXiv: 0710.3846
• Knuth Relations for the Hyperoctahedral Groups,  Journal of Algebraic Combinatorics, 29(4):509-535, 2009.
• Module Structure of Cells in Unequal Parameter Hecke Algebras, Nagoya Mathematical Journal, 198(2010). arXiv:0902.1907
• Sign under the domino Robinson-Schensted maps, Annals of Combinatorics 18 (2014). arXiv:1301.1356.
• On the Sign Representations for the Complex Reflection Groups G(r,p,n) (with A. Mbirika and W. Silver), Beitrage zur Algebra und Geometrie 57, 2016. arXiv:1303.5021.
• Kazhdan-Lusztig left cells in type B for intermediate parameters (with E. Howse). arXiv:1902.09301.