Courant Instructor / PIRE Fellow at the Courant Institute at New York University.
Email: lagatta at cims dot nyu dot edu
Office: 927 Warren Weaver Hall
Office Phone: 212-992-9920
Office Hours: TBA
Curriculum Vitae (2010)
I am especially interested in disordered systems: models where the underlying geometry in a problem is random. My dissertation research focused on random Riemannian geometry. We begin with the flat metric on d-dimensional Euclidean space, then we take a random perturbation of this geometry. Under the new random metric, we study classical geometric objects like balls and geodesics. At large scales, the geometry remains Euclidean (though perhaps stretched or scaled). An example of this phenomenon is the Shape Theorem, which states that balls under the random metric grow asymptotically like Euclidean balls (this is a geometric analogue of the Law of Large Numbers). At finer scales, however, the geometry is quite different. In analogy with the Central Limit Theorem, one might expect that the fluctuations of the random ball of radius r are of the order r1/2. This does not seem to be the case; instead, the fluctuations of the ball are of order rχ, where χ is a universal exponent depending only on the dimension of the model. When we restrict our attention to perturbations of the 2-dimensional Euclidean plane, we expect that χ = 1/3.
I am also interested in applying these ideas to problems in the social sciences. We can model social situations as interacting particle systems from statistical physics. Each "actor" in a situation is represented by a particle, and the social landscape is described by a network (or graph) which connects these particles. In most examples, the information about the strength of social ties is at best incomplete. To model this uncertainty, we overlay the edges of the network with random weights.
Particles can be entirely reactive, mindlessly maximizing some utility function, or they can have more sophisticated, strategic behavior. We model this strategic behavior using game theory. Of course, actors cannot be purely rational calculating machines: we humans are subject to physiological and neurological impulses, and often make decisions based on a simple heuristic. Rather than including these smaller scale processes in a complex, unwieldy model, we simplify matters by quantifying the lower level uncertainty as an exogenous noise term, analogous to tempature in a physical system. The random social landscape is then a disordered system, and many tools and techniques from mathematical physics can be applied to these problems.
When the random edge weights are all positive (or zero), cooperative behavior is encouraged; this is analogous to a disordered ferromagnet in physics. When the random edge weights can also be negative, this encourages a complicated blend of cooperative and antagonistic behavior. In physics this is called a spin glass, and is a rich, interesting model from the mathematical, physical and social points of view.
All of my manuscripts are publicly available on the arXiv.
• T. LaGatta. Continuous Disintegrations of Gaussian Processes. Accepted for publication in Theory of Probability and Its Applications, 2011.
• T. LaGatta and J. Wehr. A Shape Theorem for Riemannian First-Passage Percolation. J. Math. Phys., 51(5), 2010.
• T. LaGatta. Dissertation: Geodesics of Random Riemannian Metrics. May 2010.
• A. Smith, T. LaGatta and B. Bueno de Mesquita. Prizes, Groups and Pivotal Voting in a Poisson Voting Game. Submieted for publication, 2011.
• T. LaGatta and J. Wehr. Geodesics of Random Riemannian Metrics I: Random Perturbations of Euclidean Space. Work in progress, 2011.
• T. LaGatta and J. Wehr. Geodesics of Random Riemannian Metrics II: Minimizing Geodesics. Work in progress, 2011.
• S. Kim and T. LaGatta. Bounded Geodesics in Random Riemannian Geometry. Work in progress, 2011.
• T. LaGatta and D. Sanders. An Efficient Algorithm for the Lorentz Lattice Gas. Work in progress, 2011.
• T. LaGatta and S. Tyson. A Generalization of Sequential Equilibrium. Work in progress, 2011.
• T. LaGatta, B. Bakker and A. Smith. A Bayesian Voting Model with Aggregate Uncertainty. Work in progress, 2011.
• Transformations & Geometry (V63.0270)
• Spring 2011: Probability & Statistics (V63.0235-1)
• Fall 2010: Discrete Mathematics (V63.0120-2) — Course Syllabus
• Curvature, with pretty pictures of geodesics in various different environments. 2008.