Professional Skills for Mathematics is an introduction to technical and communication skills utilized in upper level mathematics courses with additional focus on resume building and professional development.
The course will follow Francis Bonahon's book Low Dimensional Geometry: From Euclidean Surfaces to Hyperbolic Knots. This book was written for undergraduates and brings students up to speed on the many developments in the geometry of 3-manifolds over the last 40 years, beginning with rather elementary constructions. Because the book was written for an undergraduate course, our syllabus will more or less follow the table of contents.
Topics: Dirichlet series, arithmetic functions, Euler products, Zeta and L-functions functions, functional equations and explicit formulae (for psi and other functions), zero-free regions, Prime Number Theorem, Dirichlet's theorem on primes, exponential sums, Siegel zeros and Siegel's Theorem, Landau-Page Theorem, large sieve, and Bombieri-Vinogradov Theorem
Contact structures on 3-manifolds have become a central part of low-dimensional topology. This course will discuss several techniques to study contact structures with a focus on convex surface theory. We will show how to classify contact structures on certain 3-manifolds and how to study special knots inside contact 3-manifolds.
This course will fill the gap between the computational and engineering aspects of neuroscience covered in BMED 7610 and ECE/BMED6790 by covering mathematical analysis and simulation of neural systems across single cells, networks, and populations, employing methods from dynamical systems, network science, and stochastic processes. The topics will include single-neuron excitability and bifurcation, network structure and synchrony, and statistical dynamics of large neural populations.
In short, I plan to cover major parts of chapters I, II, IV before covering some special topics (time permitting). Possible special topics include polytopes, lattice point enumeration (with applications to integer programming), applications in semidefinite optimization and moment problems in analysis.