The seminar activities are partially supported by the National Science Foundation.

Talks of the Spring 2018 semester are available here.

For earlier seminars, see the old webpage.

This is joint work with Luca Cardelli, Max Tschaikowski, and Andrea Vandin.

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There will be no prerequisites in model theory.

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Model order reduction for large-scale nonlinear systems is a key enabler for design, uncertainty quantification and control of complex systems. I will discuss a beneficial detour to deriving efficient reduced-order models for nonlinear systems. First, the nonlinear model is lifted to a model with more structure via variable transformations and the introduction of auxiliary variables. The lifted model is equivalent to the original model - it uses a change of variables, but introduces no approximations. When discretized, the lifted model yields a polynomial system of either ordinary differential equations or differential algebraic equations, depending on the problem and lifting transformation. In order to obtain computationally inexpensive models, we then proceed with reducing those lifted systems. Proper orthogonal decomposition (POD) is applied to the lifted models, yielding a reduced-order model for which all reduced-order operators can be pre-computed. We show several examples in form of a FitzHugh-Nagumo PDE and a tubular reactor PDE model, and show how this approach opens new pathways for rigorous analysis and input-independent model reduction via the introduction of the lifted problem structure.

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Resistance to chemotherapy is a major impediment to successful cancer treatment that has been extensively studied over the past three decades. Classically, resistance is thought to arise primarily through random genetic mutations, after which mutated cells expand via Darwinian selection. However, recent experimental evidence suggests this evolution to resistance need not occur randomly, but instead may be induced by the application of the drug. Indeed, phenotype switching via epigenetic alterations is just recently beginning to be understood. In this work, we present a mathematical model to that describes both random and induced resistance. We discuss issues related to both structural and practical identifiability of model parameters. A time-optimal control problem is formulated and analyzed utilizing differential-geometric techniques. Specifically, the control structure is precisely characterized, and therapy outcome is analyzed for different levels of resistance induction through a combination of analytic and numerical results. Existence results are also discussed, as well as further extensions to combination therapies are also considered, and questions of combination vs. sequential therapy are studied.

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This is joint work with Jean-Yves Thibon, Houyi Yu and Jianqiang Zhao.

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Delay-differential equations are actively used in areas of applied mathematics ranging from mathematical biology to electrical engineering. Elimination of unknowns is a fundamental tool for studying solutions of equations (linear, polynomial, differential, etc.). In the talk, the first elimination algorithm for a system of delay-differential equations will be presented. For this algorithm, we develop new effective methods in differential algebraic geometry and combine them with parts of the approach taken in the first elimination algorithm for systems of difference equations designed recently by Ovchinnikov, Pogudin, and Scanlon.

This is a joint work with Wei Li, Alexey Ovchinnikov, and Thomas Scanlon