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Events happening on February 26, 2019

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Feb. 26, 2019 10am (Math 350)	Geometry/Analysis Nathan Glatt-Holtz (Tulane University) X We develop ‘scaling’ and ‘saturation’ methods on general state spaces which can be used to solve low mode control problems for certain nonlinear partial differential equations. These methods naturally generalize ideas of Jurdjevic and Kupka from the finite-dimensional setting of ordinary differential equations. Our approach is highlighted with applications to specific questions concerning the Korteweg-de-Vries (KdV), Euler and Boussinesq equations. In particular, we derive support properties and unique ergodicity results for certain randomly-forced versions of these equations. Scalings and saturation in infinite-dimensional control problems with applications to stochastic partial differential equations Sponsored by the Meyer Fund
Feb. 26, 2019 11am (MATH 350)	Number Theory Damien Roy (University of Ottawa) X The theorem of Lindemann-Weierstrass asserts that the exponentials of distinct algebraic numbers are linearly independent over the field of rational numbers. The proof uses a construction of simultaneous rational approximations to such exponentials values, which goes back to Hermite. In this talk, we show that, from an adelic perspective, these approximations are essentially best possible. The point of view partly explains the nature of the algebraic numbers whose exponentials have a structured continuous fraction expansion. We conclude with a conjecture regarding simultaneous approximations to such values in adèle rings. Simultaneous rational approximation to exponentials of algebraic numbers
Feb. 26, 2019 2pm (MATH 350)	Lie Theory Bruce Sagan (Michigan State University) X The Lucas sequence is a sequence of polynomials in $s, t$ defined recursively by $0=0$ , $1=1$ , and $n=sn-1+tn-2$ for $n≥2$ . On specialization of $s$ and $t$ one can recover the Fibonacci numbers, the nonnegative integers, and the $q$ -integers $n]q$ . Given a quantity which is expressed in terms of products and quotients of positive integers, one obtains a Lucas analogue by replacing each factor of $n$ in the expression with ${n}$ . It is then natural to ask if the resulting rational function is actually a polynomial in $s$ and $t$ and, if so, what it counts. Using lattice paths, we give a combinatorial model for the Lucas analogue of binomial coefficients. This is joint work with Curtis Bennett, Juan Carrillo, and John Machacek. We then give an algebraic method for proving polynomiality using a connection with cyclotomic polynomials. This part of the talk is joint work with Jordan Tirrell and based on an idea of Richard Stanley. Finally we, also consider Catalan numbers and their relatives, such as those for finite Coxeter groups. Combinatorial and algebraic interpretations of Lucas analogues Sponsored by the Meyer Fund
Feb. 26, 2019 3pm (MATH 350)	Topology Amit Patel (CSU) X In this pretalk, I will give an introduction to classical persistent homology. I will highlight ideas MacPherson and I generalize to the setting of constructible maps to manifolds. Persistent Homology (Pretalk)
Feb. 26, 2019 4pm (MATH 350)	Topology Amit Patel (CSU) X We are interested in developing a theory of persistence for maps X —> M that are constructible with respect to a stratification of M. I will present ideas from the first paper in this direction. We give lower bounds for the homology of the fibers of a map to a manifold. Using new sheaf theoretic methods, we show that these lower bounds persist over whole open sets of the manifold, and that they are stable under perturbations of the map. This generalizes certain ideas of persistent homology to higher dimensions. Persistent Local Systems

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