Math Department
https://wp0.vanderbilt.edu/math
College of Arts and Science | Vanderbilt UniversityFri, 18 Oct 2019 21:10:07 +0000en-UShourly1https://wordpress.org/?v=4.9.8146940276Well-Posedness and Dispersive Decay of Small Data Solutions to the Benjamin-Ono Equation
https://wp0.vanderbilt.edu/math/2019/10/talk-title-tba-311/
Fri, 18 Oct 2019 21:10:07 +0000https://wp0.vanderbilt.edu/math/?p=9184This result represents a first step toward understanding the long time dynamics of solutions for the Benjamin-Ono equation on the real line. While this problem is known to be both completely integrable and globally well-posed in L^{2}, much less seems to be known concerning its long time dynamics. Here, we prove that for small localized data the solutions have (nearly) dispersive dynamics almost globally in time. An additional objective is to revisit the L^{2} theory for the Benjamin-Ono equation and provide a simpler, self-contained approach.
]]>9184An Extension of Yau’s Theorem to Asymptotically Conical Manifolds (and Orbifolds)
https://wp0.vanderbilt.edu/math/2019/10/an-extension-of-yaus-theorem-to-asymptotically-conical-manifolds-and-orbifolds/
Fri, 18 Oct 2019 20:10:55 +0000https://wp0.vanderbilt.edu/math/?p=9507Yau’s original solution to Calabi’s conjecture states that, given a prescribed form representing the first Chern class on a compact Kahler manifold, it is possible to find a Kahler metric whose Ricci form is the given one, and moreover, this metric is unique. Yau’s solution involves solving a Monge-Ampere equation via a continuity method, and uses certain a priori estimates to obtain regularity on the candidate solution. I’ll discuss a variant of this theorem in the setting where the Kahler manifold is asymptotic to a cone. The precise existence results in this setting depend on the linearized equation, and in particular on the Fredholm index of the linearized operator. In the compact case, the index is always zero, but in this asymptotic setting, the index varies as a step function of the decay rate of the prescribed Ricci form. I’ll discuss some small improvements on an existence statement from Conlon-Hein in the case where Fredholm index is the first negative value. The techniques involved are linear.
]]>9507Extremal Primes for Elliptic Curves Without Complex Multiplication
https://wp0.vanderbilt.edu/math/2019/10/talk-title-tba-323/
Fri, 18 Oct 2019 20:00:40 +0000https://wp0.vanderbilt.edu/math/?p=9283Fix an elliptic curve E over Q. An “extremal prime” for E is a prime p of good reduction such that the number of rational points on E modulo p is maximal or minimal in relation to the Hasse bound. In this talk, I will discuss what is known and conjectured about the number of extremal primes p≤X and give the first non-trivial upper bound for the number of such primes when E is a curve without complex multiplication. The result is conditional on the hypothesis that all the symmetric power L-functions associated to E are automorphic and satisfy the Generalized Riemann Hypothesis. In order to obtain this bound, we use explicit equidistribution for the Sato-Tate measure as in recent work of Rouse and Thorner, and refine certain intermediate estimates taking advantage of the fact that extremal primes have a very small Sato-Tate measure.

]]>9283Quantum Computation and Universal Algebra
https://wp0.vanderbilt.edu/math/2019/10/quantum-computation-and-universal-algebra/
Thu, 17 Oct 2019 21:10:20 +0000https://wp0.vanderbilt.edu/math/?p=9451Digital computers perform calculations on individual strings of bits. Quantum computers extend classical digital computers by making use of quantum phenomena to perform calculations on bits which are in a state of superposition. For some classes of problems, this allows for superpolynomial speedup over classical algorithms. In the first half of the talk we give an introduction to quantum computation and to the various classes of problems exhibiting superpolynomial speedup. In the second half, we propose generalizations of these problems to the domain of universal algebras and present some results for the quantum tractability of these generalizations. Tea at 3:30pm in SC 1425. (Contact person: Ralph McKenzie)
]]>9451The Irreducibility of Monodromy is a Mapping Torus Invariant
https://wp0.vanderbilt.edu/math/2019/10/talk-title-tba-315/
Wed, 16 Oct 2019 21:10:48 +0000https://wp0.vanderbilt.edu/math/?p=9198An immediate corollary of Nielsen-Thurston classification of surface homeomorphisms is that if two surface homeomorphisms f and g have homeomorphic mapping tori, then f is pseudo-Anosov if and only if g is pseudo-Anosov. Using hyperbolization theorem and rigidity results, the hypothesis can be weakened to quasi-isometric mapping tori. We show an analogous result for free group automorphisms: if two free group automorphisms have isomorphic mapping tori, then the first automorphism is fully irreducible and atoroidal if and only if the other is fully irreducible and atoroidal. This answers a question posed by Dowdall-Kapovich-Leininger.
]]>9198Graduate Student Tea
https://wp0.vanderbilt.edu/math/2019/10/graduate-student-tea-82/
Wed, 16 Oct 2019 20:30:19 +0000https://wp0.vanderbilt.edu/math/?p=95269526Retirement Celebration for Distinguished Professor Ralph McKenzie
https://wp0.vanderbilt.edu/math/2019/10/retirement-celebration-for-distinguished-professor-ralph-mckenzie/
Wed, 16 Oct 2019 20:10:40 +0000https://wp0.vanderbilt.edu/math/?p=9539We invite you to attend in showing Distinguished Professor Ralph McKenzie our tremendous appreciation at a retirement celebration. Professor McKenzie joined the Math Department in 1994. He will be greatly missed by colleagues and students alike. We hope that you are able to join in celebrating his scholarly achievements. (Contact Person: KT Griffis and Jeannie Wagner)
]]>9539An Environmental Model of Honey Bee Colony Collapse Due to Pesticide Contamination
https://wp0.vanderbilt.edu/math/2019/10/talk-title-tba-330/
Tue, 15 Oct 2019 23:00:29 +0000https://wp0.vanderbilt.edu/math/?p=9305A model of honey bee colony collapse is developed, based on the contamination of forager bees in environmental regions contaminated with pesticides. An important feature of the model is the daily homing capacity each day of foragers bees. The model consists of difference equations describing the daily homing of uncontaminated and contaminated forager bees, with an increased homing failure of contaminated bees. The model quantifies colony collapse in terms of the fraction of contaminated bees subject to this increased homing failure. If the fraction is sufficiently high, then the hive falls below a viability threshold population size that leads to rapid disintegration. If the fraction is sufficiently low, then the hive can rise above the viability threshold and attain a stable population level.
]]>9305Qualifying Examination: Finitely Generated Infinite Torsion Subgroups of Groups with Cubic Dehn Function
https://wp0.vanderbilt.edu/math/2019/10/qualifying-examination-2/
Tue, 15 Oct 2019 17:00:55 +0000https://wp0.vanderbilt.edu/math/?p=94479447Supernilpotence is Not Super Nilpotence
https://wp0.vanderbilt.edu/math/2019/10/supernilpotence-is-not-super-nilpotence/
Mon, 14 Oct 2019 21:10:45 +0000https://wp0.vanderbilt.edu/math/?p=9523Supernilpotence is a generalization of nilpotence using a recently developed theory of higher-arity commutators for universal algebras. Many important structural properties have been shown to be associated with supernilpotence, and the exact relationship between nilpotence and supernilpotence has been the subject of investigation. We construct an algebra which is not solvable (and hence not nilpotent) but which is supernilpotent, thereby showing that in general supernilpotence does not imply nilpotence.
]]>9523