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Mathematics Department

Basic Notions Seminar

Fall 2018

All talks are from 1200-1300 in the Seminar room, unless otherwise specified.

  • Nov
    13
  • Radial Basis Function and Polynomial Interpolation for Approximating the Action of Linear Operators
    Jonah Reeger
    USNA, Mathematics
    Time: 12:00 PM

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    In this talk we explore the benefits of enriching the polynomial basis with Radial Basis Functions when approximating the action of linear operators. Such approximations are finding their way into large-scale simulations that require geometric flexibility and high-orders of accuracy.
  • Oct
    18
  • Can I get a witness (set)?
    Dan Bates
    USNA, Math
    Time: 12:00 PM

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    When trying to describe the solutions of a polynomial system, we have to choose a data type for each "piece" of the solution set. For example, a solution set might consist of a surface, two lines, and five points. Some might choose to describe each of these components by providing a list of polynomials that specifically pick out the component. In the world of numerical algebraic geometry, the goal is to produce a "witness set" for each component. The primary goal for this talk is to introduce this type of problem (polynomial system) and this type of solution (witness set) through examples, though we will likely also get into some of the geometric tools around these ideas.
  • Oct
    11
  • Completeness, Incompleteness, Consistency and Independence: Gödel's revolutionizing of logic
    Darren Creutz
    USNA, Mathematics
    Time: 12:00 PM

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    Most mathematicians have heard of Gödel's incompleteness theorem but few can say what it says beyond the imprecise and somewhat misleading "there are true but unprovable statements". What does that actually mean? What is "true" here as it is evidently different than "provable"? The answer to this lies in the far more important completeness theorem, also due to Gödel, which is at the heart of the modern study of logic. We will discuss the completeness theorem at a very high level, assuming no background, and its connection to the proper formulation of the incompleteness theorem as well as how these ideas lead to proofs about consistency and independence, specifically the famous theorems of Gödel and Cohen that (1) both the axiom of choice and its negation are consistent with ZF and (2) that the continuum hypothesis is independent of ZFC. The overarching theme will be the contributions of Gödel to the field of logic and how his work laid out the path of virtually all of modern set theory.
  • Sep
    25
  • Polymatroids and Chuns
    Carolyn Chun
    USNA, Math
    Time: 12:00 PM

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    Nearly 10 years after Deborah Chun's paper on excluded minors for deletion-contraction polymatroids appeared, I am working with Joseph Bonin and Steven Noble on some excluded minor characterizations of certain classes of polymatroids related to D. Chun's work. In this talk, I define a polymatroid and present some of D. Chun's results. I will also give some motivating background and relate this subject to ongoing work in matroid theory. Note: A polymatroid is NOT a kind of matroid.
  • Sep
    13
  • A Decomposition Method for Solving Large Linear (And Integer) Programs
    Rob Curry
    USNA, Math
    Time: 12:00 PM

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    When linear programs (LPs) become too large to explicitly consider all variables, conventional LP solution techniques may not be computationally viable. In many such cases, though, a majority of variables will be non-basic and assume a value of zero for an optimal LP solution. To take advantage of this observation, decomposition methods may be deployed to consider only a subset of all possible variables. In this talk, I describe the details of one such method, column generation (CG). I detail how CG involves the implicit pricing of non-basic variables to generate new column variables, and I show how CG can prove the optimality of a current solution. Furthermore, I will also discuss the branch-and-price algorithm (BP) that deploys CG within a branch-and-bound tree to solve integer linear programs having a large number of variables. I describe the strengths, weaknesses, and challenges involved with deploying CG and BP algorithms. Finally, I outline a few classes of problems for which CG and BP might be useful.
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