| Talk_id | Date | Speaker | Title |
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4114
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Tuesday 9/12
4:10 PM
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Bruce Sagan, Michigan State University
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Bounding Roots of Polynomials
- Bounding Roots of Polynomials
- 09/12/2017
- 4:10 PM - 5:00 PM
- C304 Wells Hall
- Bruce Sagan, Michigan State University
We will present methods for bounding the modulus of the complex roots of a polynomial. These include the Cauchy bound and the use of Newton polynomials, the latter also being useful in interpolation problems. No background outside of elementary calculus will be assumed. In a subsequent talk, we will use these techniques to make progress on a conjecture about the roots of a polynomial of combinatorial interest.
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4123
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Tuesday 9/19
4:10 PM
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Bruce Sagan, Michigan State University
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Roots of descent polynomials
- Roots of descent polynomials
- 09/19/2017
- 4:10 PM - 5:00 PM
- C304 Wells Hall
- Bruce Sagan, Michigan State University
Let S_n be the symmetric group of all permutations p = p_1 ... p_n of the numbers 1, ..., n. The descent set of p is the set of indices i such that p_i > p_{i+1}. Given a set of positive integers I we let d(I;n) be the number of permutations in S_n with descent set I. In 1915 MacMahon proved that d(I;n) is a polynomial in n, but its properties do not seem to have been much studied until now. We apply the method of Newton bases from the previous lecture to make progress on a conjecture about the location of the roots of d(I;n) where n is now a complex number. This is joint work with Alexander Diaz-Lopez, Pamela Harris, Erik Insko, and Mohamed Omar.
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5137
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Tuesday 9/26
4:10 PM
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Robert Davis, MSU
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A Combinatorial Description of Schur Functions
- A Combinatorial Description of Schur Functions
- 09/26/2017
- 4:10 PM - 5:00 PM
- C304 Wells Hall
- Robert Davis, MSU
A symmetric function is a formal power series in countably many variables that is fixed under any permutation of the indices of the variables. The set of symmetric functions forms a vector space (actually, a graded algebra), and so it is natural to look for convenient bases of the space. In this talk, we will describe four "easy" bases and one more subtle, but much more important, basis, called the basis of Schur functions. We will give the combinatorial definition of Schur functions and highlight some of its uses in various branches of mathematics.
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5138
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Tuesday 10/3
4:10 PM
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Robert Davis, MSU
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Follow the Rules!
- Follow the Rules!
- 10/03/2017
- 4:10 PM - 5:00 PM
- C304 Wells Hall
- Robert Davis, MSU
Schur functions are among the most useful bases for symmetric functions, but they come at the cost of making certain computations much less obvious than when done in other bases. In particular, how can we determine the coefficients of a product of Schur functions in the basis of Schur functions? There are many equivalent ways to computing these numbers; this talk will discuss jeu de taqin, which is an equivalence among skew tableaux, and apply it to obtain a formulation of the Littlewood-Richardson rule, which answers our question. If time allows, we will discuss other important formulations of this rule.
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