NTS Spring 2012/Abstracts: Difference between revisions
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Abstract: In this talk we consider a linear algebraic object called a tridiagonal pair. This object originated in algebraic graph theory, and has connections to orthogonal polynomials and representation theory. In our discussion we aim at a general mathematical audience; no prior experience with the above topics is assumed. Our main results are joint work with Tatsuro Ito and Kazumasa Nomura. | |||
Abstract: In | |||
and | |||
joint work with | |||
< | The concept of a tridiagonal pair is best explained by starting with a special case called a Leonard pair. Let '''F''' denote a field, and let ''V'' denote a vector space over '''F''' with finite positive dimension. By a Leonard pair on ''V'' we mean a pair of linear transformations ''A'' : ''V'' → ''V'' and ''A''<sup>∗</sup> : ''V'' → ''V'' that satisfy the following two conditions: | ||
#There exists a basis for ''V'' with respect to which the matrix representing ''A'' is diagonal and the matrix representing ''A''<sup>∗</sup> is irreducible tridiagonal; | |||
#There exists a basis for ''V'' with respect to which the matrix representing ''A''<sup>∗</sup> is diagonal and the matrix representing ''A'' is irreducible tridiagonal. | |||
for | |||
We recall that a tridiagonal matrix is irreducible whenever each entry on the superdiagonal is nonzero and each entry on the subdiagonal is nonzero. The name "Leonard pair" is motivated by a connection to a 1982 theorem of the combinatorialist Doug Leonard involving the ''q''-Racah and related polynomials of the Askey scheme. Leonard’s theorem was heavily influenced by the work of Eiichi Bannai and Tatsuro Ito on ''P''- and ''Q''- polynomial association schemes, and the work of Richard Askey on orthogonal polynomials. These works in turn were influenced by the work of Philippe Delsarte on coding theory, dating from around 1973. | |||
The central result about Leonard pairs is that they are in bijection with the orthogonal polynomials that make up the terminating branch of the Askey scheme. This branch consists of the ''q''-Racah, ''q''-Hahn, dual ''q''-Hahn, ''q''-Krawtchouk, dual ''q''-Krawtchouk, quantum ''q''-Krawtchouk, affine ''q''-Krawtchouk, Racah, Hahn, dual-Hahn, Krawtchouk, and the Bannai/Ito polynomials. The bijection makes it possible to develop a uniform theory of these polynomials starting from the Leonard pair axiom, and we have done this over the past several years. | |||
< | A tridiagonal pair is a generalization of a Leonard pair and defined as follows. Let ''V'' denote a vector space over '''F''' with finite positive dimension. A ''tridiagonal pair on V'' is a pair of linear transformations ''A'' : ''V'' → ''V'' and ''A''<sup>∗</sup> : ''V'' → ''V'' that satisfy the following four conditions: | ||
{ | <ol> | ||
<li>Each of ''A'', ''A''<sup>∗</sup> is diagonalizable on ''V''; | |||
<li>There exists an ordering {''V<sub>i</sub>''}<sub>''i''=0,...,''d''</sub> of the eigenspaces of ''A'' such that | |||
::''A''<sup>*</sup>''V<sub>i</sub>'' ⊆ ''V''<sub>''i''−1</sub> + ''V<sub>i</sub>'' + ''V''<sub>''i''+1</sub> (0 ≤ ''i'' ≤ ''d''), | |||
where ''V''<sub>−1</sub> = 0, ''V''<sub>''d''+1</sub> = 0; | |||
<li>There exists an ordering {''V<sub>i</sub>''<sup>*</sup>}<sub>''i''=0,...,δ</sub> of the eigenspaces of ''A''<sup>*</sup> such that | |||
::''AV<sub>i</sub>''<sup>*</sup> ⊆ ''V''<sup>*</sup><sub>''i''−1</sub> + ''V''<sup>*</sup><sub>''i''</sub> + ''V''<sup>*</sup><sub>''i''+1</sub> (0 ≤ ''i'' ≤ δ), | |||
where ''V''<sup>*</sup><sub>−1</sub> = 0, ''V''<sup>*</sup><sub>''d''+1</sub> = 0; | |||
<li>There is no subspace ''W'' ⊆ ''V'' such that ''AW'' ⊆ W, ''A''<sup>*</sup>''W'' ⊆ ''W'', ''W'' ≠ 0, ''W'' ≠ ''V''. | |||
</ol> | |||
It turns out that ''d'' = δ and this common value is called the diameter of the pair. | |||
A Leonard pair is the same thing as a tridiagonal pair for which the eigenspaces | |||
''V'' and ''V''<sup>∗</sup> all have dimension 1. | |||
Tridiagonal pairs arise naturally in the theory of ''P''- and ''Q''-polynomial association schemes, in connection with irreducible modules for the subconstituent algebra. They also appear in recent work of Pascal Baseilhac on the Ising model and related structures in statistical mechanics. | |||
In this talk we will summarize the basic facts about a tridiagonal pair, describing | |||
features such as the eigenvalues, dual eigenvalues, shape, tridiagonal relations, | |||
split decomposition, and parameter array. We will then focus on a special case | |||
said to be sharp and defined as follows. Referring to the tridiagonal pair ''A'', ''A''<sup>∗</sup> in the above definition, it turns out that for 0 ≤ ''i'' ≤ ''d'' the dimensions of ''V<sub>i</sub>'', ''V''<sub>''d''−''i''</sub>, ''V''<sup>*</sup><sub>''i''</sub>, ''V''<sup>*</sup><sub>''d''−''i''</sub> coincide; the pair ''A'', ''A''<sup>∗</sup> is called ''sharp'' whenever ''V''<sub>0</sub> has dimension 1. It is known that if '''F''' is algebraically closed then ''A'', ''A''<sup>∗</sup> is sharp. | |||
In our main result we classify the sharp tridiagonal pairs up to isomorphism. | |||
|} | |} | ||
</center> | </center> | ||
<br> | <br> | ||
== March 29 == | == March 29 == |
Revision as of 21:18, 18 March 2012
February 2
Evan Dummit (Madison) |
Title: Kakeya sets over non-archimedean local rings |
Abstract: In a forthcoming paper with Marci Habliscek, we constructed a Kakeya set over the formal power series ring F_{q}[[t ]], answering a question posed by Ellenberg, Oberlin, and Tao. My talk will be devoted to explaining some of the older history of the Kakeya problem in analysis and the newer history of the Kakeya problem in combinatorics, including my joint work with Marci. In particular, I will give Dvir's solution of the Kakeya problem over finite fields, and explain the problem's extension to other classes of rings. |
February 16
Tonghai Yang (Madison) |
Title: A little linear algebra on CM abelian surfaces |
Abstract: In this talk, I will discuss an interesting Hermitian form structure on the space of special endormorphisms of a CM abelian surface, and how to use it make a moduli problem and prove an arithmetic Siegel–Weil formula over a real quadratic field. This is a joint work with Ben Howard. |
February 23
Christelle Vincent (Madison) |
Title: Drinfeld modular forms |
Abstract: We will begin by introducing the Drinfeld setting, and in particular Drinfeld modular forms and their connection to the geometry of Drinfeld modular curves. We will then present some results about Drinfeld modular forms that we obtained in the process of computing certain geometric points on Drinfeld modular curves. More precisely, we will talk about Drinfeld modular forms modulo P, for P a prime ideal in F_{q}[T ], and about Drinfeld quasi-modular forms. |
March 1
Shamgar Gurevich (Madison) |
Title: Computing the Matched Filter in Linear Time |
Abstract: In the digital radar problem we design a function (waveform) S(t) in the Hilbert space H = C(Z/p) of complex valued functions on Z/p = {0, ..., p − 1}, the integers modulo a prime number p ≫ 0. We transmit the function S(t) using the radar to the object that we want to detect. The wave S(t) hits the object, and is reflected back via the echo wave R(t) in H, which has the form R(t) = exp{2πiωt/p}⋅S(t+τ) + W(t), where W(t) in H is a white noise, and τ, ω in Z/p, encode the distance from, and velocity of, the object. Problem (digital radar problem) Extract τ, ω from R and S. In the lecture I first introduce the classical matched filter (MF) algorithm that suggests the 'traditional' way (using fast Fourier transform) to solve the digital radar problem in order of p^{2}⋅log(p) operations. I will then explain how to use techniques from group representation theory and arithmetic to design (construct) waveforms S(t) which enable us to introduce a fast matched filter (FMF) algorithm, that we call the "flag algorithm", which solves the digital radar problem in a much faster way of order of p⋅log(p) operations. I will demonstrate additional applications to mobile communication, and global positioning system (GPS). This is a joint work with A. Fish (Math, Madison), R. Hadani (Math, Austin), A. Sayeed (Electrical Engineering, Madison), and O. Schwartz (Electrical Engineering and Computer Science, Berkeley). The lecture is suitable for general math/engineering audience. |
March 8
Zev Klagsbrun (Madison) |
Title: Erdős–Kac Type Theorems |
Abstract: In its most popular formulation, the Erdős–Kac Theorem gives a distribution on the number of distinct primes factors (ω(n)) of the numbers up to N. Variants of the Erdős–Kac Theorem yield distributions on additive functions in a surprising number of settings. This talk will outline the basics of the theory by focusing on some results of Granville and Soundararajan that allow one to easily prove Erdős–Kac type results for a variety of problems as well as present a recent result of my own using the Granville and Soundararajan framework. The lecture is suitable for general math audience. |
March 15
Yongqiang Zhao (Madison) |
Title: On the Roberts conjecture |
Abstract: Let N(X) = #{K | [K:Q] = 3, disc(K) ≤ X} be the counting function of cubic fields of bounded discriminant. The Roberts Conjecture is about the second term of this counting function. This conjecture was thought to be hard and dormant for some time. However, recently, four very different approaches to this problem were developed independently by Bhargava, Shankar and Tsimerman, Hough, Taniguchi and Thorne, and myself. In this talk, I will mention the first three approaches very briefly, then focus on my results on the function field case. I will give an outline of the proof. If time permits, I will indicate how the geometry feeds back to the number field case, in particular, how one could possibly define a new invariant for cubic fields. |
March 22
Paul Terwilliger (Madison) |
Title: Introduction to tridiagonal pairs |
Abstract: In this talk we consider a linear algebraic object called a tridiagonal pair. This object originated in algebraic graph theory, and has connections to orthogonal polynomials and representation theory. In our discussion we aim at a general mathematical audience; no prior experience with the above topics is assumed. Our main results are joint work with Tatsuro Ito and Kazumasa Nomura. The concept of a tridiagonal pair is best explained by starting with a special case called a Leonard pair. Let F denote a field, and let V denote a vector space over F with finite positive dimension. By a Leonard pair on V we mean a pair of linear transformations A : V → V and A^{∗} : V → V that satisfy the following two conditions:
We recall that a tridiagonal matrix is irreducible whenever each entry on the superdiagonal is nonzero and each entry on the subdiagonal is nonzero. The name "Leonard pair" is motivated by a connection to a 1982 theorem of the combinatorialist Doug Leonard involving the q-Racah and related polynomials of the Askey scheme. Leonard’s theorem was heavily influenced by the work of Eiichi Bannai and Tatsuro Ito on P- and Q- polynomial association schemes, and the work of Richard Askey on orthogonal polynomials. These works in turn were influenced by the work of Philippe Delsarte on coding theory, dating from around 1973. The central result about Leonard pairs is that they are in bijection with the orthogonal polynomials that make up the terminating branch of the Askey scheme. This branch consists of the q-Racah, q-Hahn, dual q-Hahn, q-Krawtchouk, dual q-Krawtchouk, quantum q-Krawtchouk, affine q-Krawtchouk, Racah, Hahn, dual-Hahn, Krawtchouk, and the Bannai/Ito polynomials. The bijection makes it possible to develop a uniform theory of these polynomials starting from the Leonard pair axiom, and we have done this over the past several years. A tridiagonal pair is a generalization of a Leonard pair and defined as follows. Let V denote a vector space over F with finite positive dimension. A tridiagonal pair on V is a pair of linear transformations A : V → V and A^{∗} : V → V that satisfy the following four conditions:
It turns out that d = δ and this common value is called the diameter of the pair. A Leonard pair is the same thing as a tridiagonal pair for which the eigenspaces V and V^{∗} all have dimension 1. Tridiagonal pairs arise naturally in the theory of P- and Q-polynomial association schemes, in connection with irreducible modules for the subconstituent algebra. They also appear in recent work of Pascal Baseilhac on the Ising model and related structures in statistical mechanics. In this talk we will summarize the basic facts about a tridiagonal pair, describing features such as the eigenvalues, dual eigenvalues, shape, tridiagonal relations, split decomposition, and parameter array. We will then focus on a special case said to be sharp and defined as follows. Referring to the tridiagonal pair A, A^{∗} in the above definition, it turns out that for 0 ≤ i ≤ d the dimensions of V_{i}, V_{d−i}, V^{*}_{i}, V^{*}_{d−i} coincide; the pair A, A^{∗} is called sharp whenever V_{0} has dimension 1. It is known that if F is algebraically closed then A, A^{∗} is sharp. In our main result we classify the sharp tridiagonal pairs up to isomorphism. |
March 29
David P. Roberts (U. Minnesota Morris) | ||||||
Title: Lightly ramified number fields with Galois group S.M_{12}.A | ||||||
Abstract: Two of the most important invariants of an irreducible polynomial f(x) ∈ Z[x ] are its Galois group G and its field discriminant D. The inverse Galois problem asks one to find a polynomial f(x) having any prescribed Galois group G. Refinements of this problem ask for D to be small in various senses, for example of the form ± p^{a} for the smallest possible prime p. The talk will discuss this problem in general, with a focus on the technique of specializing three-point covers for solving instances of it. Then it will pursue the cases of the Mathieu group M_{12}, its automorphism group M_{12}.2, its double cover 2.M_{12}, and the combined extension 2.M_{12}.2. Among the polynomials found is
with e = 11. This polynomial has Galois group G = 2.M_{12}.2 and field discriminant 11^{88}. It makes M_{12} the first of the twenty-six sporadic simple groups Γ known to have a polynomial with Galois group G involving Γ and field discriminant D the power of a single prime dividing |Γ |. |
April 12
Chenyan Wu (Minnesota) |
Title: tba |
Abstract: tba |
April 19
Robert Guralnick (U. Southern California) |
Title: A variant of Burnside and Galois representations which are automorphic |
Abstract: Wiles, Taylor, Harris and others used the notion of a big representation of a finite group to show that certain representations are automorphic. Jack Thorne recently observed that one could weaken this notion of bigness to get the same conclusions. He called this property adequate. An absolutely irreducible representation V of a finite group G in characteristic p is called adequate if G has no p-quotients, the dimension of V is prime to p, V has non-trivial self extensions and End(V) is generated by the linear span of the elements of order prime to p in G. If G has order prime to p, all of these conditions hold—the last condition is sometimes called Burnside's Lemma. We will discuss a recent result of Guralnick, Herzig, Taylor and Thorne showing that if p > 2 dim V + 2, then any absolutely irreducible representation is adequate. We will also discuss some examples showing that the span of the p'-elements in End(V) need not be all of End(V). |
April 26
Frank Thorne (U. South Carolina) |
Title: tba |
Abstract: tba |
May 3
Alina Cojocaru (U. Illinois at Chicago) |
Title: tba |
Abstract: tba |
May 10
Samit Dasgupta (UC Santa Cruz) |
Title: tba |
Abstract: tba |
Organizer contact information
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