Date: Monday, January 30, 2012
Location: 3088 East Hall (4:00 PM to 5:00 PM)
Title: Q-analogues, Part II: A "Steampunk" version of Fleck's congruence.
Abstract: Some binomial coefficient identities are proper/strong in the sense that they read as "THIS equals THAT," where THIS could look fairly complicated, while THAT strikes us as non-intimidating or elementary; other identities are improper/weak, not reading as "THIS equals THAT," such as modular congruences. For instance, compare (1) "the alternating sum across row n>0 of Pascal's triangle equals 0" with (2) Lucas' theorem. As noted in the previous seminar, the q-binomial coefficients are polynomial cousins of traditional binomial
coefficients. If you insert these relatives into classical binomial coefficient identities, perturbing the latter results, then q-analogues of those identities can be unveiled; for example, (1) becomes the Gaussian Formula, and (2) the q-Lucas theorem. Let's call a result of this process a "q-binomial perturbation."
In 1913, Fleck proved that if p is any prime integer, there is a simple formula for bounding (from below) the multiplicity of p in a "modularly-defined" sum of signed binomial coefficients. This result is known as Fleck's congruence. Algebraist Andrew Schultz and I manufactured a q-binomial perturbation of Fleck's sum, and we asked if this perturbation admits a congruence property like Fleck's congruence. It does. In fact, our q-binomial congruence synthesizes each of Fleck's result, the Gaussian Formula and a third binomial coefficient identity. This new congruence (along with some context and questions, time permitting) will be the subject of my talk. I will aim to provide sufficient background for the talk to be student-friendly (even for fellow first-years!), and to embed some humor as well.
Speaker: Robert Walker
Institution: U Michigan
Event Organizer:
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