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DTSTART;TZID=America/Los_Angeles:20211005T123000
DTEND;TZID=America/Los_Angeles:20211005T132000
DTSTAMP:20260406T100508
CREATED:20210906T215040Z
LAST-MODIFIED:20210906T215040Z
UID:2301-1633437000-1633440000@colleges.claremont.edu
SUMMARY:Critical points of toroidal Belyi maps (Edray Goins\, Pomona)
DESCRIPTION:A Belyi map $\beta: \mathbb{P}^1(\mathbb{C}) \to \mathbb{P}^1(\mathbb{C})$ is a rational function with at most three critical values; we may assume these values are $\{ 0\, \\, 1\, \\, \infty \}$.  Replacing $\mathbb{P}^1$ with an elliptic curve $E: \ y^2 = x^3 + A \\, x + B$\, there is a similar definition of a Belyi map $\beta: E(\mathbb{C}) \to \mathbb{P}^1(\mathbb{C})$.  Since $E(\mathbb{C}) \simeq \mathbb T^2(\mathbb {R})$ is a torus\, we call $(E\, \beta)$ a Toroidal \Belyi pair. \n\n\nThere are many examples of Belyi maps $\beta: E(\mathbb{C}) \to \mathbb P^1(\mathbb{C})$ associated to elliptic curves; several can be found online at LMFDB. Given such a Toroidal Belyi map of degree $N$\, the inverse image $G = \beta^{-1} \bigl( \{ 0\, \\, 1\, \\, \infty \} \bigr)$ is a set of $N$ elements which contains the critical points of the \Belyi map. In this project\, we investigate when $G$ is contained in $E(\mathbb{C})_{\text{tors}}$. \n\n\nThis is work done as part of the Pomona Research in Mathematics Experience (NSA H98230-21-1-0015).
URL:https://colleges.claremont.edu/ccms/event/critical-points-of-toroidal-belyi-maps-edray-goins-pomona/
LOCATION:On Zoom
CATEGORIES:Algebra / Number Theory / Combinatorics Seminar
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DTSTART;TZID=America/Los_Angeles:20211012T123000
DTEND;TZID=America/Los_Angeles:20211012T132000
DTSTAMP:20260406T100508
CREATED:20210831T181118Z
LAST-MODIFIED:20211006T002703Z
UID:2267-1634041800-1634044800@colleges.claremont.edu
SUMMARY:New norms on matrices induced by polynomials (Angel Chavez\, Pomona)
DESCRIPTION:The complete homogeneous symmetric (CHS) polynomials can be used to define a  family of norms on Hermitian matrices. These ‘CHS norms’ are peculiar in the sense that they depend only on the eigenvalues of a matrix and not its singular values (as opposed to the Ky-Fan and Schatten norms). We will first give a general overview behind the construction of these norms (as well as their extensions to all n x n complex matrices). The construction and validation of these norms will take us on a tour of probability theory\, convexity analysis\, partition combinatorics and trace polynomials in noncommuting variables. We then discuss open problems and potential for future work. This talk is based on joint work with Konrad Aguilar\, Stephan Garcia and Jurij Volčič.
URL:https://colleges.claremont.edu/ccms/event/antc-seminar-angel-chavez-pomona/
LOCATION:On Zoom
CATEGORIES:Algebra / Number Theory / Combinatorics Seminar
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DTSTART;TZID=America/Los_Angeles:20211026T123000
DTEND;TZID=America/Los_Angeles:20211026T132000
DTSTAMP:20260406T100508
CREATED:20210822T191915Z
LAST-MODIFIED:20211024T022430Z
UID:2210-1635251400-1635254400@colleges.claremont.edu
SUMMARY:Damerell's theorem: p-adic version\, supersingular case (Pavel Guerzhoy\, University of Hawaii)
DESCRIPTION:It is widely believed that Weierstrass ignored Eisenstein’s theory of elliptic functions and developed an alternative treatment\, which is now standard\, because of a convergence issue. In particular\, the Eisenstein series of weight two does not converge absolutely while Eisenstein’s theory assigned a value to this series.\n\nIt is now well-known that the quantity which Eisentsein assigned to this series is not only correct\, but it has interesting interpretations and attracted much attention. It has been proved by Damerell in 1970 that this quantity is an algebraic number if the underlying elliptic curve has complex multiplication.\n\nIn 1976\, N. Katz interpreted Damerell’s theorem in terms of DeRham cohomology; that allowed for a p-adic approach to this algebraic number. This p-adic version of Damerell’s theorem was instrumental in Katz’s theory of p-adic modular forms and p-adic L-functions of CM-fields. The approach\, by design\, works for those primes which split in the CM-field.\n\nIn this talk\, we offer a modification of Katz’ p-adic approach to the weight two Eisenstein series which works uniformly well for all primes of good reduction\, both inert and splitting in the CM-field.
URL:https://colleges.claremont.edu/ccms/event/antc-seminar-pavel-guerzhoy-university-of-hawaii-2/
LOCATION:On Zoom
CATEGORIES:Algebra / Number Theory / Combinatorics Seminar
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