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DTSTART;TZID=America/Los_Angeles:20260303T121500
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DTSTAMP:20260421T172312
CREATED:20260302T023221Z
LAST-MODIFIED:20260302T023221Z
UID:4013-1772540100-1772543400@colleges.claremont.edu
SUMMARY:On a new version of Siegel’s lemma  (Lenny Fukshansky\, CMC)
DESCRIPTION:The classical Siegel’s lemma (1929) asserts the existence of a nontrivial integer solution to an underdetermined integer homogeneous linear system\, whose “size” is small as compared to the size of the coefficients of the system. Far-reaching generalizations of this theorem\, producing a full basis for the solution space\, were obtained over number fields by Bombieri & Vaaler (1983)\, and over the field of algebraic numbers by Roy & Thunder (1996)\, where the “size” was measured by a height function. We obtain a new version of Siegel’s lemma\, bridging the Bombieri & Vaaler and Roy & Thunder results in two ways: (1) our basis lies over a fixed number field as in Bombieri & Vaaler’s theorem; (2) our height-bound does not depend on the number field in question as in Roy & Thunder’s theorem. Our result does not imply the previously established ones and is not implied by them\, and our basis has some additional interesting properties. Our method is quite different from the previous ones\, using only linear algebra. Joint work with Max Forst.
URL:https://colleges.claremont.edu/ccms/event/on-a-new-version-of-siegels-lemma-lenny-fukshansky-cmc/
LOCATION:Estella 2099
CATEGORIES:Algebra / Number Theory / Combinatorics Seminar
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BEGIN:VEVENT
DTSTART;TZID=America/Los_Angeles:20260310T123000
DTEND;TZID=America/Los_Angeles:20260310T131000
DTSTAMP:20260421T172312
CREATED:20260119T182717Z
LAST-MODIFIED:20260307T023545Z
UID:3961-1773145800-1773148200@colleges.claremont.edu
SUMMARY:Hecke algebras and motives (Robert Cass\, CMC)
DESCRIPTION:Hecke algebras play a central role in both number theory and representation theory. While some Hecke algebras have explicit descriptions in terms of generators and relations\, others are understood through structure constants that encode multiplicities in tensor products of representations. In this talk\, I will discuss several projects with Thibaud van den Hove and Jakob Scholbach aimed at using geometry and motives to give a uniform categorification of Hecke algebras. Along the way\, we will encounter the geometric Satake equivalence\, Gaitsgory’s central functor\, and Iwahori-Whittaker models.
URL:https://colleges.claremont.edu/ccms/event/antc-talk-robert-cass-cmc-2/
LOCATION:Estella 2099
CATEGORIES:Algebra / Number Theory / Combinatorics Seminar
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BEGIN:VEVENT
DTSTART;TZID=America/Los_Angeles:20260324T121500
DTEND;TZID=America/Los_Angeles:20260324T131000
DTSTAMP:20260421T172312
CREATED:20260209T235439Z
LAST-MODIFIED:20260226T211019Z
UID:3991-1774354500-1774357800@colleges.claremont.edu
SUMMARY:Computing certificates for complete positivity (Achill Schürmann\, University of Rostock)
DESCRIPTION:A key problem in computer proofs based on solutions from copositive optimization\, is checking whether or not a given quadratic form is completely positive or not. In this talk we describe the first known algorithm for arbitrary rational input. It is based on a suitable adaption of Voronoi’s Algorithm and the underlying theory from positive definite to copositive quadratic forms. We observe several similarities with the classical theory\, but also some differences\, in particular for three and more variables. A key element and currently the main bottleneck in our algorithm is an adapted shortest vector computation\, asking for all nonnegative integer vectors attaining the copositive minimum of a given copositive quadratic form. \n(based on joint work with Valentin Dannenberg\, Alexander Oertel\, Mathieu Dutour Sikiric and Frank Vallentin)
URL:https://colleges.claremont.edu/ccms/event/antc-talk-achill-schurmann-university-of-rostock/
LOCATION:Estella 2099
CATEGORIES:Algebra / Number Theory / Combinatorics Seminar
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DTSTART;TZID=America/Los_Angeles:20260331T121500
DTEND;TZID=America/Los_Angeles:20260331T131000
DTSTAMP:20260421T172312
CREATED:20260106T162953Z
LAST-MODIFIED:20260321T145212Z
UID:3941-1774959300-1774962600@colleges.claremont.edu
SUMMARY:Central moments of autocorrelation demerit factors of binary sequences (Daniel Katz\, CSUN)
DESCRIPTION:A low autocorrelation binary sequence of length $\ell$ is an $\ell$-tuple of $+1$s and $-1$s that does not strongly resemble any translate of itself.  Such sequences are used in communications and remote sensing for synchronization and ranging\, where translation represents time delay.  A single number that indicates how good a sequence is for such purposes\, called the merit factor\, was introduced by Golay.  Its reciprocal is the demerit factor\, which is more natural to analyze due to its connection with norms of polynomials on the complex unit circle.  We consider the uniform probability measure on the $2^\ell$ binary sequences of length $\ell$ and investigate the distribution of the demerit factors of these sequences.  Sarwate and Jedwab have respectively calculated the mean and variance of this distribution. For each positive integer $p$\, we derive a formula for the $p$th central moment of the demerit factor for the binary sequences of length $\ell$; this is $\ell^{-2 p}$ times a quasipolynomial function of $\ell$.  The derivations rely on new combinatorial techniques\, assisted by group theory and Ehrhart theory\, and show that all the central moments are strictly positive for $p\geq 2$ and $\ell \geq 4$. Jedwab’s formula for variance is confirmed\, and we go beyond previous results by also deriving an exact formula for the skewness (by hand) and for the kurtosis and the fifth moment (by computer).  We obtain asymptotic values for all central moments in the limit as the length $\ell$ of the sequences tends to infinity.
URL:https://colleges.claremont.edu/ccms/event/antc-seminar-daniel-katz-csun/
LOCATION:Estella 2099
CATEGORIES:Algebra / Number Theory / Combinatorics Seminar
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