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DTSTART;TZID=America/Los_Angeles:20220906T121500
DTEND;TZID=America/Los_Angeles:20220906T131000
DTSTAMP:20260617T170308
CREATED:20220811T001752Z
LAST-MODIFIED:20220902T173415Z
UID:2779-1662466500-1662469800@colleges.claremont.edu
SUMMARY:Monodromy groups of Belyi Lattes maps (Edray Goins\, Pomona College)
DESCRIPTION:An elliptic curve $ E: y^2 + a_1 \\, x \\, y + a_3 \\, y = x^3 + a_2 \\, x^2 + a_1 \\, x + a_6 $ is a cubic equation which has two curious properties: (1) the curve is nonsingular\, so that we can draw tangent lines to every point $ P = (x\,y) $ on the curve; and (2) the collection of complex points\, namely $ E(\mathbb C) $\, forms an abelian group under a certain binary operation $ \bigoplus: E(\mathbb C) \times E(\mathbb C) \to E(\mathbb C) $.   In particular\, for every positive integer $N$\, the map $ P \mapsto [N] P $ which adds a point $ P \in E(\mathbb C) $ to itself $N$ times is a group homomorphism.   A rational map $\gamma: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C) $ from the Riemann Sphere to itself is said to be a Latt\`{e}s Map if there are “well-behaved” maps $ \phi: E(\mathbb C) \to \mathbb P^1(\mathbb C) $ and $\psi: E(\mathbb C) \to E(\mathbb C) $ such that $\gamma \circ \phi = \phi \circ \psi$.  We are interested in those Latt\`{e}s Maps $\gamma$ which are also Bely\u{\i} Maps\, that is\, the only critical values are $ 0 $\, $ 1 $\, and $ \infty $.  Work of Zeytin classifies all such maps: For example\, if $ E: y^2 = x^3 + 1 $ then $ \phi: (x\,y) \mapsto (y+1)/2 $ while $\psi = [N] $ for some positive integer $N$.\n\nWe would like to know more about Bely\u{\i} Latt\`{e}s Maps $\gamma$.  What can we say about such maps?  What are their Dessin d’Enfants?  In some cases\, this is a bipartite graph with $ 3 \\, N^2 $ vertices.  What are their monodromy groups? Sometimes this is a group of size $ 3 \\, N^2 $.  In this talk\, we explain the complete answers to these questions\, exploiting the relationship between fundamental groups of Riemann surfaces and Galois groups of function fields.  This work is conducted as part of the Pomona Research in Mathematics Experience (DMS-2113782).
URL:https://colleges.claremont.edu/ccms/event/monodromy-groups-of-belyi-lattes-maps-edray-goins-pomona-college/
LOCATION:Estella 1021 (Emmy Noether Room)\, Pomona College\, Claremont\, CA\, 91711\, United States
CATEGORIES:Algebra / Number Theory / Combinatorics Seminar
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DTSTART;TZID=America/Los_Angeles:20220913T121500
DTEND;TZID=America/Los_Angeles:20220913T131000
DTSTAMP:20260617T170308
CREATED:20220902T001706Z
LAST-MODIFIED:20220906T231347Z
UID:2814-1663071300-1663074600@colleges.claremont.edu
SUMMARY:Kriz's theorem via dynamics of linear operators (Yunied Puig de Dios\, CMC)
DESCRIPTION:The existence of a set $A\subset \N_0$ of positive upper Banach density such that $A-A:=\{m-n:m\, n\in A\, m>n\}$ does not contain a set of the form $S-S$ with $S$ a piecewise syndetic is in essence the content of a popular result due to K\v r\'{i}\v z in 1987. Since then at least four different proofs of this result have been given\, and all of them give basically the example originally exhibited by K\v r\'{i}\v z when viewed appropriately. We obtain a generalization of K\v r\'{i}\v z’s result. Our approach differs completely from the previous ones\, as this would be the first proof of K\v r\'{i}\v z’s Theorem which does not rely on Lov\'{a}sz’s Theorem for chromatic numbers of Kneser graphs. Furthermore\, it is done via operator theory\, namely using dynamics of bounded linear operators on infinite-dimensional complex separable Banach spaces. As a consequence\, our example is genuinely different from the one exhibited  originally by K\v r\'{i}\v z.
URL:https://colleges.claremont.edu/ccms/event/krizs-theorem-via-dynamics-of-linear-operators-yunied-puig-de-dios-cmc/
LOCATION:Davidson Lecture Hall\, CMC\, 340 E 9th St\, Claremont\, CA\, 91711\, United States
CATEGORIES:Algebra / Number Theory / Combinatorics Seminar
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BEGIN:VEVENT
DTSTART;TZID=America/Los_Angeles:20220920T121500
DTEND;TZID=America/Los_Angeles:20220920T131000
DTSTAMP:20260617T170308
CREATED:20220811T002022Z
LAST-MODIFIED:20220906T231455Z
UID:2780-1663676100-1663679400@colleges.claremont.edu
SUMMARY:Arithmetical structures (Luis Garcia Puente\, Colorado College)
DESCRIPTION:An arithmetical structure on a finite\, connected graph G without loops is given by an assignment of positive integers to the vertices such that\, at each vertex\, the integer there is a divisor of the sum of the integers at adjacent vertices\, counted with multiplicity if the graph is not simple. Alternatively\,  an arithmetical structure on G is a pair  of positive integer vectors (d\,r) such that  Mr = 0\, where M = diag(d) – A  is a square matrix whose diagonal entries are given by the vector d\, and whose off-diagonal elements are given by the negative adjacency matrix of G. Arithmetical structures were first introduced by Lorenzini in 1989; matrices of the form (diag(d) – A) arise in algebraic geometry as intersection matrices of degenerating curves.  However\, they also naturally appear in the context of algebraic graph theory as matrices of the form  (diag(d) – A)  generalize the Laplacian matrix of a graph.\n\nIn this talk\, I will give an introduction to the topic. We will discuss some combinatorial\, structural and computational aspects of arithmetical structures. In particular\, we will count the number of distinct arithmetical structures on certain graph families such as path\, cycle\, complete and bident graphs. For paths\, we will show that arithmetical structures are enumerated by the Catalan numbers. For cycles\, we prove that arithmetical structures are enumerated by the binomial coefficients C(2n-1\,n-1).  We will also discuss results about the associated critical group of an arithmetical structure\, i.e.\,  the cokernel of the matrix M.   This talk will be accessible to undergraduate students with some knowledge of linear algebra and discrete mathematics.
URL:https://colleges.claremont.edu/ccms/event/antc-talk-luis-garcia-puente-colorado-college/
LOCATION:Davidson Lecture Hall\, CMC\, 340 E 9th St\, Claremont\, CA\, 91711\, United States
CATEGORIES:Algebra / Number Theory / Combinatorics Seminar
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DTSTART;TZID=America/Los_Angeles:20220927T121500
DTEND;TZID=America/Los_Angeles:20220927T131000
DTSTAMP:20260617T170308
CREATED:20220906T160640Z
LAST-MODIFIED:20220922T053209Z
UID:2836-1664280900-1664284200@colleges.claremont.edu
SUMMARY:Spinning switches on a wreath product (Peter Kagey\, HMC)
DESCRIPTION:This talk discusses a puzzle called “Spinning Switches\,” based on a problem popularized by Martin Gardner in his February 1979 column of “Mathematical Games”. This puzzle can be generalized to a two-player game on a finite wreath products. This talk will provide a classification of several families of these generalized puzzles\, including a full classification in the case of Abelian groups.
URL:https://colleges.claremont.edu/ccms/event/antc-talk-peter-kagey-hmc/
LOCATION:Davidson Lecture Hall\, CMC\, 340 E 9th St\, Claremont\, CA\, 91711\, United States
CATEGORIES:Algebra / Number Theory / Combinatorics Seminar
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