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DTSTART;TZID=America/Los_Angeles:20260921T161500
DTEND;TZID=America/Los_Angeles:20260921T171500
DTSTAMP:20260913T231319Z
CREATED:20260913T231319Z
LAST-MODIFIED:20260913T231319Z
UID:10002329-1790007300-1790010900@colleges.claremont.edu
SUMMARY:Number Theory meets Differential Equations: Using Homotopy Continuation to Compute Shabat Polynomials
DESCRIPTION:A Belyĭ pair \(\bigl( S\,\\, z(w) \bigr)\) is a compact\, connected Riemann surface \(S\) together with a rational function \(z \colon S \to \mathbb{P}^1(\mathbb{C})\) which has at most three critical values \(\bigl\{ 0\,\\, 1\,\\, \infty \bigr\}\). The preimage \(z^{-1}\bigl( [0\,1] \bigr) \subseteq S \hookrightarrow \mathbb{R}^3\) is a bipartite graph called a Dessin d’Enfant. When \(S = \mathbb{P}^1(\mathbb{C}) \simeq S^2(\mathbb{R})\) is the Riemann sphere\, a tree—namely\, a planar graph with one face—can be realized as the Dessin d’Enfant of a polynomial. Named in honor of Georgii Borisovich Shabat\, a Belyĭ map \(z \colon S \to \mathbb{P}^1(\mathbb{C})\) which is also a polynomial is called a Shabat polynomial. \nIn this project\, we seek to compute more examples of Shabat polynomials. We introduce a trick which reduces the computation of such polynomials to finding solutions to a system of polynomial equations. We explore computing these solutions via two methods: (1) Newton’s method\, to find approximate values\, and (2) the homotopy continuation method\, to deform known solutions into desired yet unknown ones. Simply put\, in this talk we give an introduction to using differential equations to answer questions in number theory. \nThis project was completed through the Research Experience for Undergraduate Faculty (REUF) at AIM/ICERM in Summers 2025 and 2026 (DMS‑2015462).
URL:https://colleges.claremont.edu/ccms/event/number-theory-meets-differential-equations-using-homotopy-continuation-to-compute-shabat-polynomials/
LOCATION:Emmy Noether Room\, Estella 1021\, Pomona College\,\, 610 N. College Ave.\, Claremont\, CA\, 91711\, United States
CATEGORIES:Applied Math Seminar
ORGANIZER;CN="Ami Radunskaya":MAILTO:aradunskaya@pomona.edu
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DTSTART;TZID=America/Los_Angeles:20260922T121500
DTEND;TZID=America/Los_Angeles:20260922T131000
DTSTAMP:20260917T215711Z
CREATED:20260917T215711Z
LAST-MODIFIED:20260917T215711Z
UID:10002333-1790079300-1790082600@colleges.claremont.edu
SUMMARY:Hilbert basis for conical semigroups and indecomposable elements (Lenny Fukshansky\, CMC)
DESCRIPTION:The unique minimal generating set for a semigroup of lattice points in an n-dimensional polyhedral cone is called its Hilbert basis. In the case when the cone is spanned by lattice points\, this Hilbert basis is finite\, however its cardinality can be much larger than n. Nevertheless\, a conjecture of Sebö states that each point in this semigroup can be spanned by no more than n elements of the Hilbert basis. We will brief discuss this conjecture and its current status\, and then shift to the situation where the cone in question is the positive orthant in R^n. In this case the Hilbert basis consists of the so-called indecomposable elements\, notation coming from the theory of universal quadratic forms over algebraic number fields. We will classify lattices that have finite versus infinite Hilbert basis of indecomposables. We will then discuss an analogue of Sebö’s conjecture for indecomposables in planar lattices and finish by presenting a counting estimate on the number of indecomposables of bounded norm in this 2-dimensional case. Joint work with Filiana Kostopoulou.
URL:https://colleges.claremont.edu/ccms/event/hilbert-basis-for-conical-semigroups-and-indecomposable-elements-lenny-fukshansky-cmc/
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:20260922T161500
DTEND;TZID=America/Los_Angeles:20260922T171500
DTSTAMP:20260916T004846Z
CREATED:20260916T004846Z
LAST-MODIFIED:20260916T004846Z
UID:10002332-1790093700-1790097300@colleges.claremont.edu
SUMMARY:Claremont Topology Seminar: Sam Nelson (CMC)
DESCRIPTION:Speaker: Sam Nelson (CMC) \nTitle: Permutation Jones Polynomials \nAbstract: The Jones Polynomial is an important invariant of knots with applications in physics\, quantum computation and more. In this talk I will introduce a family of generalizations of the Jones polynomial associated to permutations of finite sets and discuss their properties. \n_______ \nWe especially welcome all undergraduates and graduate students to attend topology seminar! \nThe Seminar Room Information will be added soon.
URL:https://colleges.claremont.edu/ccms/event/claremont-topology-seminar-sam-nelson-cmc-2/
LOCATION:CA
CATEGORIES:Topology Seminar
ORGANIZER;CN="Bahar Acu":MAILTO:Bahar_Acu@pitzer.edu
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DTSTART;TZID=America/Los_Angeles:20260923T161500
DTEND;TZID=America/Los_Angeles:20260923T171500
DTSTAMP:20260921T164909Z
CREATED:20260921T164635Z
LAST-MODIFIED:20260921T164909Z
UID:10002336-1790180100-1790183700@colleges.claremont.edu
SUMMARY:CCMS Colloquium: The Stochastic Logistic Map: Theory and Applications — Kimberly Ayers (CSU San Marcos)
DESCRIPTION:Flyer for Kim’s talk. \nAbstract: The logistic map is one of the most famous examples in dynamical systems. Originally introduced as a simple model of population growth\, it exhibits a surprisingly rich range of behavior: as a single parameter varies\, stable equilibria give way to periodic orbits and eventually to chaos. But in real populations\, quantities such as reproductive and death rates are unlikely to remain constant over time. What happens when we allow this parameter to vary randomly from one generation to the next? \nIn this talk\, we explore a stochastic version of the logistic map in which the parameter is chosen randomly at each iteration. Introducing randomness changes the questions we ask: rather than studying individual fixed points or periodic orbits\, we investigate probability distributions that describe the long-term behavior of the system. We will see how tools from dynamical systems and Markov processes can be used to show that\, under appropriate conditions\, the stochastic system settles into a unique long-term statistical behavior that is independent of its initial state. We conclude with simulations illustrating this invariant distribution and discuss several questions about how the system’s behavior changes as the amount and type of randomness vary. \n\n\n\nFor questions about the event\, please contact your CCMS Colloquium chairs: \nProf. Robert Cass (Robert.Cass@ClaremontMcKenna.edu) or Prof. Qidi Peng (Qidi.Peng@cgu.edu) \nFor logistical questions\, contact Maxwell Kooiker (Maxwell.Kooiker@cgu.edu)\, the CCMS Colloquium Assistant.
URL:https://colleges.claremont.edu/ccms/event/ccms-colloquium-the-stochastic-logistic-map-theory-and-applications-kimberly-ayers-csu-san-marcos/
LOCATION:Burkle 16\, Claremont Graduate University\, 1021 N. Dartmouth Ave.\, Claremont\, CA\, 91711\, United States
CATEGORIES:Colloquium
ORGANIZER;CN="Qidi Peng":MAILTO:Qidi.Peng@cgu.edu
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