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DTSTART;TZID=America/New_York:20220301T090000
DTEND;TZID=America/New_York:20220301T100000
DTSTAMP:20260727T171611
CREATED:20240214T045733Z
LAST-MODIFIED:20240304T060140Z
UID:10002534-1646125200-1646128800@cmsa.fas.harvard.edu
SUMMARY:Rational Polypols
DESCRIPTION:Abstract: Eugene Wachspress introduced polypols as real bounded semialgebraic sets in the plane that generalize polygons. He aimed to generalize barycentric coordinates from triangles to arbitrary polygons and further to polypols. For this\, he defined the adjoint curve of a rational polypol. In the study of scattering amplitudes in physics\, positive geometries are real semialgebraic sets together with a rational canonical form. We combine these two worlds by providing an explicit formula for the canonical form of a rational polypol in terms of defining equations of the adjoint curve and the facets of the polypol. For the special case of polygons\, we show that the adjoint curve is hyperbolic and provide an explicit description of its nested ovals. Finally\, we discuss the map that associates the adjoint curve to a given rational polypol\, in particular the cases where this map is finite. For instance\, using monodromy we find that a general quartic curve is the adjoint of 864 heptagons. \nThis talk is based on joint work with R. Piene\, K. Ranestad\, F. Rydell\, B. Shapiro\, R. Sinn\,  M.-S. Sorea\, and S. Telen.
URL:https://cmsa.fas.harvard.edu/event/3-1-2022-combinatorics-physics-and-probability-seminar/
CATEGORIES:Combinatorics Physics and Probability
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/CMSA-Combinatorics-Physics-and-Probability-Seminar-3.01.2022.png
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BEGIN:VEVENT
DTSTART;TZID=America/New_York:20220308T090000
DTEND;TZID=America/New_York:20220308T100000
DTSTAMP:20260727T171611
CREATED:20240214T064241Z
LAST-MODIFIED:20240304T090552Z
UID:10002550-1646730000-1646733600@cmsa.fas.harvard.edu
SUMMARY:Greedy maximal independent sets via local limits
DESCRIPTION:Abstract: The random greedy algorithm for finding a maximal independent set in a graph has been studied extensively in various settings in combinatorics\, probability\, computer science\, and chemistry. The algorithm builds a maximal independent set by inspecting the graph’s vertices one at a time according to a random order\, adding the current vertex to the independent set if it is not connected to any previously added vertex by an edge. \nIn this talk\, I will present a simple yet general framework for calculating the asymptotics of the proportion of the yielded independent set for sequences of (possibly random) graphs\, involving a valuable notion of local convergence. I will demonstrate the applicability of this framework by giving short and straightforward proofs for results on previously studied families of graphs\, such as paths and various random graphs\, and by providing new results for other models such as random trees. \nIf time allows\, I will discuss a more delicate (and combinatorial) result\, according to which\, in expectation\, the cardinality of a random greedy independent set in the path is no larger than that in any other tree of the same order. \nThe talk is based on joint work with Michael Krivelevich\, Tamás Mészáros and Clara Shikhelman.
URL:https://cmsa.fas.harvard.edu/event/3-8-2022-combinatorics-physics-and-probability-seminar/
CATEGORIES:Combinatorics Physics and Probability
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/CMSA-Combinatorics-Physics-and-Probability-Seminar-3.08.2022.png
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BEGIN:VEVENT
DTSTART;TZID=America/New_York:20220315T090000
DTEND;TZID=America/New_York:20220315T100000
DTSTAMP:20260727T171611
CREATED:20240214T065321Z
LAST-MODIFIED:20240304T085303Z
UID:10002551-1647334800-1647338400@cmsa.fas.harvard.edu
SUMMARY:Moduli space of tropical curves\, graph Laplacians and physics
DESCRIPTION:Abstract: I will first review the construction of the moduli space of tropical curves (or metric graphs)\, and its relation to graph complexes. The graph Laplacian may be interpreted as a tropical version of the classical Torelli map and its determinant is the Kirchhoff graph polynomial (also called 1st Symanzik)\, which is one of the two key components in Feynman integrals in high energy physics.The other component is the so-called 2nd Symanzik polynomial\, which is defined for graphs with external half edges and involves particle masses (edge colourings). I will explain how this too may be interpreted as the determinant of a generalised graph Laplacian\, and how it leads to a volumetric interpretation of a certain class of Feynman integrals.
URL:https://cmsa.fas.harvard.edu/event/3-15-2022-combinatorics-physics-and-probability-seminar/
CATEGORIES:Combinatorics Physics and Probability
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/CMSA-Combinatorics-Physics-and-Probability-Seminar-3.15.2022-1.png
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20220322T093000
DTEND;TZID=America/New_York:20220322T103000
DTSTAMP:20260727T171611
CREATED:20240214T065544Z
LAST-MODIFIED:20240304T085053Z
UID:10002552-1647941400-1647945000@cmsa.fas.harvard.edu
SUMMARY:Flip processes
DESCRIPTION:Abstract: We introduce a class of random graph processes\, which we call \emph{flip processes}. Each such process is given by a \emph{rule} which is just a function $\mathcal{R}:\mathcal{H}_k\rightarrow \mathcal{H}_k$ from all labelled $k$-vertex graphs into itself ($k$ is fixed). The process starts with a given $n$-vertex graph $G_0$. In each step\, the graph $G_i$ is obtained by sampling $k$ random vertices $v_1\,\ldots\,v_k$ of $G_{i-1}$ and replacing the induced graph $F:=G_{i-1}[v_1\,\ldots\,v_k]$ by  $\mathcal{R}(F)$. This class contains several previously studied processes including the Erd\H{o}s–R\’enyi random graph process and the triangle removal process. \nGiven a flip process with a rule $\mathcal{R}$\, we construct time-indexed trajectories $\Phi:\Gra\times [0\,\infty)\rightarrow\Gra$ in the space of graphons. We prove that for any $T > 0$ starting with a large finite graph $G_0$ which is close to a graphon $W_0$ in the cut norm\, with high probability the flip process will stay in a thin sausage around the trajectory $(\Phi(W_0\,t))_{t=0}^T$ (after rescaling the time by the square of the order of the graph). \nThese graphon trajectories are then studied from the perspective of dynamical systems. Among others\, we study continuity properties of these trajectories with respect to time and the initial graphon\, existence and stability of fixed points and speed of convergence (whenever the infinite time limit exists). We give an example of a flip process with a periodic trajectory. This is joint work with Frederik Garbe\, Matas \v Sileikis and Fiona Skerman (arXiv:2201.12272). \nWe also study several specific families flip processes. This is joint work with Pedro Ara\’ujo\, Eng Keat Hng and Matas \v{S}ileikis (in preparation).\nA brief introduction to the necessary bits of the theory of graph limits will be given in the talk.
URL:https://cmsa.fas.harvard.edu/event/3-22-2022-combinatorics-physics-and-probability-seminar/
CATEGORIES:Combinatorics Physics and Probability
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/CMSA-Combinatorics-Physics-and-Probability-Seminar-3.15.2022-1-1.png
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