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BEGIN:VEVENT
DTSTART;TZID=America/New_York:20220517T093000
DTEND;TZID=America/New_York:20220517T103000
DTSTAMP:20260404T215841
CREATED:20240214T072604Z
LAST-MODIFIED:20240304T055019Z
UID:10002559-1652779800-1652783400@cmsa.fas.harvard.edu
SUMMARY:Hypergraph Matchings Avoiding Forbidden Submatchings
DESCRIPTION:Abstract:  In 1973\, Erdős conjectured the existence of high girth (n\,3\,2)-Steiner systems. Recently\, Glock\, Kühn\, Lo\, and Osthus and independently Bohman and Warnke proved the approximate version of Erdős’ conjecture. Just this year\, Kwan\, Sah\, Sawhney\, and Simkin proved Erdős’ conjecture. As for Steiner systems with more general parameters\, Glock\, Kühn\, Lo\, and Osthus conjectured the existence of high girth (n\,q\,r)-Steiner systems. We prove the approximate version of their conjecture.  This result follows from our general main results which concern finding perfect or almost perfect matchings in a hypergraph G avoiding a given set of submatchings (which we view as a hypergraph H where V(H)=E(G)). Our first main result is a common generalization of the classical theorems of Pippenger (for finding an almost perfect matching) and Ajtai\, Komlós\, Pintz\, Spencer\, and Szemerédi (for finding an independent set in girth five hypergraphs). More generally\, we prove this for coloring and even list coloring\, and also generalize this further to when H is a hypergraph with small codegrees (for which high girth designs is a specific instance). Indeed\, the coloring version of our result even yields an almost partition of K_n^r into approximate high girth (n\,q\,r)-Steiner systems.  If time permits\, I will explain some of the other applications of our main results such as to rainbow matchings.  This is joint work with Luke Postle.
URL:https://cmsa.fas.harvard.edu/event/5-17-2022-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-05.17.22.png
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20220503T093000
DTEND;TZID=America/New_York:20220503T103000
DTSTAMP:20260404T215841
CREATED:20240214T072025Z
LAST-MODIFIED:20240304T055241Z
UID:10002557-1651570200-1651573800@cmsa.fas.harvard.edu
SUMMARY:The threshold for stacked triangulations
DESCRIPTION:Abstract: Consider a bootstrap percolation process that starts with a set of `infected’ triangles $Y \subseteq \binom{[n]}3$\, and a new triangle f gets infected if there is a copy of K_4^3 (= the boundary of a tetrahedron) in which f is the only not-yet infected triangle.\nSuppose that every triangle is initially infected independently with probability p=p(n)\, what is the threshold probability for percolation — the event that all triangles get infected? How many new triangles do get infected in the subcritical regime? \nThis notion of percolation can be viewed as a simplification of simple-connectivity. Namely\, a stacked triangulation of a triangle is obtained by repeatedly subdividing an inner face into three faces.\nWe ask: for which $p$ does the random simplicial complex Y_2(n\,p) contain\, for every triple $xyz$\, the faces of a stacked triangulation of $xyz$ whose internal vertices are arbitrarily labeled in [n]. \nWe consider this problem in every dimension d>=2\, and our main result identifies a sharp probability threshold for percolation\, showing it is asymptotically (c_d*n)^(-1/d)\, where c_d is the growth rate of the Fuss–Catalan numbers of order d. \nThe proof hinges on a second moment argument in the supercritical regime\, and on Kalai’s algebraic shifting in the subcritical regime. \nJoint work with Eyal Lubetzky.
URL:https://cmsa.fas.harvard.edu/event/5-3-2022-cmsa-combinatorics-physics-and-probability-seminar/
LOCATION:Hybrid
CATEGORIES:Combinatorics Physics and Probability
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/CMSA-Combinatorics-Physics-and-Probability-Seminar-05.03.22-1.png
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20220426T090000
DTEND;TZID=America/New_York:20220426T100000
DTSTAMP:20260404T215841
CREATED:20240214T071014Z
LAST-MODIFIED:20240304T055455Z
UID:10002555-1650963600-1650967200@cmsa.fas.harvard.edu
SUMMARY:Algebraic Statistics with a View towards Physics
DESCRIPTION:Abstract: We discuss the algebraic geometry of maximum likelihood estimation from the perspective of scattering amplitudes in particle physics. A guiding examples the moduli space of n-pointed rational curves. The scattering potential plays the role of the log-likelihood function\, and its critical points are solutions to rational function equations. Their number is an Euler characteristic. Soft limit degenerations are combined with certified numerical methods for concrete computations. \n**This talk will be hybrid. Talk will be held at CMSA (20 Garden St) Room G10. \nAll non-Harvard affiliated visitors to the CMSA building will need to complete this covid form prior to arrival. \nLINK TO FORM
URL:https://cmsa.fas.harvard.edu/event/4-26-2022-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-04.26.22.png
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20220419T093000
DTEND;TZID=America/New_York:20220419T103000
DTSTAMP:20260404T215841
CREATED:20240214T070331Z
LAST-MODIFIED:20240304T084706Z
UID:10002554-1650360600-1650364200@cmsa.fas.harvard.edu
SUMMARY:Some combinatorics of Wilson loop diagrams
DESCRIPTION:Abstract: Wilson loop diagrams can be used to study amplitudes in N=4 SYM.  I will set them up and talk about some of their combinatorial aspects\, such as how many Wilson loop diagrams give the same positroid and how to combinatorially read off the dimension and the denominators for the integrands. \n**This talk will be hybrid. Talk will be held at CMSA (20 Garden St) Room G10. \nAll non-Harvard affiliated visitors to the CMSA building will need to complete this covid form prior to arrival. \nLINK TO FORM
URL:https://cmsa.fas.harvard.edu/event/4-19-2022-combinatorics-physics-and-probability-seminar/
LOCATION:Hybrid
CATEGORIES:Combinatorics Physics and Probability
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/CMSA-Combinatorics-Physics-and-Probability-Seminar-04.19.22.png
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20220412T093000
DTEND;TZID=America/New_York:20220412T103000
DTSTAMP:20260404T215841
CREATED:20240214T065956Z
LAST-MODIFIED:20240304T060444Z
UID:10002553-1649755800-1649759400@cmsa.fas.harvard.edu
SUMMARY:BCFW recursion relations and non-planar positive geometry
DESCRIPTION:Abstract: There is a close connection between the scattering amplitudes in planar N=4 SYM theory and the cells in the positive Grassmannian. In the context of BCFW recursion relations the tree-level S-matrix is represented as a sum of planar on-shell diagrams (aka plabic graphs) and associated with logarithmic forms on the Grassmannian cells of certain dimensionality. In this talk\, we explore non-adjacent BCFW shifts which naturally lead to non-planar on-shell diagrams and new interesting subspaces inside the real Grassmannian. \n**This talk will be hybrid. Talk will be held at CMSA (20 Garden St) Room G10. \nAll non-Harvard affiliated visitors to the CMSA building will need to complete this covid form prior to arrival. \nLINK TO FORM
URL:https://cmsa.fas.harvard.edu/event/4-12-2022-combinatorics-physics-and-probability-seminar/
CATEGORIES:Combinatorics Physics and Probability
ATTACH;FMTTYPE=image/jpeg:https://cmsa.fas.harvard.edu/media/CMSA-Combinatorics-Physics-and-Probability-Seminar-04.12.22-1583x2048-1.jpg
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20220322T093000
DTEND;TZID=America/New_York:20220322T103000
DTSTAMP:20260404T215841
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
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20220315T090000
DTEND;TZID=America/New_York:20220315T100000
DTSTAMP:20260404T215841
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:20220308T090000
DTEND;TZID=America/New_York:20220308T100000
DTSTAMP:20260404T215841
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
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20220301T090000
DTEND;TZID=America/New_York:20220301T100000
DTSTAMP:20260404T215841
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
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20220215T090000
DTEND;TZID=America/New_York:20220215T100000
DTSTAMP:20260404T215841
CREATED:20240213T104814Z
LAST-MODIFIED:20240304T100739Z
UID:10002456-1644915600-1644919200@cmsa.fas.harvard.edu
SUMMARY:Equiangular lines and regular graphs
DESCRIPTION:Abstract: In 1973\, Lemmens and Seidel asked to determine N_alpha(r)\, the maximum number of equiangular lines in R^r with common angle arccos(alpha). Recently\, this problem has been almost completely settled when r is exponentially large relative to 1/alpha\, with the approach both relying on Ramsey’s theorem\, as well as being limited by it. In this talk\, we will show how orthogonal projections of matrices with respect to the Frobenius inner product can be used to overcome this limitation\, thereby obtaining significantly improved upper bounds on N_alpha(r) when r is polynomial in 1/alpha. In particular\, our results imply that N_alpha(r) = Theta(r) for alpha >= Omega(1 / r^1/5). \nOur projection method generalizes to complex equiangular lines in C^r\, which may be of independent interest in quantum theory. Applying this method also allows us to obtain\nthe first universal bound on the maximum number of complex equiangular lines in C^r with common Hermitian angle arccos(alpha)\, an extension of the Alon-Boppana theorem to dense regular graphs\, which is tight for strongly regular graphs corresponding to r(r+1)/2 equiangular lines in R^r\, an improvement to Welch’s bound in coding theory.
URL:https://cmsa.fas.harvard.edu/event/2-15-2022-combinatorics-physics-and-probability-seminar/
CATEGORIES:Combinatorics Physics and Probability
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20220208T090000
DTEND;TZID=America/New_York:20220208T100000
DTSTAMP:20260404T215841
CREATED:20240213T104820Z
LAST-MODIFIED:20240304T105941Z
UID:10002457-1644310800-1644314400@cmsa.fas.harvard.edu
SUMMARY:Invariant theory for maximum likelihood estimation
DESCRIPTION:Abstract:  I will talk about work to uncover connections between invariant theory and maximum likelihood estimation. I will describe how norm minimization over a torus orbit is equivalent to maximum likelihood estimation in log-linear models. We will see the role played by polytopes and discuss connections to scaling algorithms. Based on joint work with Carlos Améndola\, Kathlén Kohn\, and Philipp Reichenbach.
URL:https://cmsa.fas.harvard.edu/event/2-8-2022-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-2.8.2022-1.png
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20220203T090000
DTEND;TZID=America/New_York:20220203T100000
DTSTAMP:20260404T215841
CREATED:20240304T105610Z
LAST-MODIFIED:20240304T105610Z
UID:10002902-1643878800-1643882400@cmsa.fas.harvard.edu
SUMMARY:The Amplituhedron BCFW Triangulation
DESCRIPTION:Abstract:  The (tree) amplituhedron was introduced in 2013 by Arkani-Hamed and Trnka in their study of N=4 SYM scattering amplitudes. A central conjecture in the field was to prove that the m=4 amplituhedron is triangulated by the images of certain positroid cells\, called the BCFW cells. In this talk I will describe a resolution of this conjecture. The seminar is based on a recent joint work with Chaim Even-Zohar and Tsviqa Lakrec.
URL:https://cmsa.fas.harvard.edu/event/2-3-2022-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-2.3.2022.png
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20220125T090000
DTEND;TZID=America/New_York:20220125T100000
DTSTAMP:20260404T215841
CREATED:20240213T111559Z
LAST-MODIFIED:20240304T105047Z
UID:10002485-1643101200-1643104800@cmsa.fas.harvard.edu
SUMMARY:Adventures in Perturbation Theory
DESCRIPTION:Abstract: Recent years have seen tremendous advances in our understanding of perturbative quantum field theory—fueled largely by discoveries (and eventual explanations and exploitation) of shocking simplicity in the mathematical form of the predictions made for experiment. Among the most important frontiers in this progress is the understanding of loop amplitudes—their mathematical form\, underlying geometric structure\, and how best to manifest the physical properties of finite observables in general quantum field theories. This work is motivated in part by the desire to simplify the difficult work of doing Feynman integrals. I review some of the examples of this progress\, and describe some ongoing efforts to recast perturbation theory in terms that expose as much simplicity (and as much physics) as possible.
URL:https://cmsa.fas.harvard.edu/event/1-25-2022-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-01.25.2022-1.png
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20211214T093000
DTEND;TZID=America/New_York:20211214T103000
DTSTAMP:20260404T215841
CREATED:20240213T112343Z
LAST-MODIFIED:20240304T102729Z
UID:10002495-1639474200-1639477800@cmsa.fas.harvard.edu
SUMMARY:The longest induced path in a sparse random graph
DESCRIPTION:Abstract: A long-standing problem in random graph theory has been to determine asymptotically the length of a longest induced path in sparse random graphs. Independent work of Luczak and Suen from the 90s showed the existence of an induced path of roughly half the optimal size\, which seems to be a barrier for certain natural approaches. Recently\, in joint work with Draganic and Krivelevich\, we solved this problem. In the talk\, I will discuss the history of the problem and give an overview of the proof.
URL:https://cmsa.fas.harvard.edu/event/12-14-21-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-12.14.2021.png
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20211207T093000
DTEND;TZID=America/New_York:20211207T103000
DTSTAMP:20260404T215841
CREATED:20240213T070713Z
LAST-MODIFIED:20240213T070713Z
UID:10002160-1638869400-1638873000@cmsa.fas.harvard.edu
SUMMARY:The singularity probability of random symmetric matrices
DESCRIPTION:Abstract: Let M_n be drawn uniformly from all n by n symmetric matrices with entries in {-1\,1}. In this talk I’ll consider the following basic question: what is the probability that M_n is singular? I’ll discuss recent joint work with Marcelo Campos\, Marcus Michelen and Julian Sahasrabudhe where we show that this probability is exponentially small. I hope to make the talk accessible to a fairly general audience.
URL:https://cmsa.fas.harvard.edu/event/the-singularity-probability-of-random-symmetric-matrices/
CATEGORIES:Combinatorics Physics and Probability
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/CMSA-Combinatorics-Physics-and-Probability-Seminar-12.07.2021.png
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20211130T093000
DTEND;TZID=America/New_York:20211130T103000
DTSTAMP:20260404T215841
CREATED:20240213T065738Z
LAST-MODIFIED:20240213T065738Z
UID:10002146-1638264600-1638268200@cmsa.fas.harvard.edu
SUMMARY:Resistance curvature – a new discrete curvature on graphs
DESCRIPTION:Abstract: The last few decades have seen a surge of interest in building towards a theory of discrete curvature that attempts to translate the key properties of curvature in differential geometry to the setting of discrete objects and spaces. In the case of graphs there have been several successful proposals\, for instance by Lin-Lu-Yau\, Forman and Ollivier\, that replicate important curvature theorems and have inspired applications in a variety of practical settings.\nIn this talk\, I will introduce a new notion of discrete curvature on graphs\, which we call the resistance curvature\, and discuss some of its basic properties. The resistance curvature is defined based on the concept of effective resistance which is a metric between the vertices of a graph and has many other properties such as a close relation to random spanning trees. The rich theory of these effective resistances allows to study the resistance curvature in great detail; I will for instance show that “Lin-Lu-Yau >= resistance >= Forman curvature” in a specific sense\, show strong evidence that the resistance curvature converges to zero in expectation for Euclidean random graphs\, and give a connectivity theorem for positively curved graphs. The resistance curvature also has a naturally associated discrete Ricci flow which is a gradient flow and has a closed-form solution in the case of vertex-transitive and path graphs.\nFinally\, if time permits I will draw a connection with the geometry of hyperacute simplices\, following the work of Miroslav Fiedler.\nThis work was done in collaboration with Renaud Lambiotte.
URL:https://cmsa.fas.harvard.edu/event/resistance-curvature-a-new-discrete-curvature-on-graphs/
CATEGORIES:Combinatorics Physics and Probability
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/CMSA-Combinatorics-Physics-and-Probability-Seminar-11.30.2021-1.png
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20211123T093000
DTEND;TZID=America/New_York:20211123T103000
DTSTAMP:20260404T215841
CREATED:20240213T065330Z
LAST-MODIFIED:20240226T111143Z
UID:10002138-1637659800-1637663400@cmsa.fas.harvard.edu
SUMMARY:Prague dimension of random graphs
DESCRIPTION:Abstract: The Prague dimension of graphs was introduced by Nesetril\, Pultr and Rodl in the 1970s: as a combinatorial measure of complexity\, it is closely related to clique edges coverings and partitions. Proving a conjecture of Furedi and Kantor\, we show that the Prague dimension of the binomial random graph is typically of order n/(log n) for constant edge-probabilities. The main new proof ingredient is a Pippenger-Spencer type edge-coloring result for random hypergraphs with large uniformities\, i.e.\, edges of size O(log n).
URL:https://cmsa.fas.harvard.edu/event/11-23-21-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-11.23.21-1.png
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20211026T090000
DTEND;TZID=America/New_York:20211026T100000
DTSTAMP:20260404T215841
CREATED:20240213T113529Z
LAST-MODIFIED:20240304T101126Z
UID:10002507-1635238800-1635242400@cmsa.fas.harvard.edu
SUMMARY:The n-queens problem
DESCRIPTION:Abstract: The n-queens problem asks how many ways there are to place n queens on an n x n chessboard so that no two queens can attack one another\, and the toroidal n-queens problem asks the same question where the board is considered on the surface of a torus. Let Q(n) denote the number of n-queens configurations on the classical board and T(n) the number of toroidal n-queens configurations. The toroidal problem was first studied in 1918 by Pólya who showed that T(n)>0 if and only if n is not divisible by 2 or 3. Much more recently Luria showed that T(n) is at most ((1+o(1))ne^{-3})^n and conjectured equality when n is not divisible by 2 or 3. We prove this conjecture\, prior to which no non-trivial lower bounds were known to hold for all (sufficiently large) n not divisible by 2 or 3. We also show that Q(n) is at least ((1+o(1))ne^{-3})^n for all natural numbers n which was independently proved by Luria and Simkin and\, combined with our toroidal result\, completely settles a conjecture of Rivin\, Vardi and Zimmerman regarding both Q(n) and T(n). \nIn this talk we’ll discuss our methods used to prove these results. A crucial element of this is translating the problem to one of counting matchings in a 4-partite 4-uniform hypergraph. Our strategy combines a random greedy algorithm to count `almost’ configurations with a complex absorbing strategy that uses ideas from the methods of randomised algebraic construction and iterative absorption. \nThis is joint work with Peter Keevash.
URL:https://cmsa.fas.harvard.edu/event/10-26-2021-combinatorics-physics-and-probability-seminar/
CATEGORIES:Combinatorics Physics and Probability
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20211019T090000
DTEND;TZID=America/New_York:20211019T100000
DTSTAMP:20260404T215841
CREATED:20240213T114112Z
LAST-MODIFIED:20240304T100424Z
UID:10002511-1634634000-1634637600@cmsa.fas.harvard.edu
SUMMARY:10/19/2021 Combinatorics\, Physics and Probability Seminar
DESCRIPTION:Title: Ising model\, total positivity\, and criticality \nAbstract: The Ising model\, introduced in 1920\, is one of the most well-studied models in statistical mechanics. It is known to undergo a phase transition at critical temperature\, and has attracted considerable interest over the last two decades due to special properties of its scaling limit at criticality.\nThe totally nonnegative Grassmannian is a subset of the real Grassmannian introduced by Postnikov in 2006. It arises naturally in Lusztig’s theory of total positivity and canonical bases\, and is closely related to cluster algebras and scattering amplitudes.\nI will give some background on the above objects and then explain a precise relationship between the planar Ising model and the totally nonnegative Grassmannian\, obtained in our recent work with P. Pylyavskyy. Building on this connection\, I will give a new boundary correlation formula for the critical Ising model
URL:https://cmsa.fas.harvard.edu/event/10-19-2021-combinatorics-physics-and-probability-seminar/
CATEGORIES:Combinatorics Physics and Probability
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20211012T090000
DTEND;TZID=America/New_York:20211012T100000
DTSTAMP:20260404T215841
CREATED:20240213T114547Z
LAST-MODIFIED:20240304T100222Z
UID:10002513-1634029200-1634032800@cmsa.fas.harvard.edu
SUMMARY:10/12/2021 Combinatorics\, Physics and Probability Seminar
DESCRIPTION:Title: On counting algebraically defined graphs \nAbstract: For many classes of graphs that arise naturally in discrete geometry (for example intersection graphs of segments or disks in the plane)\, the edges of these graphs can be defined algebraically using the signs of a finite list of fixed polynomials. We investigate the number of n-vertex graphs in such an algebraically defined class of graphs. Warren’s theorem (a variant of a theorem of Milnor and Thom) implies upper bounds for the number of n-vertex graphs in such graph classes\, but all the previously known lower bounds were obtained from ad hoc constructions for very specific classes. We prove a general theorem giving a lower bound for this number (under some reasonable assumptions on the fixed list of polynomials)\, and this lower bound essentially matches the upper bound from Warren’s theorem.
URL:https://cmsa.fas.harvard.edu/event/10-12-2021-combinatorics-physics-and-probability-seminar/
CATEGORIES:Combinatorics Physics and Probability
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BEGIN:VEVENT
DTSTART;TZID=America/New_York:20211005T090000
DTEND;TZID=America/New_York:20211005T100000
DTSTAMP:20260404T215841
CREATED:20240213T113617Z
LAST-MODIFIED:20240304T085033Z
UID:10002508-1633424400-1633428000@cmsa.fas.harvard.edu
SUMMARY:10/5/2021 Combinatorics\, Physics and Probability Seminar
DESCRIPTION:Title: Geodesic Geometry on Graphs \nAbstract: In a graph G = (V\, E) we consider a system of paths S so that for every two vertices u\,v in V there is a unique uv path in S connecting them. The path system is said to be consistent if it is closed under taking subpaths\, i.e. if P is a path in S then any subpath of P is also in S. Every positive weight function w: E–>R^+ gives rise to a consistent path system in G by taking the paths in S to be geodesics w.r.t. w. In this case\, we say w induces S. We say a graph G is metrizable if every consistent path system in G is induced by some such w. \nWe’ll discuss the concept of graph metrizability\, and\, in particular\, we’ll see that while metrizability is a rare property\, there exists infinitely many 2-connected metrizable graphs. \nJoint work with Nati Linial.
URL:https://cmsa.fas.harvard.edu/event/10-5-2021-combinatorics-physics-and-probability-seminar/
CATEGORIES:Combinatorics Physics and Probability
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20210928T130000
DTEND;TZID=America/New_York:20210928T130000
DTSTAMP:20260404T215841
CREATED:20240214T045955Z
LAST-MODIFIED:20240304T060348Z
UID:10002535-1632834000-1632834000@cmsa.fas.harvard.edu
SUMMARY:9/28/2021 Combinatorics\, Physics and Probability Seminar
DESCRIPTION:Title: The hypersimplex and the m=2 amplituhedron \nAbstract: I’ll discuss a curious correspondence between the m=2 amplituhedron\, a 2k-dimensional subset of Gr(k\, k+2)\, and the hypersimplex\, an (n-1)-dimensional polytope in R^n. The amplituhedron and hypersimplex are both images of the totally nonnegative Grassmannian under some map (the amplituhedron map and the moment map\, respectively)\, but are different dimensions and live in very different ambient spaces. I’ll talk about joint work with Matteo Parisi and Lauren Williams in which we give a bijection between decompositions of the amplituhedron and decompositions of the hypersimplex (originally conjectured by Lukowski–Parisi–Williams). Along the way\, we prove the sign-flip description of the m=2 amplituhedron conjectured by Arkani-Hamed–Thomas–Trnka and give a new decomposition of the m=2 amplituhedron into Eulerian-number-many chambers (inspired by an analogous hypersimplex decomposition).
URL:https://cmsa.fas.harvard.edu/event/9-28-2021-combinatorics-physics-and-probability-seminar/
CATEGORIES:Combinatorics Physics and Probability
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20210921T093000
DTEND;TZID=America/New_York:20210921T093000
DTSTAMP:20260404T215841
CREATED:20240214T050308Z
LAST-MODIFIED:20240304T060511Z
UID:10002536-1632216600-1632216600@cmsa.fas.harvard.edu
SUMMARY:Surfacehedra and the Binary Positive Geometry of Particle and “String” Amplitudes
DESCRIPTION:Speaker: Nima Arkani-Hamed\, IAS \nTitle: Surfacehedra and the Binary Positive Geometry of Particle and “String” Amplitudes
URL:https://cmsa.fas.harvard.edu/event/9-21-2021-combinatorics-physics-and-probability-seminar/
LOCATION:Virtual
CATEGORIES:Combinatorics Physics and Probability
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20210911T093000
DTEND;TZID=America/New_York:20210911T103000
DTSTAMP:20260404T215841
CREATED:20240222T111949Z
LAST-MODIFIED:20240222T112111Z
UID:10002807-1631352600-1631356200@cmsa.fas.harvard.edu
SUMMARY:Gradient flows on totally nonnegative flag varieties
DESCRIPTION:Abstract: One can view a partial flag variety in C^n as an adjoint orbit inside the Lie algebra of n x n skew-Hermitian matrices. We use the orbit context to study the totally nonnegative part of a partial flag variety from an algebraic\, geometric\, and dynamical perspective. We classify gradient flows on adjoint orbits in various metrics which are compatible with total positivity. As applications\, we show how the classical Toda flow fits into this framework\, and prove that a new family of amplituhedra are homeomorphic to closed balls. This is joint work with Anthony Bloch.
URL:https://cmsa.fas.harvard.edu/event/11-9-21-combinatorics-physics-and-probability-seminar/
CATEGORIES:Combinatorics Physics and Probability
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