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DTSTART;TZID=America/New_York:20241104T163000
DTEND;TZID=America/New_York:20241104T173000
DTSTAMP:20241016T202352Z
CREATED:20240903T195045Z
LAST-MODIFIED:20241016T202352Z
UID:10003436-1730737800-1730741400@live-hu-cmsa-222.pantheonsite.io
SUMMARY:The mathematics of evolution
DESCRIPTION:Colloquium \nSpeaker: Martin Nowak (Harvard) \nTitle: The mathematics of evolution \nAbstract: All living systems are guided by evolutionary dynamics. Evolution is a search process which occurs in populations of reproducing individuals. The three fundamental forces of evolution are mutation\, selection and cooperation. I will present basic ideas in the mathematical description of evolutionary dynamics\, including quasi-species theory\, evolutionary game theory\, and evolutionary graph theory. I will discuss specific problems such as origin of life\, emergence of complexity\, mechanisms of cooperation\, evolution of cancer and how to overcome resistance to targeted therapy. \n 
URL:https://live-hu-cmsa-222.pantheonsite.io/event/colloquium-11424/
LOCATION:CMSA Room G10\, CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Colloquium
ATTACH;FMTTYPE=image/png:https://live-hu-cmsa-222.pantheonsite.io/media/CMSA-Colloquium-11.4.2024.docx.png
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20241021T163000
DTEND;TZID=America/New_York:20241021T173000
DTSTAMP:20241016T144838Z
CREATED:20240903T195022Z
LAST-MODIFIED:20241016T144838Z
UID:10003435-1729528200-1729531800@live-hu-cmsa-222.pantheonsite.io
SUMMARY:Higher Vapnik–Chervonenkis theory
DESCRIPTION:Colloquium \nSpeaker: Artem Chernikov\, University of Maryland \nTitle: Higher Vapnik–Chervonenkis theory \nAbstract: Finite VC-dimension\, a combinatorial property of families of sets\, was discovered simultaneously by Vapnik and Chervonenkis in probabilistic learning theory\, and by Shelah in model theory (where it is called NIP). It plays an important role in several areas including machine learning\, combinatorics\, mathematical logic\, functional analysis and topological dynamics. We develop aspects of higher-order VC-theory\, in particular establishing a generalization of the epsilon-net theorem for families of sets (and functions) on n-fold product spaces with bounded VC_n-dimension (i.e. there is a bound on the sizes of n-dimensional boxes that can be shattered). We obtain some applications in combinatorics and in model theory\, including a strong version of Szemerdi’s regularity lemma for hypergraphs omitting a fixed finite n-partite n-hypergraph. Joint work with Henry Towsner.
URL:https://live-hu-cmsa-222.pantheonsite.io/event/colloquium-102124/
LOCATION:CMSA Room G10\, CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Colloquium
ATTACH;FMTTYPE=image/png:https://live-hu-cmsa-222.pantheonsite.io/media/CMSA-Colloquium-10.21.2024.png
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20241007T163000
DTEND;TZID=America/New_York:20241007T173000
DTSTAMP:20241003T160128Z
CREATED:20240903T194924Z
LAST-MODIFIED:20241003T160128Z
UID:10003433-1728318600-1728322200@live-hu-cmsa-222.pantheonsite.io
SUMMARY:Local complexity measures in modern parameterized function classes for supervised learning
DESCRIPTION:Colloquium \nSpeaker: Elisenda Grigsby\, Boston College \nTitle: Local complexity measures in modern parameterized function classes for supervised learning \nAbstract: The parameter space for any fixed architecture of neural networks serves as a proxy during training for the associated class of functions – but how faithful is this representation? For any fixed feedforward ReLU network architecture\, it is well-known that many different parameter settings can determine the same function. It is less well-known that the degree of this redundancy is inhomogeneous across parameter space. I’ll discuss two locally-applicable complexity measures for ReLU network classes and what we know about the relationship between them: (1) the local functional dimension\, and (2) a local version of VC dimension called persistent pseudodimension. The former is easy to compute on finite batches of points\, the latter should give local bounds on the generalization gap. I’ll speculate about how this circle of ideas might help guide our understanding of the double descent phenomenon. All of the work described in this talk is joint with Kathryn Lindsey. Some portions are also joint with Rob Meyerhoff\, David Rolnick\, and Chenxi Wu.
URL:https://live-hu-cmsa-222.pantheonsite.io/event/colloquium-10724/
LOCATION:CMSA Room G10\, CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Colloquium
ATTACH;FMTTYPE=application/pdf:https://live-hu-cmsa-222.pantheonsite.io/media/CMSA-Colloquium-10.7.2024.docx.pdf
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20240923T163000
DTEND;TZID=America/New_York:20240923T173000
DTSTAMP:20240918T190927Z
CREATED:20240903T194207Z
LAST-MODIFIED:20240918T190927Z
UID:10003431-1727109000-1727112600@live-hu-cmsa-222.pantheonsite.io
SUMMARY:Symmetry groups in infinite dimensions
DESCRIPTION:Colloquium \nSpeaker: Lisa Carbone\, Rutgers University \nTitle: Symmetry groups in infinite dimensions \nAbstract: The study of many physical theories requires an understanding of symmetries of infinite dimensional Lie algebras. The construction of groups of automorphisms for infinite dimensional Lie algebras is challenging\, but there is well established theory for the class of Kac-Moody algebras. A generalization of Kac-Moody algebras known as Borcherds algebras arise in string theory models\, but the methods for constructing Kac-Moody groups break down for this more general class. We discuss the challenges that arise and describe several approaches to constructing groups for Borcherds algebras. Our main example is the Monster Lie algebra which plays an important role in the solution of Monstrous Moonshine and which is a symmetry algebra of a model of the compactified Heterotic String.
URL:https://live-hu-cmsa-222.pantheonsite.io/event/colloquium-92324/
LOCATION:CMSA Room G10\, CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Colloquium
ATTACH;FMTTYPE=image/png:https://live-hu-cmsa-222.pantheonsite.io/media/CMSA-Colloquium-09.23.2024.png
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20240916T163000
DTEND;TZID=America/New_York:20240916T173000
DTSTAMP:20240916T163127Z
CREATED:20240903T193540Z
LAST-MODIFIED:20240916T163127Z
UID:10003430-1726504200-1726507800@live-hu-cmsa-222.pantheonsite.io
SUMMARY:Periodic pencils of flat connections and their p-curvature
DESCRIPTION:Colloquium \nSpeaker: Pavel Etingof (MIT) \nTitle: Periodic pencils of flat connections and their p-curvature \n A periodic pencil of flat connections on a smooth algebraic variety  is a linear family of flat connections  \, where  are local coordinates on  and  are matrix-valued regular functions. A pencil is periodic if it is generically invariant under the shifts  up to isomorphism. I will explain that periodic pencils have many remarkable properties\, and there are many interesting examples of them\, e.g. Knizhnik-Zamolodchikov\, Dunkl\, Casimir connections and equivariant quantum connections for conical symplectic resolutions with finitely many torus fixed points. I will also explain that in characteristic \, the -curvature operators  of a periodic pencil  are isospectral to the commuting endomorphisms \, where  is the Frobenius twist of . This allows us to compute the eigenvalues of the -curvature for the above examples\, and also to show that a periodic pencil of connections always has regular singularites. This is joint work with Alexander Varchenko. \n(Abstract link (pdf)
URL:https://live-hu-cmsa-222.pantheonsite.io/event/colloquium_91624/
LOCATION:CMSA Room G10\, CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Colloquium
ATTACH;FMTTYPE=image/png:https://live-hu-cmsa-222.pantheonsite.io/media/CMSA-Colloquium-09.16.2024.png
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BEGIN:VEVENT
DTSTART;TZID=America/New_York:20240909T163000
DTEND;TZID=America/New_York:20240909T173000
DTSTAMP:20240903T152309Z
CREATED:20240827T200454Z
LAST-MODIFIED:20240903T152309Z
UID:10003406-1725899400-1725903000@live-hu-cmsa-222.pantheonsite.io
SUMMARY:Combinatorics and geometry of the amplituhedron
DESCRIPTION:Colloquium \nSpeaker: Lauren Williams\, Harvard University \nTitle: Combinatorics and geometry of the amplituhedron \nAbstract: The amplituhedron is a geometric object introduced by Arkani-Hamed and Trnka to compute scattering amplitudes in N=4 super Yang Mills theory. It generalizes interesting objects such as cyclic polytopes and the positive Grassmannian. It has connections to tropical geometry\, cluster algebras\, and combinatorics (plane partitions\, Catalan numbers). I’ll give a gentle introduction to the amplituhedron\, then survey some recent progress on some of the main conjectures about the amplituhedron: the Magic Number Conjecture\, the BCFW tiling conjecture\, and the Cluster Adjacency conjecture.  Based on joint works withEvan-Zohar\, Lakrec\, Parisi\, Sherman-Bennett\, and Tessler.
URL:https://live-hu-cmsa-222.pantheonsite.io/event/colloquium_9924/
LOCATION:CMSA Room G10\, CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Colloquium
ATTACH;FMTTYPE=image/png:https://live-hu-cmsa-222.pantheonsite.io/media/CMSA-Colloquium-09.09.2024.png
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20240529T090000
DTEND;TZID=America/New_York:20240531T170000
DTSTAMP:20240624T164905Z
CREATED:20240105T071351Z
LAST-MODIFIED:20240624T164905Z
UID:10001120-1716973200-1717174800@live-hu-cmsa-222.pantheonsite.io
SUMMARY:Amplituhedra\, Cluster Algebras\, and Positive Geometry
DESCRIPTION:Amplituhedra\, Cluster Algebras\, and Positive Geometry \nDates: May 29-31\, 2024 \nLocation: Harvard CMSA\, 20 Garden Street\, Cambridge MA 02138 & via Zoom \nIn recent years\, a remarkable paradigm shift has occurred in understanding quantum observables in particle physics and cosmology\, revealing their emergence from underlying novel mathematical objects known as positive geometries. The conference will center on the amplituhedron—the first and major example of a positive geometry. Building on the work of Lusztig and Postnikov on the positive Grassmannian\, the physicists Arkani-Hamed and Trnka introduced the amplituhedron in 2013 as a geometric object that “explains” the so-called BCFW recurrence for scattering amplitudes in N = 4 super Yang Mills theory (SYM). Simultaneously\, cluster algebras\, originally introduced by Fomin and Zelevinsky to study total positivity\, have been revealed to have a crucial role in describing singularities of N = 4 SYM scattering amplitudes. Thus\, one can use ideas from quantum field theory (QFT) to connect cluster algebras to positive geometries\, and in particular to the amplituhedron. Additionally\, QFT can also be used to discover new examples of positive geometries. The conference will bring together a wide range of mathematicians and physicists both to draw new connections within algebraic combinatorics and geometry and to advance our physical understanding of scattering amplitudes and QFT. \nThe conference features: Introductory Lectures\, an Open Problems Forum\, Emerging Scholars Talks\, and talks by experts in the fields. \n  \nConference Videos (Youtube Playlist) \n  \nConfirmed Speakers: \n\nEvgeniya Akhmedova\, Weizmann Institute of Science\nNima Arkani-Hamed\, IAS\nPaolo Benincasa\, MPI\nNick Early\, Weizmann Institute of Science\nCarolina Figueiredo\, Princeton University\nYu-tin Huang\, National Taiwan University\nDani Kaufman\, University of Copenhagen\nChia-Kai Kuo\, National Taiwan University\nThomas Lam\, University of Michigan\nYelena Mandelshtam\, UC Berkeley\nShruti Paranjape\, UC Davis\nLizzie Pratt\, UC Berkeley\nLecheng Ren\, Brown University\nSebastian Seemann\, KU Leuven\nKhrystyna Serhiyenko\, University of Kentucky\nMelissa Sherman-Bennett\, MIT & UC Davis\nMarcus Spradlin\, Brown University\nRan Tessler\, Weizmann Institute of Science\nHugh Thomas\, Université du Québec à Montréal\nJaroslav Trnka\, UC Davis\nAnastasia Volovich\, Brown University\n\nOrganizers: \n\nMatteo Parisi\, Harvard CMSA\nLauren Williams\, Harvard Mathematics\n\nParticipants (PDF) \nThis event is co-funded by the National Science Foundation. \nLimited funding to help defray travel expenses is available for graduate students and recent PhDs. If you are a graduate student or postdoc and would like to apply for support\, please register above and send an email to amplituhedra@cmsa.fas.harvard.edu no later than Friday\, April 19\, 2024. \nPlease include your name\, address\, current status\, university affiliation\, citizenship\, and area of study. F1 visa holders are eligible to apply for support. If you are a graduate student\, please send a brief letter of recommendation from a faculty member to explain the relevance of the conference to your studies or research. If you are a postdoc\, please include a copy of your CV. \n\nSCHEDULE (pdf download) \nWednesday\, May 29\, 2024\n8:30 – 9:00 am\nRegistration and Breakfast \n9:00 – 10:00 am\nJaroslav Trnka\, UC Davis\nTitle: Amplituhedron\nAbstract: I will review basics of the Amplituhedron\, connection to the positive Grassmannian on the mathematical side\, and the scattering amplitudes on the physics side. \n10:00 – 10:15 am\nCoffee Break \n10:15 – 11:15 am\nvia Zoom\nKhrystyna Serhiyenko\, University of Kentucky\nTitle: Introduction to Cluster Algebras\nAbstract: Cluster algebras is a class of commutative rings with an intricate combinatorial structure. They were introduced by Fomin and Zelevinsky in 2002 to study total positivity and canonical basis in Lie Theory\, but quickly evolved into a highly active research area with surprising connections to numerous other areas of mathematics and physics.\nIn this course we will introduce cluster algebras and discuss their basic properties including positivity and Laurent phenomenon. We will also review cluster structures coming from coordinate rings of Grassmannians and the combinatorics of plabic graphs. \n11:15 – 11:30 am\nCoffee Break \n11:30 – 12:30 pm\nThomas Lam\, University of Michigan\nTitle: Introductory Lecture on Positive Geometries\nAbstract: Positive geometries are semialgebraic spaces that appear in the study of scattering amplitudes. Examples include polytopes\, totally nonnegative parts of flag varieties\, and conjecturally\, the amplituhedron. We will give a broad introduction to positive geometries\, and to their canonical forms. \n12:30 – 2:00 pm\nLunch Break \n2:00 – 2:50 pm\nAnastasia Volovich\, Brown University\nTitle: Scattering Amplitudes and Cluster Algebras\nAbstract: I will review some of the deep connections between cluster algebras and the (loop level) scattering amplitudes in N=4 super Yang-Mills theory\, focusing on the cases of n=6 and 7 particles where the corresponding Grassmannian cluster algebras Gr(4\,n) are finite and certain features of the amplitudes are known or believed to be true to all loop order. \n2:50 – 3:00 pm\nCoffee Break \n3:00 – 3:50 pm\nMarcus Spradlin\, Brown University\nTitle: Scattering Amplitudes\, Positive Geometry and the Amplituhedron\nAbstract: I will review the status of (loop level) scattering amplitudes in N=4 super Yang-Mills theory for n>7\, where the corresponding Grassmannian cluster algebras Gr(4\,n) are infinite and novel features emerge\, notably the appearance of certain “marginally positive” algebraic functions of cluster variables. \n3:50 – 4:00 pm\nCoffee Break \n4:00 – 4:30 pm\nCarolina Figueiredo\, Princeton University\nTitle: All-order splits and multi-soft limits for particle and string amplitude\nAbstract: The most important aspects of scattering amplitudes have long been thought to be associated with their poles. Recently a very different sort of “split” factorizations for a wide range of particle and string tree amplitudes have been discovered away from poles. In this talk\, I will explain how natural properties of the binary geometry of the curve integral formulation for scattering amplitudes for Tr$(\Phi^3)$ theory give a simple\, conceptual origin for these splits\, that generalizes them to all orders in the topological expansion. I will also explain how the splits allow us to access and compute loop-integrated multi-soft limits for particle and string amplitudes in Tr$(\Phi^3)$ theory\, the non-linear sigma model and Yang-Mills theory. \n4:30 – 5:15 pm\nYelena Mandelshtam\, UC Berkeley\nTitle: Combinatorics of m=1 Grasstopes\nAbstract: A Grasstope is a linear projection of the totally nonnegative Grassmannian to a smaller Grassmannian. This is a generalization of the amplituhedron\, a geometric object of great importance to calculating scattering amplitudes in physics. The amplituhedron is a Grasstope arising from a totally positive linear map. While amplituhedra are relatively well-studied\, much less is known about general Grasstopes. In this talk\, I will discuss combinatorics and geometry of Grasstopes in the m=1 case. In particular\, I will show that they can be characterized as unions of cells of a hyperplane arrangement satisfying a certain sign variation condition and argue that amplituhedra are (in a certain sense) minimal Grasstopes. This is based on joint work with Dmitrii Pavlov and Lizzie Pratt. \n5:30 – 6:30 pm\nWelcome Reception \n  \nThursday\, May 30\, 2024 \n8:30 – 9:00 am\nBreakfast \n9:00 – 10:00 am\nNima Arkani-Hamed\, IAS\nTitle: Surface Kinematics and THE all-loop integrand for gluon amplitudes \n10:00 – 10:30 am\nCoffee Break \n10:30 – 11:20 am\nHugh Thomas\, Université du Québec à Montréal\nTitle: u-equations from finite dimensional algebras\nAbstract: In this talk\, I will explain how to write down and solve a system of u-equations associated to any finite dimensional algebra with finitely many indecomposable representations. These vastly generalize the system of equations written down by Koba and Nielsen in 1969\, which from our point of view are associated to the representation theory of a Dynkin type A quiver. I will discuss features of the resulting solution spaces\, including connections to tau-tilting theory\, and the relationships that exist among different spaces of solutions. I will also say something about how different choices of finite-dimensional algebra put us in (i) the setting of cluster algebras\, (ii) the Grassmannian combinatorics of non-kissing complexes\, or (iii) the curves-on-surfaces model directly relevant to amplitudes. This talk reports on joint work with Nima Arkani-Hamed\, Hadleigh Frost\, Pierre-Guy Plamondon\, and Giulio Salvatori. \n11:20 – 11:30 am\nCoffee Break \n11:30 – 12:20 pm\nDani Kaufman\, University of Copenhagen\nTitle: Affine Cluster Algebras\nAbstract: Affine cluster algebras form the simplest examples of non-finite type cluster algebras. While they have infinitely many clusters\, they are still mutation finite and have essentially one mutation sequence which produces infinitely many clusters. I will give an introduction to these cluster algebras by comparing them with finite cluster algebras. I will also show how some structures similar to finite type cluster algebras appear “in the limit” along this infinite mutation sequence. If time I will also mention how the “infinite cluster variables” which live in the limit are related to the algebraic letters appearing in the symbol alphabet for 8 particle N=4 SYM amplitudes. \n12:30-12:45 pm\nGroup Photo\, 20 Garden Street\, front entrance stairs outside building \n12:45 – 2:00 pm\nLunch Break \n2:00 – 2:50 pm\nvia Zoom\nRan Tessler\, Weizmann Institute of Science\nTitle: The magic number for the m=2 amplituhedron\nAbstract: We will start by reviewing the amplituhedron and its tilings.\nWe will then show that all tilings of the m=2 amplituhedron have the same cardinality (“the magic number”)\, proving the m=2 case of a conjecture that the same holds for all even-m amplituhedra. If time permits we will discuss related results and consequences.\nBased on a joint work with Parisi\, Sherman-Bennett and Williams. \n2:50 – 3:00 pm\nCoffee Break \n3:00 – 3:50 pm\nMelissa Sherman-Bennett\, MIT & UC Davis\nTitle: Cluster algebras and tilings of amplituhedra\nAbstract: Physicists Arkani-Hamed and Trnka introduced the amplituhedron to better understand scattering amplitudes in N=4 super Yang-Mills theory. The amplituhedron is the image of the totally nonnegative Grassmannian under the “amplituhedron map”. Examples of amplituhedra include cyclic polytopes\, the totally nonnegative Grassmannian itself\, and cyclic hyperplane arrangements. Of primary interest to physics are tilings of amplituhedra\, which are roughly analogous to subdivisions of polytopes. I will discuss joint work with Even-Zohar\, Lakrec\, Parisi\, Tessler and Williams on BCFW tilings of m=4 amplituhedra and the surprising connection between these tilings and the cluster algebra structure of the Grassmannian. \n3:50 – 4:00 pm\nCoffee Break \n4:00 – 5:30 pm\nOpen Problems Forum \n6:00 – 8:00 pm\nConference Dinner (by invitation) \n  \nFriday\, May 31\, 2024 \n8:30 – 9:00 am\nBreakfast \n9:00 – 10:00 am\nYu-tin Huang\, National Taiwan University\nTitle: Chambers and all loop geometry for four-point correlators\nAbstract: The all loop amplituhedron for N=4 SYM (and ABJM theory) can be recast into the notion of loop fibration over tree geometry. This leads to a further dissection of the tree geometry into “chambers”\, whose boundaries represents when the associated loop-form changes. In this talk I will present a new geometry associated with the all loop four-point correlator of N=4 SYM\, where similar description is present. Interestingly\, at four-loops\, this gives a first example where the chamber form is rational even though it’s loop form contains elliptic integrals. \n10:00 – 10:15 am\nCoffee Break \n10:15 – 12:30 am\nEmerging Scholar Talks \n10:15 – 10:40 am\nEvgeniya Akhmedova\, Weizmann Institute of Science\nTitle: The tropical amplituhedron\nAbstract: The Amplituhedron is a geometric object discovered recently by Arkani-Hamed and Trnka\, that provides a completely new direction for calculating scattering amplitudes in quantum field theory. We define a tropical analogue of this object\, the tropicial amplituhedron and study its structure and boundaries. It can be considered as both the tropical limit of the amplituhedron and a generalization of the tropical positive Grassmannian. \n10:40 – 11:10 am\nLizzie Pratt\, UC Berkeley\nTitle: The Chow-Lam Form\nAbstract: The classical Chow form encodes any projective variety by one equation. We introduce the Chow-Lam form for subvarieties of a Grassmannian. By evaluating the Chow-Lam form at twistor coordinates\, we obtain universal projection formulas\, which were pioneered by Thomas Lam for positroid varieties in the study of amplituhedra. This is joint work with Bernd Sturmfels. \n11:10– 11:30 am\nSebastian Seemann\, KU Leuven\nTitle: Vandermonde cells as positive geometries\nAbstract: Vandermonde cells represent semialgebraic subsets of R^n\, characterized as the image of a simplex under the Vandermonde map. However\, within the realm of positive geometry\, several challenges arise in establishing canonical forms for these cells. These include issues such as non-normal boundaries\, non-transversal intersections\, and singularities of boundary curves. Even more difficulties appear when considing the limiting Vandermonde cell\, which is not semi-algebraic and thus doesn’t fit within the standard framework of positive geometries. In this presentation\, I will first review the notion of Polypols and their canonical forms\, examining the complexities encountered when dealing with Vandermonde cells. In particular\, I will explain what goes wrong in the case of Vandermonde cells and which obstructions we can deal with. \n11:30 – 11:40 am\nCoffee break \n11:40 – 12:10 pm\nChia-Kai Kuo\, National Taiwan University\nTitle: Geometric transition from maximal SYM to ABJM\nAbstract: Recently\, the ABJM amplituhedron has been proposed\, encoding all-loop and all-multiplicity ABJM amplitudes. It is constructed by slightly modifying the original definition. In this talk\, I will explore the significance of these modifications in transitioning theoretical models from super Yang-Mills theory to ABJM theory. A key focus will be on how symplectic reduction and the overall sign change in the positivity conditions ensure the consistency of ABJM amplitudes. Additionally\, I will discuss some distinct features of this geometry. \n12:10– 12:30 pm\nLecheng Ren\, Brown University\nTitle: Symbol alphabets from tensor diagrams\nAbstract: We propose to use tensor diagrams and the Fomin-Pylyavskyy conjectures to explore the connection between symbol alphabets of n-particle amplitudes in planar N= 4 Yang-Mills theory and certain polytopes associated to the Grassmannian Gr(4\, n). We show how to assign a web (a planar tensor diagram) to each facet of these polytopes. Webs with no inner loops are associated to cluster variables (rational symbol letters). For webs with a single inner loop we propose and explicitly evaluate an associated web series that contains information about algebraic symbol letters. In this manner we reproduce the results of previous analyses of n ≤ 8\, and find that the polytope C(4\,9) encodes all rational letters\, and all square roots of the algebraic letters\, of known nine-particle amplitudes. \n12:30 – 2:00 pm\nLunch Break \n2:00 – 2:50 pm\nvia Zoom\nPaolo Benincasa\,  MPI\nTitle: Cosmological Polytopes & Beyond\nAbstract: Together with being the source of the most profound questions in fundamental physics\, cosmology turns out to be an arena from where novel combinatorial structures emerge. In this talk\, I will give a gentle introduction to the cosmological polytopes\, describing the so-called Bunch-Davies wavefunction for a large class of scalar theories\, and how it can be used to define and characterize less conventional objects\, named optical polytopes and weighted cosmological polytopes\, which provide examples of non-convex and weighted geometries respectively. \n2:50 – 3:00 pm\nCoffee Break \n3:00 – 3:45 pm\nShruti Paranjape\, UC Davis\nTitle: Loops in a loop expansion\nAbstract: In a paper by Arkani-Hamed\, Henn and Trnka\, it was shown that the amplituhedron construction of N=4 SYM can be recast in terms of negative geometries with a certain hierarchy of loops (closed cycles) in the space of loop momentum twistors. Furthermore\, using differential equation methods\, it was possible to calculate and resum integrated expressions and obtain strong coupling results. In this talk\, we provide a more general framework for the loops of loops expansion and outline a powerful method for the determination of differential forms for higher-order geometries. In particular\, we will focus on the case of 1 closed cycle in loop space and select integrated results. \n3:45 – 4:30 pm\nNick Early\, Weizmann Institute of Science\nTitle: Minimal Kinematics on $\mathcal{M}_{0\,n}$\, and beyond\nAbstract: Minimal Kinematics (MK) identifies kinematic degenerations of the CHY scattering potential where the critical points are given by rational formulas. These rest on the Horn uniformization of Kapranov-Huh; they are specified combinatorially by 2-trees. On the other hand\, Planar Kinematics (PK) identifies the locus in $M_{0\,n}$ which is fixed by cyclic permutation.  Combining MK and PK realizes a maximally thin relative of the associahedron known as the PK polytope; it is a reflexive polytope\, and its polar dual\, the root polytope\, has volume a Catalan number. In this talk\, we start by exploring MK and PK on the moduli space $M_{0\,n}$.  We explain how this story generalizes to moduli spaces $X(k\,n)$ of points in projective space $\mathbb{P}^{k-1}$\, to CEGM amplitudes and beyond. \n4:30 – 5:00 pm\nCoffee and Farewell \n  \n \n  \nAbout the image: \n\nLeft: the 3-dimensional associahedron\, Fomin and Zelevinsky\n\nCenter: artistic depiction of the amplituhedron\, Gilmore\nRight: Schlegel diagram of a hypersimplex\, Ziegler
URL:https://live-hu-cmsa-222.pantheonsite.io/event/amplituhedra2024/
LOCATION:CMSA Room G10\, CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Conference
ATTACH;FMTTYPE=image/png:https://live-hu-cmsa-222.pantheonsite.io/media/amplituhedron_cluster-algebras_posgeometry.png
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