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BEGIN:VEVENT
DTSTART;TZID=America/New_York:20250512T163000
DTEND;TZID=America/New_York:20250512T173000
DTSTAMP:20260515T164123
CREATED:20250407T140851Z
LAST-MODIFIED:20250506T191033Z
UID:10003734-1747067400-1747071000@cmsa.fas.harvard.edu
SUMMARY:Factorizations for data analysis
DESCRIPTION:Colloquium \nSpeaker: Anna Seigal\, Harvard University \nTitle: Factorizations for data analysis \nAbstract: We can find structure in data by factoring it into building blocks\, which should be interpretable for the context at hand. A classical example is principal component analysis (PCA)\, which uses the eigendecomposition of the covariance matrix to find axes of variation in a dataset. Starting from PCA\, I will discuss matrix and tensor factorizations for data analysis\, and the linear and multilinear algebra that underpins their theoretical properties. We will see examples from causal inference\, independent component analysis\, and dimensionality reduction.
URL:https://cmsa.fas.harvard.edu/event/colloquium-51225/
LOCATION:CMSA Room G10\, CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Colloquium
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/CMSA-Colloquium-5.12.2025.png
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20250505T163000
DTEND;TZID=America/New_York:20250505T173000
DTSTAMP:20260515T164123
CREATED:20250407T140808Z
LAST-MODIFIED:20250501T134436Z
UID:10003733-1746462600-1746466200@cmsa.fas.harvard.edu
SUMMARY:Thinking Outside the Ballot Box
DESCRIPTION:Colloquium \nSpeaker: Ariel Procaccia\, Harvard University \nTitle: Thinking Outside the Ballot Box \nAbstract: How should one design unprecedented democratic processes capable of handling enormous sets of alternatives like all possible policies\, bills\, or statements? I argue that this challenge can be addressed through a framework called generative social choice\, which fuses the rigor of social choice theory with the flexibility and power of large language models. I then explore an application of generative social choice to the problem of identifying a proportionally representative slate of opinion statements. This includes a discussion of desired properties\, an algorithm that provably achieves them\, an implementation using GPT\, and insights from an end-to-end pilot. By providing guarantees\, generative social choice could alleviate concerns about AI-driven democratic innovation and help unlock its potential.
URL:https://cmsa.fas.harvard.edu/event/colloquium-5525/
LOCATION:CMSA Room G10\, CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Colloquium
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/CMSA-Colloquium-5.5.2025.png
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20250428T163000
DTEND;TZID=America/New_York:20250428T173000
DTSTAMP:20260515T164123
CREATED:20241209T171137Z
LAST-MODIFIED:20250423T174326Z
UID:10003637-1745857800-1745861400@cmsa.fas.harvard.edu
SUMMARY:Bass-Note Spectra of locally uniform geometries
DESCRIPTION:Colloquium \nSpeaker: Peter Sarnak\, IAS & Princeton University \nTitle: Bass-Note Spectra of locally uniform geometries \nAbstract: We formulate and report on the problem of the Bass-Note Spectrum of an invariant operator as one varies over locally uniform geometries. In the Euclidean setting this recasts classical problems of Mahler from the geometry of numbers in a new light. For certain operators homogeneous dynamics can be used decisively. In the non-Euclidean setting of hyperbolic manifolds we review some recent developments using the conformal bootstrap method and of random covers to study the Bass-Note spectra. We highlight the theme and impact of rigidity.
URL:https://cmsa.fas.harvard.edu/event/colloquium-42825/
LOCATION:CMSA Room G10\, CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Colloquium
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/CMSA-Colloquium-4.28.2025.png
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20250421T163000
DTEND;TZID=America/New_York:20250421T173000
DTSTAMP:20260515T164123
CREATED:20241209T163847Z
LAST-MODIFIED:20250418T142045Z
UID:10003636-1745253000-1745256600@cmsa.fas.harvard.edu
SUMMARY:Modeling the emergence of complex cortical structure from simple precursors in the brain: maps\, hierarchies\, and modules
DESCRIPTION:Colloquium \nSpeaker: Ila Fiete\, MIT \nTitle: Modeling the emergence of complex cortical structure from simple precursors in the brain: maps\, hierarchies\, and modules \nAbstract: Modular and hierarchical structures are ubiquitous in the brain. Two distinct hypotheses for such morphogenesis involve genetic specification (the positional information hypothesis) or spontaneous structure emergence from symmetry breaking (the pattern formation hypothesis). Indeed\, there is rich evidence supporting both hypotheses in different systems\, and more recently evidence that both systems might interact\, for instance with genetic specification providing an initial but relatively low-information scaffold of positional guidance and pattern formation constructing sharper structures by bootstrapping from this guidance. In this talk\, I will consider the emergence of two systems in the brain: the visual processing hierarchy with topographic structure\, and a modular cognitive circuit consisting of functionally independent grid cell networks that compute spatial location from velocity cues as animals move and navigate the world. I will describe how simple activity-driven growth and competition rules can lead to the emergence of topographically ordered sensory processing hierarchies\, and how genetically specified smooth gradients with purely local recurrent interactions on two scales can lead to global module emergence. In sum\, simple growth rules\, local interactions and smooth gradients can interact to produce rich emergent order on multiple scales in the form of maps\, modules\, and hierarchies\, with predictions that bridge scales from genes to connectivity to function.
URL:https://cmsa.fas.harvard.edu/event/colloquium-42125/
LOCATION:CMSA Room G10\, CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Colloquium
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/CMSA-Colloquium-4.21.2025.png
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20250414T163000
DTEND;TZID=America/New_York:20250414T173000
DTSTAMP:20260515T164123
CREATED:20241209T163821Z
LAST-MODIFIED:20250410T204704Z
UID:10003635-1744648200-1744651800@cmsa.fas.harvard.edu
SUMMARY:Quantum K-theory at roots of unity
DESCRIPTION:Colloquium \nSpeaker: Andrey Smirnov\, University of North Carolina at Chapel Hill \nTitle: Quantum K-theory at roots of unity \nAbstract: In this talk\, I will discuss a version of quantum K-theory introduced by A.Okounkov\, which can be defined through quasimap counts. In this framework\, the quantum K-theory ring is obtained as a specialization of the equivariant quasimap count at $q=1$\, where $q$ is the equivariant parameter associated with the torus action on the source of the quasimaps. A related\, but less explored\, structure emerges when $q$ is specialized at the roots of unity. I will outline the key ideas behind this construction and its implications. As an application\, I’ll also describe the spectrum of $p$-curvature for the quantum connection\, which offers a new proof of a recent result by P.Etingof and A.Varchenko. This talk is based on joint work with P. Koroteev.
URL:https://cmsa.fas.harvard.edu/event/colloquium-41425/
LOCATION:CMSA Room G10\, CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Colloquium
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/CMSA-Colloquium-4.14.2025.png
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20250407T163000
DTEND;TZID=America/New_York:20250407T173000
DTSTAMP:20260515T164123
CREATED:20241209T163727Z
LAST-MODIFIED:20250401T191454Z
UID:10003634-1744043400-1744047000@cmsa.fas.harvard.edu
SUMMARY:3-d Mirror Symmetry
DESCRIPTION:Colloquium \nSpeaker: Ben Webster\, University of Waterloo & Perimeter Institute \nTitle: 3-d Mirror Symmetry \nAbstract: I’ll give an introduction (or update\, for those who’ve been introduced) to 3d mirror symmetry from the perspective of a mathematician. \n 
URL:https://cmsa.fas.harvard.edu/event/colloquium-4725/
LOCATION:CMSA Room G10\, CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Colloquium
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/CMSA-Colloquium-4.7.2025.docx-1.png
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20250324T163000
DTEND;TZID=America/New_York:20250324T173000
DTSTAMP:20260515T164123
CREATED:20241209T163216Z
LAST-MODIFIED:20250321T163829Z
UID:10003631-1742833800-1742837400@cmsa.fas.harvard.edu
SUMMARY:The Toda Lattice as a Soliton Gas
DESCRIPTION:Colloquium \nSpeaker: Amol Aggarwal\, Columbia University \nTitle: The Toda Lattice as a Soliton Gas \nAbstract: A basic tenet of integrable systems is that\, under sufficiently irregular initial data\, they can be thought of as dense collections of many solitons\, or “soliton gases.” In this talk we focus on the Toda lattice\, which is an archetypal example of an integrable Hamiltonian dynamical system. We explain how the system\, under certain random initial data\, can be interpreted through solitons\, and provide a framework for studying how these solitons asymptotically evolve in time. The arguments use ideas from random matrix theory\, particularly the analysis of Lyapunov exponents governing the decay rates of eigenvectors of random tridiagonal matrices.
URL:https://cmsa.fas.harvard.edu/event/colloquium-32425/
LOCATION:CMSA Room G10\, CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Colloquium
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/CMSA-Colloquium-3.24.2025.docx.final_.png
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20250303T163000
DTEND;TZID=America/New_York:20250303T173000
DTSTAMP:20260515T164123
CREATED:20241209T163145Z
LAST-MODIFIED:20250218T153212Z
UID:10003630-1741019400-1741023000@cmsa.fas.harvard.edu
SUMMARY:Large value estimates in number theory and computer science
DESCRIPTION:Colloquium \nSpeaker: Larry Guth\, MIT \nTitle: Large value estimates in number theory and computer science \nAbstract: A large value estimate for a matrix M is a simple type of estimate in quantitative linear algebra. Estimates of this type appear in many parts of math\, both pure and applied. One example is the large value problem for Dirichlet polynomials from analytic number theory\, which is related to estimates about the zeroes of the Riemann zeta function. We will also give some examples from computer science. Many large value problems are difficult. On the pure math side\, the sharp conjecture about large values of Dirichlet polynomials has been open for a long time and is out of reach of current methods. On the computer science side\, we don’t know any efficient algorithm to approximately solve the large value problem for a given matrix M. Many experts think that such an algorithm does not exist. In this talk we will survey how large value estimates come up\, the known methods for working on them\, and some of the obstacles to fully understanding them. \n 
URL:https://cmsa.fas.harvard.edu/event/colloquium-3325/
LOCATION:CMSA Room G10\, CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Colloquium
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/CMSA-Colloquium-3.3.2025.png
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20250210T163000
DTEND;TZID=America/New_York:20250210T173000
DTSTAMP:20260515T164123
CREATED:20240903T195201Z
LAST-MODIFIED:20250130T165640Z
UID:10003438-1739205000-1739208600@cmsa.fas.harvard.edu
SUMMARY:AI in math and theoretical physics: Status and prospects
DESCRIPTION:Colloquium \nSpeaker: Michael Douglas\, Harvard CMSA \nTitle: AI in math and theoretical physics: status and prospects \nAbstract: AI is making great progress and has the potential to change how we work in unprecedented ways. In this talk I will survey a few recent works which illustrate the state of the art\, some from my own research\, some developed at the CMSA’s recent program on Mathematics and Machine Learning. I will then report on current developments in AI and speculate on how they will affect our work in the next few years. \n 
URL:https://cmsa.fas.harvard.edu/event/colloquium-21025/
LOCATION:CMSA Room G10\, CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Colloquium
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/CMSA-Colloquium-2.10.2025.png
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20250203T163000
DTEND;TZID=America/New_York:20250203T173000
DTSTAMP:20260515T164123
CREATED:20240903T194951Z
LAST-MODIFIED:20250130T165403Z
UID:10003434-1738600200-1738603800@cmsa.fas.harvard.edu
SUMMARY:Rational approximation and the AAA algorithm
DESCRIPTION:Colloquium \nSpeaker: Nick Trefethen\, Harvard University \nTitle: Rational approximation and the AAA algorithm \nApproximation by rational functions used to be mainly a theoretical subject\, but with the introduction of the AAA algorithm in 2018\, it became computationally practical and indeed easy. The implications for what we can do numerically are enormous. This talk will outline the algorithm and demonstrate its application to a collection of problems. We can also use it to demonstrate the potential theory that underlies the theory of rational approximation\, a topic that goes back to Joseph Walsh here at Harvard a century ago.
URL:https://cmsa.fas.harvard.edu/event/colloquium-2325/
LOCATION:CMSA Room G10\, CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Colloquium
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/CMSA-Colloquium-2.3.2025.png
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20241202T163000
DTEND;TZID=America/New_York:20241202T173000
DTSTAMP:20260515T164123
CREATED:20240903T195308Z
LAST-MODIFIED:20241126T142827Z
UID:10003440-1733157000-1733160600@cmsa.fas.harvard.edu
SUMMARY:Computability on $\mathbb R$ and other continuum-size structures
DESCRIPTION:Colloquium \nSpeaker: Russell Miller\, CUNY \nTitle: Computability on $\mathbb R$ and other continuum-size structures \nAbstract: We begin by recalling the notion of a computable function on the real numbers $\mathbb R$\, developed independently by Gregorczyk and Lacombe over sixty years ago. Using this notion\, we note that the real numbers that are themselves computable form a countable subfield of $\mathbb R$ with exactly the same first-order properties as $\mathbb R$ itself. (Logicians would therefore call it an \emph{elementary subfield}.) So\, in a first-order sense\, everything that happens in $\mathbb R$ is already exemplified in this much nicer subfield. However\, even when one knows that an existential statement holds for all parameters\, it may be impossible (both in $\mathbb R$ and in the subfield) to give a computable procedure for producing witnesses. Similar results hold in $\mathbb C$. \nWe will then turn to a different continuum-sized structure: the absolute Galois group $\operatorname{Gal}(\mathbb Q)$ of the rational numbers. Once again the computable elements of this group form a subgroup\, but now it is an open problem whether the group and the subgroup have the same first-order theory\, let alone whether this is an elementary subgroup. (If they do have the same theory\, this would put nice upper bounds on the complexity of the theory of $\operatorname{Gal}(\mathbb Q)$.) However\, using joint work with Kundu\, we can show that once again there is no computable procedure for producing witnesses to the truth of (true) existential statements\, either in the full group or in the subgroup.
URL:https://cmsa.fas.harvard.edu/event/colloquium-12224/
LOCATION:CMSA Room G10\, CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Colloquium
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/CMSA-Colloquium-12.2.2024.png
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20241125T163000
DTEND;TZID=America/New_York:20241125T173000
DTSTAMP:20260515T164123
CREATED:20240903T195237Z
LAST-MODIFIED:20241119T192853Z
UID:10003439-1732552200-1732555800@cmsa.fas.harvard.edu
SUMMARY:Mathematical Structures of Scattering Amplitudes
DESCRIPTION:Colloquium \nSpeaker: Anastasia Volovich\, Brown University \nTitle: Mathematical Structures of Scattering Amplitudes \nAbstract: Planar N=4 Yang-Mills scattering amplitudes have been computed to very high loop order. They have many remarkable properties that have sparked interest from mathematicians working on combinatorics\, algebraic geometry\, and number theory. At the same time\, several methods that have been developed for N=4 Yang-Mills can often be applied to more general quantum field theories\, including QCD. I will overview some of these exciting developments.
URL:https://cmsa.fas.harvard.edu/event/colloquium-112524/
LOCATION:CMSA Room G10\, CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Colloquium
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/CMSA-Colloquium-11.25.2024.png
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20241104T163000
DTEND;TZID=America/New_York:20241104T173000
DTSTAMP:20260515T164123
CREATED:20240903T195045Z
LAST-MODIFIED:20241016T202352Z
UID:10003436-1730737800-1730741400@cmsa.fas.harvard.edu
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://cmsa.fas.harvard.edu/event/colloquium-11424/
LOCATION:CMSA Room G10\, CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Colloquium
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/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:20260515T164123
CREATED:20240903T195022Z
LAST-MODIFIED:20241016T144838Z
UID:10003435-1729528200-1729531800@cmsa.fas.harvard.edu
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://cmsa.fas.harvard.edu/event/colloquium-102124/
LOCATION:CMSA Room G10\, CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Colloquium
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/CMSA-Colloquium-10.21.2024.png
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20241007T163000
DTEND;TZID=America/New_York:20241007T173000
DTSTAMP:20260515T164123
CREATED:20240903T194924Z
LAST-MODIFIED:20241003T160128Z
UID:10003433-1728318600-1728322200@cmsa.fas.harvard.edu
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://cmsa.fas.harvard.edu/event/colloquium-10724/
LOCATION:CMSA Room G10\, CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Colloquium
ATTACH;FMTTYPE=application/pdf:https://cmsa.fas.harvard.edu/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:20260515T164123
CREATED:20240903T194207Z
LAST-MODIFIED:20240918T190927Z
UID:10003431-1727109000-1727112600@cmsa.fas.harvard.edu
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://cmsa.fas.harvard.edu/event/colloquium-92324/
LOCATION:CMSA Room G10\, CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Colloquium
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/CMSA-Colloquium-09.23.2024.png
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20240916T163000
DTEND;TZID=America/New_York:20240916T173000
DTSTAMP:20260515T164123
CREATED:20240903T193540Z
LAST-MODIFIED:20240916T163127Z
UID:10003430-1726504200-1726507800@cmsa.fas.harvard.edu
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://cmsa.fas.harvard.edu/event/colloquium_91624/
LOCATION:CMSA Room G10\, CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Colloquium
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/CMSA-Colloquium-09.16.2024.png
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=America/New_York:20240909T163000
DTEND;TZID=America/New_York:20240909T173000
DTSTAMP:20260515T164123
CREATED:20240827T200454Z
LAST-MODIFIED:20240903T152309Z
UID:10003406-1725899400-1725903000@cmsa.fas.harvard.edu
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://cmsa.fas.harvard.edu/event/colloquium_9924/
LOCATION:CMSA Room G10\, CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Colloquium
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BEGIN:VEVENT
DTSTART;TZID=America/New_York:20240529T090000
DTEND;TZID=America/New_York:20240531T170000
DTSTAMP:20260515T164123
CREATED:20240105T071351Z
LAST-MODIFIED:20240624T164905Z
UID:10001120-1716973200-1717174800@cmsa.fas.harvard.edu
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://cmsa.fas.harvard.edu/event/amplituhedra2024/
LOCATION:CMSA Room G10\, CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Conference
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