### Third Annual Yip Lecture

Andrew Strominger will give the Third Annual Yip Lecture on February 2, 2023. Time: 7:00-8:00...

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Andrew Strominger will give the Third Annual Yip Lecture on February 2, 2023. Time: 7:00-8:00...

Read MoreThe CMSA will host a series of three 90-minute lectures on the subject of machine...

Read More(Dates TBA) In March 2023 Andrea Bertozzi (UCLA) will give two lectures hosted by the...

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November | November | November | November | 1 - WorkshopCMSA, 20 Garden Street, Cambridge, MA 02138 USA
On November 28 – Dec 1, 2022, the CMSA will host a Workshop on Representation Theory, Calabi-Yau Manifolds, and Mirror Symmetry. **Organizers:**An Huang (Brandeis University) | Siu-Cheong Lau (Boston University) | Tsung-Ju Lee (CMSA, Harvard) | Andrew Linshaw (University of Denver)**Scientific Advisor:**Shing-Tung Yau (Harvard, Tsinghua)**Location:**Room G10, CMSA, 20 Garden Street, Cambridge MA 02138Directions and Recommended Lodging The conference will be hybrid: it will be both in-person and online. **Registration is required.****Zoom webinar registration form: Zoom Webinar.**The workshop is partially supported by Simons and NSF Grant DMS-2227199. There are funds available for participant support, which will be allocated in keeping with guidelines – students, recent PhDs, underrepresented groups, and people with no other federal support get priority. Requests for support should be sent to the email address: rtcyms2022@gmail.com. The subject of your email should be “Request for support.” Please write a short description about your research and also attach your CV to the email. Speakers: - Tomoyuki Arakawa (Kyoto)
- Thomas Creutzig (Edmonton)
- Jonathan Mboyo Esole (Northeastern)
- Fei Han (National University of Singapore)
- Shinobu Hosono (Gakushuin University)
- Flor Orosz Hunziker (Colorado)
- Cuipo Jiang (Shanghai)
- Shashank Kanade (Denver)
- Matt Kerr (Washington University in St. Louis)
- Carl Lian (Humboldt-Universität zu Berlin)
- Nai-Chung Conan Leung (CUHK)
- Ivan Loseu (Yale)
- Robert McRae (Tsinghua University)
- Anne Moreau (Université Paris-Saclay, Orsay)
- Tony Pantev (University of Pennsylvania)
- Mauricio Romo (Tsinghua University)
- Bailin Song (USTC)
- Cumrun Vafa (Harvard University)
- Chin-Lung Wang (National Taiwan University)
- Weiqiang Wang (Virginia)
- Yaping Yang (University of Melbourne)
- Shing-Tung Yau (Tsinghua University)
- Chenglong Yu (Tsinghua University)
- Gufang Zhao (University of Melbourne)
**Schedule**(Eastern Time)11/28 (Monday) 08:30am – 08:55am Refreshments 08:55am – 09:00am Opening remarks by Horng-Tzer Yau 09:00am – 09:45am Shing-Tung Yau* **Title:**The Hull-Strominger system through conifold transitions**Abstract:**In this talk I discuss the geometry of C-Y manifolds outside of the Kähler regime and especially describe the Hull-Strominger system through the conifold transitions.10:00am – 10:45am Chenglong Yu* **Title:**Commensurabilities among Lattices in PU(1,n)**Abstract:**In joint work with Zhiwei Zheng, we study commensurabilities among certain subgroups in PU(1,n). Those groups arise from the monodromy of hypergeometric functions. Their discreteness and arithmeticity are classified by Deligne and Mostow. Thurston also obtained similar results via flat conic metrics. However, the classification of the lattices among them up to conjugation and finite index (commensurability) is not completed. When n=1, it is the commensurabilities of hyperbolic triangles. The cases of n=2 are almost resolved by Deligne-Mostow and Sauter’s commensurability pairs, and commensurability invariants by Kappes-Möller and McMullen. Our approach relies on the study of some higher dimensional Calabi-Yau type varieties instead of complex reflection groups. We obtain some relations and commensurability indices for higher n and also give new proofs for existing pairs in n=2.11:00am – 11:45am Thomas Creutzig* **Title:**Shifted equivariant W-algebras**Abstract:**The CDO of a compact Lie group is a family of VOAs whose top level is the space of functions on the Lie group. Similar structures appear at the intersections of boundary conditions in 4-dimensional gauge theories, I will call these new families of VOAs shifted equivariant W-algebras. I will introduce these algebras, construct them and explain how they can be used to quickly prove the GKO-coset realization of principal W-algebras.11:45am – 1:30 pm Lunch 01:30pm – 02:15pm Cumrun Vafa **Title:**Reflections on Mirror Symmetry**Abstract:**In this talk I review some of the motivations leading to the search and discovery of mirror symmetry as well as some of the applications it has had.02:30pm – 03:15pm Jonathan Mboyo Esole **Title:**Algebraic topology and matter representations in F-theory**Abstract:**Recently, it was observed that representations appearing in geometric engineering in F-theory all satisfy a unique property: they correspond to characteristic representations of embedding of Dynkin index one between Lie algebras. However, the reason why that is the case is still being understood. In this talk, I will present new insights, giving a geometric explanation for this fact using K-theory and the topology of Lie groups and their classifying spaces. In physics, this will be interpreted as conditions on the charge of instantons and the classifications of Wess-Zumino-Witten terms.03:15pm – 03:45 pm Break 03:45pm – 04:30pm Weiqiang Wang **Title:**A Drinfeld presentation of affine i-quantum groups**Abstract:**A quantum symmetric pair of affine type (U, U^i) consists of a Drinfeld-Jimbo affine quantum group (a quantum deformation of a loop algebra) U and its coideal subalgebra U^i (called i-quantum group). A loop presentation for U was formulated by Drinfeld and proved by Beck. In this talk, we explain how i-quantum groups can be viewed as a generalization of quantum groups, and then we give a Drinfeld type presentation for the affine quasi-split i-quantum group U^i. This is based on joint work with Ming Lu (Sichuan) and Weinan Zhang (Virginia).04:45pm – 05:30pm Tony Pantev **Title:**Decomposition, anomalies, and quantum symmetries**Abstract:**Decomposition is a phenomenon in quantum physics which converts quantum field theories with non-effectively acting gauge symmetries into equivalent more tractable theories in which the fields live on a disconnected space. I will explain the mathematical content of decomposition which turns out to be a higher categorical version of Pontryagin duality. I will examine how this duality interacts with quantum anomalies and secondary quantum symmetries and will show how the anomalies can be canceled by homotopy coherent actions of diagrams of groups. I will discuss in detail the case of 2-groupoids which plays a central role in anomaly cancellation, and will describe a new duality operation that yields decomposition in the presence of anomalies. The talk is based on joint works with Robbins, Sharpe, and Vandermeulen.11/29 (Tuesday) Refreshments 09:00am – 09:45am Robert MacRae* **Title:**Rationality for a large class of affine W-algebras**Abstract:**One of the most important results in vertex operator algebras is Huang’s theorem that the representation category of a “strongly rational” vertex operator algebra is a semisimple modular tensor category. Conversely, it has been conjectured that every (unitary) modular tensor category is the representation category of a strongly rational (unitary) vertex operator algebra. In this talk, I will describe my results on strong rationality for a large class of affine W-algebras at admissible levels. This yields a large family of modular tensor categories which generalize those associated to affine Lie algebras at positive integer levels, as well as those associated to the Virasoro algebra.10:00am – 10:45am Bailin Song* **Title:**The global sections of chiral de Rham complexes on compact Calabi-Yau manifolds**Abstract:**Chiral de Rham complex is a sheaf of vertex algebras on a complex manifold. We will describe the space of global sections of the chiral de Rham complexes on compact Calabi-Yau manifolds.11:00am – 11:45am Carl Lian* **Title:**Curve-counting with fixed domain**Abstract:**The fixed-domain curve-counting problem asks for the number of pointed curves of fixed (general) complex structure in a target variety X subject to incidence conditions at the marked points. The question comes in two flavors: one can ask for a virtual count coming from Gromov-Witten theory, in which case the answer can be computed (in principle) from the quantum cohomology of X, or one can ask for the “honest” geometric count, which tends to be more subtle. The answers are conjectured to agree in the presence of sufficient positivity, but do not always. I will give an overview of some recent results and open directions. Some of this work is joint with Alessio Cela, Gavril Farkas, and Rahul Pandharipande.11:45am – 01:30pm Lunch 01:30pm – 02:15pm Chin-Lung Wang **Title:**A blowup formula in quantum cohomology**Abstract:**We study analytic continuations of quantum cohomology $QH(Y)$ under a blowup $\phi: Y \to X$ of complex projective manifolds along the extremal ray variable $q^{\ell}$. Under $H(Y) = \phi^* H(X) plus K$ where $K = \ker \phi_*$, we show that (i) the restriction of quantum product along the $\phi^*H(X)$ direction, denoted by $QH(Y)_X$, is meromorphic in $x := 1/q^\ell$, (ii) $K$ deforms uniquely to a quantum ideal $\widetilde K$ in $QH(Y)_X$, (iii) the quotient ring $QH(Y)_X/\widetilde K$ is regular over $x$, and its restriction to $x = 0$ is isomorphic to $QH(X)$. This is a joint work (in progress) with Y.-P. Lee and H.-W. Lin.02:30pm – 03:15pm Ivan Loseu **Title:**Quantizations of nilpotent orbits and their Lagrangian subvarieties**Abstract:**I’ll report on some recent progress on classifying quantizations of the algebras of regular functions of nilpotent orbits (and their covers) in semisimple Lie algebras, as well as the classification of quantizations of certain Lagrangian subvarieties. An ultimate goal here is to understand the classification of unitary representations of real semisimple Lie groups.03:15pm – 03:45pm Break 03:45pm – 04:30pm Matt Kerr* **Title:**$K_2$ and quantum curves**Abstract:**The basic objects for this talk are motives consisting of a curve together with a $K_2$ class, and their mixed Hodge-theoretic invariants.My main objective will be to explain a connection (recently proved in joint work with C. Doran and S. Sinha Babu) between (i) Hodge-theoretically distinguished points in the moduli of such motives and (ii) eigenvalues of operators on L^2(R) obtained by quantizing the equations of the curves. By local mirror symmetry, this gives evidence for a conjecture in topological string theory (due to M. Marino, A. Grassi, and others) relating enumerative invariants of toric CY 3-folds to spectra of quantum curves. 04:45pm – 05:30pm Flor Orosz Hunziker **Title:**Tensor structures associated to the N=1 super Virasoro algebra**Abstract:**We have recently shown that there is a natural category of representations associated to the N=1 super Virasoro vertex operator algebras that have braided tensor structure. We will describe this category and discuss the problem of establishing its rigidity at particular central charges. This talk is based on joint work in progress with Thomas Creutzig, Robert McRae and Jinwei Yang.11/30 (Wednesday) 08:30am – 09:00am Refreshments 09:00am – 09:45am Tomoyuki Arakawa **Title:**4D/2D duality and representation theory**Abstract:**This talk is about the 4D/2D duality discovered by Beem et*al.*rather recently in physics. It associates a vertex operator algebra (VOA) to any 4-dimensional superconformal field theory, which is expected to be a complete invariant of thl theory. The VOAs appearing in this manner may be regarded as chiralization of various symplectic singularities and their representations are expected to be closely related with the Coulomb branch of the 4D theory. I will talk about this remarkable 4D/2D duality from a representation theoretic perspective.10:00am – 10:45am Shashank Kanade **Title**: Combinatorics of principal W-algebras of type A**Abstract**: The combinatorics of principal W_r(p,p’) algebras of type A is controlled by cylindric partitions. However, very little seems to be known in general about fermionic expressions for the corresponding characters. Welsh’s work explains the case of Virasoro minimal models W_2(p,p’). Andrews, Schilling and Warnaar invented and used an A_2 version of the usual (A_1) Bailey machinery to give fermionic characters (up to a factor of (q)_\infty) of some, but not all, W_3(3,p’) modules. In a recent joint work with Russell, we have given a complete set of conjectures encompassing all of the remaining modules for W_3(3,p’), and proved our conjectures for small values of p’. In another direction, characters of W_r(p,p’) algebras also arise as appropriate limits of certain sl_r coloured Jones invariants of torus knots T(p,p’), and we expect this to provide further insights on the underlying combinatorics.11:00am – 11:45am Gufang Zhao **Title:**Quasimaps to quivers with potentials**Abstract:**This talk concerns non-compact GIT quotient of a vector space, in the presence of an abelian group action and an equivariant regular function (potential) on the quotient. We define virtual counts of quasimaps from prestable curves to the critical locus of the potential. The construction borrows ideas from the theory of gauged linear sigma models as well as recent development in shifted symplectic geometry and Donaldson-Thomas theory of Calabi-Yau 4-folds. Examples of virtual counts arising from quivers with potentials are discussed. This is based on work in progress, in collaboration with Yalong Cao.11:45am – 01:30pm Group Photo, Lunch 01:30pm – 02:15pm Yaping Yang **Title:**Cohomological Hall algebras and perverse coherent sheaves on toric Calabi-Yau 3-folds**Abstract:**Let X be a smooth local toric Calabi-Yau 3-fold. On the cohomology of the moduli spaces of certain sheaves on X, there is an action of the cohomological Hall algebra (COHA) of Kontsevich and Soibelman via “raising operators”. I will discuss the “double” of the COHA that acts on the cohomology of the moduli space by adding the “lowering operators”. We associate a root system to X. The double COHA is expected to be the shifted Yangian of this root system. We also give a prediction for the shift in terms of an intersection pairing. We provide evidence of the aforementioned expectation in various examples. This is based on my joint work with M. Rapcak, Y. Soibelman, and G. Zhao02:30pm – 03:15pm Fei Han **Title:**Graded T-duality with H-flux for 2d sigma models**Abstract:**T-duality in string theory can be realised as a transformation acting on the worldsheet fields in the two-dimensional nonlinear sigma model. Bouwknegt-Evslin-Mathai established the T-duality in a background flux for the first time upon compactifying spacetime in one direction to a principal circle by constructing the T-dual maps transforming the twisted cohomology of the dual spacetimes. In this talk, we will describe our recent work on how to promote the T-duality maps of Bouwknegt-Evslin-Mathai in two aspects. More precisely, we will introduce (1) graded T-duality, concerning the graded T-duality maps of all levels of twistings; (2) the 2-dimensional sigma model picture, concerning the double loop space of spacetimes. This represents our joint work with Mathai.03:15pm – 3:45pm Break 03:45pm – 04:30pm Mauricio Romo **Title:**Networks and BPS Counting: A-branes view point**Abstract:**I will review the countings of BPS invariants via exponential/spectral networks and present an interpretation of this counting as a count of certain points in the moduli space of A-branes corresponding to degenerate Lagrangians.04:45pm – 05:30pm Shinobu Hosono **Title:**Mirror symmetry of abelian fibered Calabi-Yau manifolds with ρ = 2**Abstract:**I will describe mirror symmetry of Calabi-Yau manifolds fibered by (1,8)-polarized abelian surfaces, which have Picard number two. Finding a mirror family over a toric variety explicitly, I observe that mirror symmetry of all related Calabi-Yau manifods arises from the corresponding boundary points, which are not necessarily toric boundary points. Calculating Gromov-Witten invariants up to genus 2, I find that the generating functions are expressed elliptic (quasi-)modular forms, which reminds us the modular anomaly equation found for elliptic surfaces. This talk is based on a published work with Hiromichi Takaki (arXiv:2103.08150).06:00pm Banquet @ Royal East Restaurant, 782 Main St, Cambridge, MA 02139 12/1 (Thursday) 08:30am – 09:00am Refreshments 09:00am – 09:45am Conan Nai Chung Leung* **Title:**Quantization of Kahler manifolds**Abstract:**I will explain my recent work on relationships among geometric quantization, deformation quantization, Berezin-Toeplitz quantization and brane quantization.10:00am – 10:45am Cuipo Jiang* **Title:**Cohomological varieties associated to vertex operator algebras**Abstract:**We define and examine the cohomological variety of a vertex algebra, a notion cohomologically dual to that of the associated variety, which measures the smoothness of the associated scheme at the vertex point. We study its basic properties. As examples, we construct a closed subvariety of the cohomological variety for rational affine vertex operator algebras constructed from finite dimensional simple Lie algebras. We also determine the cohomological varieties of the simple Virasoro vertex operator algebras. These examples indicate that, although the associated variety for a rational $C_2$-cofinite vertex operator algebra is always a simple point, the cohomological variety can have as large a dimension as possible. This talk is based on joint work with Antoine Caradot and Zongzhu Lin.11:00am – 11:45am Anne Moreau* **Title:**Action of the automorphism group on the Jacobian of Klein’s quartic curve**Abstract:**In a joint work with Dimitri Markouchevitch, we prove that the quotient variety of the 3-dimensional Jacobian of the plane Klein quartic curve by its full automorphism group of order 336 is isomorphic to the 3-dimensional weighted projective space with weights 1,2,4,7.The latter isomorphism is a particular case of the general conjecture of Bernstein and Schwarzman suggesting that a quotient of the n-dimensional complex space by the action of an irreducible complex crystallographic group generated by reflections is a weighted projective space. In this talk, I will explain this conjecture and the proof of our result. An important ingredient is the computation of the Hilbert function of the algebra of invariant theta-functions on the Jacobian. 11:45am – 11:50am Closing remarks 11:50am Free discussions and departure * = Online speaker
| 2 - Member SeminarCMSA, 20 Garden Street, Cambridge, MA 02138 USA
**Member Seminar****Speaker:**Alejandro Poveda**Title:**Compactness and Anticompactness Principles in Set Theory**Abstract:**Several fundamental properties in Topology, Algebra or Logic are expressed in terms of Compactness Principles.For instance, a natural algebraic question is the following: Suppose that G is an Abelian group whose all small subgroups are free – Is the group G free? If the answer is affirmative one says that compactness holds; otherwise, we say that compactness fails. Loosely speaking, a compactness principle is anything that fits the following slogan: Suppose that M is a mathematical structure (a group, a topological space, etc) such that all of its small substructures N have certain property $\varphi$; then the ambient structure M has property $\varphi$, as well. Oftentimes when these questions are posed for infinite sets the problem becomes purely set-theoretical and axiom-sensitive. In this talk I will survey the most paradigmatic instances of compactness and present some related results of mine. If time permits, I will hint the proof of a recent result (joint with Rinot and Sinapova) showing that stationary reflection and the failure of the Singular Cardinal Hypothesis can co-exist. These are instances of two antagonist set-theoretic principles: the first is a compactness principle while the second is an anti-compactness one. This result solves a question by M. Magidor from 1982.
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4 | 5 - Quantum Matter
**Quantum Matter Seminar****Speaker:**Junkai Dong (Harvard University)**Title:**Exact Many-Body Ground States from Decomposition of Ideal Higher Chern Bands: Applications to Chirally Twisted Graphene Multilayers**Abstract:**Motivated by the higher Chern bands of twisted graphene multilayers, we consider flat bands with arbitrary Chern number C with ideal quantum geometry. While C>1 bands differ from Landau levels, we show that these bands host exact fractional Chern insulator (FCI) ground states for short range interactions. We show how to decompose ideal higher Chern bands into separate ideal bands with Chern number 1 that are intertwined through translation and rotation symmetry. The decomposed bands admit an SU(C) action that combines real space and momentum space translations. Remarkably, they also allow for analytic construction of exact many-body ground states, such as generalized quantum Hall ferromagnets and FCIs, including flavor-singlet Halperin states and Laughlin ferromagnets in the limit of short-range interactions. In this limit, the SU(C) action is promoted to a symmetry on the ground state subspace. While flavor singlet states are translation symmetric, the flavor ferromagnets correspond to translation broken states and admit charged skyrmion excitations corresponding to a spatially varying density wave pattern. We confirm our analytic predictions with numerical simulations of ideal bands of twisted chiral multilayers of graphene, and discuss consequences for experimentally accessible systems such as monolayer graphene twisted relative to a Bernal bilayer. - Swampland Seminar17 Oxford St, Cambridge, MA 02138
**Swampland Seminar****Location:**Jefferson Physical Laboratory, Seminar room J356**Speaker:**Markus Dierigl (LMU Munich)**Title:**IIB Explored – Dualities, Bordisms, and the Swampland**Abstract:**In this talk I will discuss the application of the Cobordism Conjecture to type IIB supergravity with non-trivial duality bundle. Calculating the relevant bordism groups we find that they are highly non-trivial and would predict the presence of various global symmetries in the underlying theory. Since quantum gravity theories do not allow for global symmetries, we discuss which defects need to be included to break them completely. Interestingly, we find many backgrounds that are well-known in the F-theory literature, such as [p,q]-7-branes, non-Higgsable clusters, as well as S-folds and their generalizations to higher codimensions. Further including worldsheet reflections, predicts the existence of a new non-supersymmetric 7-brane with interesting properties and applications, which I will discuss in more detail.
| 6 - Quantum Matter
**Quantum Matter Seminar****Speaker:**Sungwoo Hong (U Chicago & KAIST)**Title:**Neutrino Masses from Generalized Symmetry Breaking**Abstract:**We explore generalized global symmetries in theories of physics beyond the Standard Model. Theories of Z′ bosons generically contain ‘non-invertible’ chiral symmetries, whose presence indicates a natural paradigm to break this symmetry by an exponentially small amount in an ultraviolet completion. For example, in models of gauged lepton family difference such as the phenomenologically well-motivated U(1)Lμ−Lτ, there is a non-invertible lepton number symmetry which protects neutrino masses. We embed these theories in gauged non-Abelian horizontal lepton symmetries, e.g. U(1)Lμ−Lτ⊂SU(3)H, where the generalized symmetries are broken nonperturbatively by the existence of lepton family magnetic monopoles. In such theories, either Majorana or Dirac neutrino masses may be generated through quantum gauge theory effects from the charged lepton Yukawas e.g. yν∼yτexp(−Sinst). These theories require no bevy of new fields nor ad hoc additional global symmetries, but are instead simple, natural, and predictive: the discovery of a lepton family Z′ at low energies will reveal the scale at which Lμ−Lτ emerges from a larger gauge symmetry.
| 7 - Topological Quantum Matter SeminarCMSA, 20 Garden Street, Cambridge, MA 02138 USA
**Topological Quantum Matter Seminar****Speaker**: Vasil Rokaj (Harvard)**Title:**Controlling Quantum Matter with Quantum Cavity Fields**Abstract:**Cavity modification of material properties and phenomena is a novel research field motivated by the advances in strong light-matter interactions [1]. For condensed matter systems it has been demonstrated experimentally that the transport properties of 2D materials can be modified via coupling to vacuum fields [2,3]. While in polaritonic chemistry it has been shown that ground state chemical properties can be controlled with cavity fields [4]. In the first part of my talk, I will present how the quantized cavity field can alter the conduction properties of a condensed matter system by focusing on the paradigmatic Sommerfeld model of the free electron gas [5]. The exact analytic solution of the Sommerfeld model in the cavity will be presented as well as its fundamental properties. Then, in the second part of the talk, I will focus on a many-particle system of cold ions in a harmonic trap coupled to the cavity field. I will show how this system couples collectively to the cavity and that hybrid states between light and matter, known as polaritons, emerge. The formation of polaritons leads to the modification of the properties of the cold ions and enhances the localization of the many-body wave function [6]. Connections to experiments will be discussed as well.[1] F. Garcia-Vidal, C. Ciuti, T. W. Ebbesen, Science, 373, 178 (2021) [2] G. L. Paravicini-Bagliani et al., Nat. Phys. 15, 186-190 (2019) [3] F. Appugliese et al., Science 375 (6584), 1030-1034 (2022) [4] T. W. Ebbesen, Acc. Chem. Res. 49, 11, 2403–2412 (2016) [5] V. Rokaj, M. Ruggenthaler, F. G. Eich, A. Rubio, Phys. Rev. Research 4, 013012 (2022) [6] V. Rokaj, S.I. Mistakidis, H.R. Sadeghpour, arXiv:2207.03436 (2022) - New Technologies in Mathematics SeminarCMSA, 20 Garden Street, Cambridge, MA 02138 USA
**New Technologies in Mathematics Seminar****Speaker:**Cyril Zhang, Microsoft Research**Title:**How do Transformers reason? First principles via automata, semigroups, and circuits**Abstract:**The current “Transformer era” of deep learning is marked by the emergence of combinatorial and algorithmic reasoning capabilities in large sequence models, leading to dramatic advances in natural language understanding, program synthesis, and theorem proving. What is the nature of these models’ internal representations (i.e. how do they represent the states and computational steps of the algorithms they execute)? How can we understand and mitigate their weaknesses, given that they resist interpretation? In this work, we present some insights (and many further mysteries) through the lens of automata and their algebraic structure.Specifically, we investigate the apparent mismatch between recurrent models of computation (automata & Turing machines) and Transformers (which are typically shallow and non-recurrent). Using tools from circuit complexity and semigroup theory, we characterize shortcut solutions, whereby a shallow Transformer with only o(T) layers can exactly replicate T computational steps of an automaton. We show that Transformers can efficiently represent these shortcuts in theory; furthermore, in synthetic experiments, standard training successfully finds these shortcuts. We demonstrate that shortcuts can lead to statistical brittleness, and discuss mitigations. Joint work with Bingbin Liu, Jordan Ash, Surbhi Goel, and Akshay Krishnamurthy. - Probability Seminar
##### Probability Seminar Fourier quasicrystals and stable polynomials1 Oxford Street, Cambridge MA 02138**Probability Seminar***Note location change:*Science Center Room 300H**Speaker:**Lior Alon (MIT)**Title:**Fourier quasicrystals and stable polynomials**Abstract:**The Poisson summation formula says that the countable sum of exp(int), over all integers n, vanishes as long as t is not an integer multiple of 2 pi. Can we find a non-periodic discrete set A, such that the sum of exp(iat), over a in A, vanishes for all t outside of a discrete set? The surprising answer is yes. Yves Meyer called the atomic measure supported on such a set a crystalline measure. Crystalline measures provide another surprising connection between physics (quasicrystals) and number theory (the zeros of the Zeta and L functions under GRH). A recent work of Pavel Kurasov and Peter Sarnak provided a construction of crystalline measures with ‘good’ convergence (Fourier quasicrystals) using stable polynomials, a family of multivariate polynomials that were previously used in proving the Lee-Yang circle theorem and the Kadison-Singer conjecture. After providing the needed background, I will discuss a recent work in progress with Cynthia Vinzant on the classification of these Kurasov-Sarnak measures and their supporting sets. We prove that these sets have well-defined gap distributions. We show that each Kurasov-Sarnak measure decomposes according to the irreducible decomposition of its associated polynomial, and the measures associated with each irreducible factor is either supported on an arithmetic progression, or its support has a bounded intersection with any arithmetic progression. Finally, we construct random Kurasov-Sarnak measures with gap distribution as close as we want to the eigenvalues spacing of a random unitary matrix.Based on joint work with Pravesh Kothari.
| 8 - General Relativity Seminar
**General Relativity Seminar****Speaker:**Allen Fang (Princeton)**Title:**A new proof for the nonlinear stability of slowly-rotating Kerr-de Sitter**Abstract:**The nonlinear stability of the slowly-rotating Kerr-de Sitter family was first proven by Hintz and Vasy in 2016 using microlocal techniques. In my talk, I will present a novel proof of the nonlinear stability of slowly-rotating Kerr-de Sitter spacetimes that avoids frequency-space techniques outside of a neighborhood of the trapped set. The proof uses vector field techniques to uncover a spectral gap corresponding to exponential decay at the level of the linearized equation. The exponential decay of solutions to the linearized problem is then used in a bootstrap proof to conclude nonlinear stability. - Active Matter SeminarLyman Lab, 17 Oxford Street, Cambridge MA 02138
**Active Matter Seminar****Speaker:**Julien Tailleur (MIT)**Title:**Self-organization of motile cells by quorum-sensing or chemotactic interactions**Abstract:**Equilibrium statistical mechanics tells us how to control the self-assembly of passive materials by tuning the competition between energy and entropy to achieve desired states of organization. Out of equilibrium, no such principles apply and self-organization principles are scarce. Active matter describes systems in which individual units dissipate energy to exert forces on their environment. Dissipation and injection of energy are then disconnected at the microscopic scale, hence driving the system strongly out of thermal equilibrium. This leads to a phenomenology markedly different from that of equilibrium systems, such as the emergence of dense phases in the absence of cohesive attractive forces, but it also leaves us without guiding principles to understand the self-organization of active matter. In this talk, I will review the progress which has been made over the past ten years to control the organization of self-propelled agents using motility control, either externally or through interactions. I will show that generic principles apply and illustrate the theoretical developments presented in the talk using recent experiments on the motility-induced self-organization of bacterial mixtures.
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