The Role of Orientational Order in Development
2023-03-30 13:00 - 14:00 (America/New_York) | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Active Matter Seminar

Speaker: Mark Bowick, Kavli Institute for Theoretical Physics, UCSB
Title: The Role of Orientational Order in Development
Abstract: Morphogenesis, the process through which genes generate form, establishes tissue scale order as a template for constructing the complex shapes of the body plan. The extensive growth required to build these ordered substrates is fueled by cell proliferation, which, naively, should disrupt order. Understanding how active morphogenetic mechanisms couple cellular and mechanical processes to generate order remains an outstanding question in animal development. I will review the statistical mechanics of orientational order and discuss the observation of a fourfold orientationally ordered phase (tetratic) in the model organism Parhyale hawaiensis. I will also discuss theoretical mechanisms for the formation of orientational order that require both motility and cell division, with support from self-propelled vertex models of tissue. The aim is to uncover a robust, active mechanism for generating global orientational order in a non-equilibrium system that then sets the stage for the development of shape and form.

Kähler-Einstein metrics on families of Fano varieties
2023-04-03 10:00 - 11:00 | Virtual
Algebraic Geometry in String Theory Seminar
Speaker: Chung-Ming Pan, Institut de Mathématiques de Toulouse
Title: Kähler-Einstein metrics on families of Fano varieties
Abstract: This talk aims to introduce a pluripotential approach to study uniform a priori estimates of Kähler-Einstein (KE) metrics on families of Fano varieties. I will first recall basic tools in the pluripotential theory and the variational approach to complex Monge-Ampère equations. I will then define a notion of convergence of quasi-plurisubharmonic functions in families of normal varieties and extend several classical properties under this context. Last, I will explain how these elements help to obtain a purely analytic proof of the openness of existing singular KE metrics and a uniform $L^infty$ estimate of KE potentials. This is joint work with Antonio Trusiani.

Black hole microstate counting from the gravitational path integral
2023-04-03 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Luca Iliesiu, Stanford
Title: Black hole microstate counting from the gravitational path integral
Abstract: Reproducing the integer count of black hole micro-states from the gravitational path integral is an important problem in quantum gravity. In the first part of the talk, I will show that, by using supersymmetric localization, the gravitational path integral for 1/16-BPS black holes in supergravity can reproduce the index obtained in the string theory construction of such black holes. A more refined argument then shows that not only the black hole index but also the total number of black hole microstates within an energy window above extremality that is polynomially suppressed in the charges also matches this string theory index. In the second part of the talk, I will present a second perspective on this state count and show how the BPS Hilbert space can be obtained by directly preparing states using the gravitational path integral. While such a preparation naively gives rise to a Hilbert space of BPS states whose dimension is much larger than expected, I will explain how non-perturbative corrections in the overlap of such states are again responsible for reproducing the correct dimension of the Hilbert space.

Optimal Dynamic Allocation: Simplicity through Information Design
2023-04-04 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Faidra Monachou
Title: Optimal Dynamic Allocation: Simplicity through Information Design
Abstract: We study dynamic nonmonetary markets where objects are allocated to unit-demand agents with private types. An agent’s value for an object is supermodular in her type and the quality of the object, and her payoff is quasilinear in her waiting cost. The social planner’s objective is a linear combination of allocative efficiency (i.e., the sum of values) and welfare (i.e., the sum of payoffs). We identify the optimal mechanism in the class of direct-revelation mechanisms that elicit agents’ types and assign them to objects over time. We show that, when the social planner can design the information disclosed to the agents about the objects, the optimal mechanism has a simple implementation: a first-come first-served waitlist with deferrals. In this implementation, the information disclosed about each object is an interval containing the object quality, rather than the exact quality. These intervals partition the quality space. We also show that when the planner’s objective weight on welfare increases, these intervals become coarser, and optimal disclosure policies less informative. A direct corollary is that mechanisms that achieve higher welfare also induce lower distributional inequality, in terms of the Lorenz order.

Sampling from the SK and mixed p-spin measures with stochastic localization
2023-04-05 15:30 - 16:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Probability Seminar

Speaker: Ahmed El Alaoui (Cornell)
Title: Sampling from the SK and mixed p-spin measures with stochastic localization
Abstract: I will present an algorithm which efficiently samples from the Sherrington-Kirkpatrick (SK) measure with no external field at high temperature. The approach is based on the stochastic localization process of Eldan, together with a subroutine for computing the mean vectors of a family of measures tilted by an appropriate external field. Conversely, we show that no ‘stable’ algorithm can approximately sample from the SK measure at low temperature. Time permitting, we discuss extensions to the p-spin model. This is based on a joint work with Andrea Montanari and Mark Sellke.

Rough solutions of the relativistic Euler equations
2023-04-06 09:30 - 10:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
General Relativity Seminar
Speaker: Sifan Yu, Vanderbilt University
Title: Rough solutions of the relativistic Euler equations
Abstract: I will discuss recent works on the relativistic Euler equations with dynamic vorticity and entropy. We use a new formulation of the equations, which has geo-analytic structures. In this geometric formulation, we decompose the flow into geometric “sound-wave part” and “transport-div-curl part”. This allows us to derive sharp results about the dynamics, including the existence of low-regularity solutions. Then, I will discuss the results of rough solutions of the relativistic Euler equations and the role that nonlinear geometric optics plays in the framework. Our main result is that the Sobolev norm $H^{2+}$ of the variables in the “wave-part” and the H”older norm $C^{0,0+}$ of the variables in the “transport-part” can be controlled in terms of initial data for short times. We note that the Sobolev norm assumption $H^{2+}$ is the optimal result for the variables in the “wave-part.” This talk will include the main ideas of the proof, as well as a comparison of the relativistic and non-relativistic scenarios.

Enhancing Detection of Topological Order by Local Error Correction
2023-04-07 10:00 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Matter Seminar
Speaker: Nishad Maskara (Harvard)
Title: Enhancing Detection of Topological Order by Local Error Correction
Abstract: The exploration of topologically-ordered states of matter is a long-standing goal at the interface of several subfields of the physical sciences. Such states feature intriguing physical properties such as long-range entanglement, emergent gauge fields and non-local correlations, and can aid in realization of scalable fault-tolerant quantum computation. However, these same features also make creation, detection, and characterization of topologically-ordered states particularly challenging. Motivated by recent experimental demonstrations, we introduce a new paradigm for quantifying topological states—locally error-corrected decoration (LED)—by combining methods of error correction with ideas of renormalization-group flow. Our approach allows for efficient and robust identification of topological order, and is applicable in the presence of incoherent noise sources, making it particularly suitable for realistic experiments. We demonstrate the power of LED using numerical simulations of the toric code under a variety of perturbations, and we subsequently apply it to an experimental realization of a quantum spin liquid using a Rydberg-atom quantum simulator. Finally, we illustrate how LED can be applied to more general phases including non-abelian topological orders.

Modular graph forms and iterated integrals in string amplitudes
2023-04-07 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar
Speaker: Oliver Schlotterer (Uppsala University)
Title: Modular graph forms and iterated integrals in string amplitudes
Abstract: I will discuss string amplitudes as a laboratory for special functions and period integrals that drive fruitful cross-talk with particle physicists and mathematicians. At genus zero, integration over punctures on a disk or sphere worldsheet generates multiple zeta values in the low-energy expansion of open- and closed-string amplitudes. At genus one, closed-string amplitudes introduce infinite families of non-holomorphic modular forms through the integration over torus punctures known as modular graph forms. The latter inspired Francis Brown’s alternative construction of non-holomorphic modular forms in the mathematics literature via iterated integrals, and I will report on recent progress in clarifying their connection with modular graph forms.

Current Developments in Mathematics Conference 2023
2023-04-07 14:00 - 2023-04-08 17:00 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
Current Developments in Mathematics 2023
Harvard University Science Center, Lecture Hall C
April 7-8, 2023
Speakers:
Amol Aggarwal – Columbia University
Bhargav Bhatt – Institute for Advanced Study, Princeton University, & University of Michigan
Paul Bourgade – New York University, Courant Institute
Vesselin Dimitrov – Institute for Advanced Study & Georgia Institute of Technology
Greta Panova – University of Southern California

For more information, and to register, please visit:
Current Developments in Mathematics 2023

Organizers: David Jerison, Paul Seidel, Nike Sun (MIT); Denis Auroux, Mark Kisin, Lauren Williams, Horng-Tzer Yau
Sponsored by the National Science Foundation, Harvard University Mathematics, Harvard University Center of Mathematical Sciences and Applications, and the Massachusetts Institute of Technology.
Harvard University is committed to maintaining a safe and healthy educational and work environment in which no member of the University community is, on the basis of sex, sexual orientation, or gender identity, excluded from participation in, denied the benefits of, or subjected to discrimination in any University program or activity. More information can be found here.

Swampland bounds on the abelian gauge sectors
2023-04-10 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Swampland Seminar
Speaker: Seung-Joo Lee (IBS Daejeon)

Title: Swampland bounds on the abelian gauge sectors
Abstract: In this talk we will derive various bounds on the 0-form and the 1-form abelian gauge sectors of gravitational effective theories in 6 dimensions with minimal supersymmetry. We will start by considering 6-dimensional F-theory vacua with at least one tensor multiplets, to bound for them the number of the (0-form) U(1) gauge factors as well as the cyclic orders of the 1-form discrete gauge factors. While the two abelian gauge sectors may look rather independent, we will observe that both are heavily constrained by the solitonic heterotic strings present in the spectrum, which provide a common intuition for the derived bounds. Building upon the heterotic intuition, we will also try extending the arena to address analogous bounds for all F-theory vacua in 6 dimensions and even beyond. If time permits, several applications and future directions of research will be discussed at the end of the talk.

Localization for random band matrices
2023-04-10 15:00 - 16:00 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
Probability Seminar
*Please note room change: Science Center 232*

Speaker: Ron Peled (Tel Aviv University)
Title: Localization for random band matrices
Abstract: I will explain an approach via “an adaptive Mermin-Wagner style shift” which proves localization of N x N Gaussian random band matrices with band width W satisfying W << N^{1/4}.
Joint work with Giorgio Cipolloni, Jeffrey Schenker and Jacob Shapiro.

Spectral gap and two-point functions in spin glasses
2023-04-11 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Changji Xu
Title: Spectral gap and two-point functions in spin glasses
Abstract: Many have worked on spin glass models over the past 50 years, including physicists, mathematicians, and computers. A question that arises is whether computers yield dependable simulation results. In this talk, I will discuss some recent mathematical progress on spectral gap and two-point functions in spin glasses models.

Unexpected Uses of Neural Networks: Field Theory and Metric Flows
2023-04-12 12:30 - 13:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: James Halverson (Northeastern University)

Title: Unexpected Uses of Neural Networks: Field Theory and Metric Flows
Abstract: We are now quite used to the idea that deep neural networks may be trained in a variety of ways to tackle cutting-edge problems in physics and mathematics, sometimes leading to rigorous results. In this talk, however, I will argue that breakthroughs in deep learning theory are also useful for making progress, focusing on applications to field theory and metric flows. Specifically, I will introduce a neural network approach to field theory with a different statistical origin, that exhibits generalized free field behavior at infinite width and interactions at finite width, and that allows for the study of symmetries via the study of correlation functions in a different duality frame. Then, I will review recent progress in approximating Calabi-Yau metrics as neural networks and cast that story into the language of neural tangent kernel theory, yielding a theoretical understanding of neural network metric flows induced by gradient descent and recovering famous metric flows, such as Perelman’s formulation of Ricci flow, in particular limits.

Large deviations of Selberg’s central limit theorem
2023-04-12 15:30 - 16:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Probability Seminar

Speaker: Emma Bailey (CUNY)
Title: Large deviations of Selberg’s central limit theorem
Abstract: Selberg’s CLT concerns the typical behaviour of the Riemann zeta function and shows that the random variable $Re log zeta(1/2 + i t)$, for a uniformly drawn $t$, behaves as a Gaussian random variable with a particular variance. It is natural to investigate how far into the tails this Gaussianity persists, which is the topic of this work. There are also very close connections to similar problems in circular unitary ensemble characteristic polynomials. It transpires that a `multiscale scheme’ can be applied to both situations to understand these questions of large deviations, as well as certain maxima and moments. In this talk I will focus more on the techniques we apply to approach this problem and I will assume no number theoretic knowledge. This is joint work with Louis-Pierre Arguin.

Resolving the photon ring
2023-04-13 09:30 - 10:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
General Relativity Seminar
Speaker: Shahar Hadar (University of Haifa)
Title: Resolving the photon ring
Abstract: In the past few years, the Event Horizon Telescope has released the first close-up interferometric images of two supermassive black holes, M87* and SgrA*. It is believed that within these images is embedded a fine, yet-unresolved brightness enhancement called the photon ring. The ring is a universal consequence of strong lensing by the black hole and thereby conveys information on its spacetime geometry, potentially providing a new independent avenue for tests of general relativity in the strong-field regime. In the talk I will briefly review the theory of the photon ring and its corresponding spacetime region, the photon shell, which governs the universal lensing structure. I will then describe some current efforts and future prospects for resolving the ring, which include both the construction of transformative new instruments and the development of novel analysis methods. Focusing on the latter, I will present an upcoming proposal to use spectro-temporal autocorrelations in signals emitted from black hole environs as a probe of strong lensing effects.

Control of actin cable length by decelerated growth and network geometry
2023-04-13 13:00 - 14:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Active Matter Seminar

Speaker: Shane McInally, Brandeis
Title: Control of actin cable length by decelerated growth and network geometry
Abstract: The sizes of many subcellular structures are coordinated with cell size to ensure that these structures meet the functional demands of the cell. In eukaryotic cells, these subcellular structures are often membrane-bound organelles, whose volume is the physiologically important aspect of their size. Scaling organelle volume with cell volume can be explained by limiting pool mechanisms, wherein a constant concentration of molecular building blocks enables subcellular structures to increase in size proportionally with cell volume. However, limiting pool mechanisms cannot explain how the size of linear subcellular structures, such as cytoskeletal filaments, scale with the linear dimensions of the cell. Recently, we discovered that the length of actin cables in budding yeast (used for intracellular transport) precisely matches the length of the cell in which they are assembled. Using mathematical modeling and quantitative imaging of actin cable growth dynamics, we found that as the actin cables grow longer, their extension rates slow (or decelerate), enabling cable length to match cell length. Importantly, this deceleration behavior is cell-length dependent, allowing cables in longer cells to grow faster, and therefore reach a longer length before growth stops at the back of the cell. In addition, we have unexpectedly found that cable length is specified by cable shape. Our imaging analysis reveals that cables progressively taper as they extend from the bud neck into the mother cell, and further, this tapering scales with cell length. Integrating observations made for tapering actin networks in other systems, we have developed a novel mathematical model for cable length control that recapitulates our quantitative experimental observations. Unlike other models of size control, this model does not require length-dependent rates of assembly or disassembly. Instead, feedback control over the length of the cable is an emergent property due to the cross-linked and bundled architecture of the actin filaments within the cable. This work reveals a new strategy that cells use to coordinate the size of their internal parts with their linear dimensions. Similar design principles may control the size and scaling of other subcellular structures whose physiologically important dimension is their length.

Fault-tolerant quantum computation via topological order on fractals and emergent symmetries
2023-04-14 10:00 - 11:30 | Hybrid – G10
Quantum Matter Seminar
Speaker: Guanyu Zhu (IBM Quantum, T. J. Watson Research Center)
Title: Fault-tolerant quantum computation via topological order on fractals and emergent symmetries
Abstract: Topological quantum error correcting codes in integer spatial dimensions have been widely studied in the field of quantum information. A remaining major challenge is to reduce the space-time overhead for universal fault-tolerant quantum computation with topological codes. In the first part of my talk, I will present a theory of topological order and quantum codes on fractals embedded in three and higher dimensions and its connection to systolic geometry. The construction of such fractal codes can hence significantly reduce the space overhead. In the second part, I will show how to perform fault-tolerant non-Clifford logical gates in such fractal codes using the idea of emergent symmetries. In particular, I will discuss the existence of higher-form symmetries corresponding to sweeping of certain codimension-2 invertible defects and exotic gapped boundaries which condense such defects.
References:
1. PRX Quantum 3 (3), 030338 (2022), Guanyu Zhu, Tomas Jochym-O’Connor, Arpit Dua
2. arXiv:2201.03568 (2022), Arpit Dua, Tomas Jochym-O'Connor, Guanyu Zhu
3. arXiv:2208.07367 (2022), Maissam Barkeshli, Yu-An Chen, Sheng-Jie Huang, Ryohei Kobayashi, Nathanan Tantivasadakarn, Guanyu Zhu

Recent progress on coupled Einstein-Yang-Mills dynamics
2023-04-18 12:00 - 13:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Puskar Mondal
Title: Recent progress on coupled Einstein-Yang-Mills dynamics
Abstract: I will discuss my recent work with prof. S.T. Yau on Einstein-Yang-Mills theory. I’ll discuss the main ideas behind the stability of the Minkowski and Milne space time under coupled gauge-gravity perturbations. Construction of the gauge invariant energies that control the suitable norms of the physical degrees of freedom in a double-null framework play an important role. I’ll sketch how the special structure of the Einstein and Yang-Mills nonlinearities play a crucial role in obtaining the stability result of non-expanding spacetimes in particular AF spacetimes.

Diagonalizing Transition Matrices of Card Shuffles
2023-04-19 15:30 - 16:30 | Science Center 232, Harvard Science Center, 1 Oxford Street, Cambridge MA 02138
Probability Seminar
Speaker: Evita Nestoridi (Stonybrook)

Title: Diagonalizing Transition Matrices of Card Shuffles
Abstract: In their seminal work, Diaconis and Shahshahani used representation theory of the symmetric group to diagonalize the transition matrix of random transpositions. More recently, Dieker and Saliola introduced another technique to diagonalize the random-to-random card shuffle. In this talk we will discuss connections between these techniques as well as application to card shuffling.

Black hole collider physics
2023-04-20 13:30 - 14:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Julio Parra Martinez, Caltech
Title: Black hole collider physics
Abstract: Despite more than a century since the development of Einstein’s theory, the general relativistic two-body problem remains unsolved. A precise description of its solution is now essential, as it is necessary for understanding the strong-gravity dynamics of compact binaries observed at LIGO/VIRGO/KAGRA and in future gravitational wave observatories. In this talk, I will describe how considering the scattering of black holes and gravitons can shed new light on this problem. I will explain how using modern ideas from collider and particle physics we can calculate scattering observables in classical gravity, and extract the basic ingredients that describe the bound binary dynamics. Such calculations have produced state-of-art predictions for current and future gravitational wave observatories, which open the door for further discovery as we enter this new era of precision gravitational physics.

A model of the cuprates: from the pseudogap metal to d-wave superconductivity and charge order
2023-04-21 10:00 - 11:30 | Hybrid – G10
Quantum Matter Seminar
Speaker: Prof. Subir Sachdev (Harvard)
Title: A model of the cuprates: from the pseudogap metal to d-wave superconductivity and charge order
Abstract: Soon after the discovery of high-temperature superconductivity in the cuprates, Anderson proposed a connection to quantum spin liquids. But observations since then have shown that the low-temperature phase diagram is dominated by conventional states, with a competition between superconductivity and charge-ordered states which break translational symmetry. We employ the “pseudogap metal” phase, found at intermediate temperatures and low hole doping, as the parent to the phases found at lower temperatures. The pseudogap metal is described as a fractionalized phase of a single-band model, with small pocket Fermi surfaces of electron-like quasiparticles whose enclosed area is not equal to the free electron value, and an underlying pi-flux spin liquid with an emergent SU(2) gauge field. This pi-flux spin liquid is now known to be unstable to confinement at sufficiently low energies. We develop a theory of the different routes to confinement of the pi-flux spin liquid and show that d-wave superconductivity, antiferromagnetism, and charge order are natural outcomes. We argue that this theory provides routes to resolving a number of open puzzles on the cuprate phase diagram.
As a side result, at half-filling, we propose a deconfined quantum critical point between an antiferromagnet and a d-wave superconductor described by a conformal gauge theory of 2 flavors of massless Dirac fermions and 2 flavors of complex scalars coupled as fundamentals to a SU(2) gauge field.
This talk is based on Maine Christos, Zhu-Xi Luo, Henry Shackleton, Mathias S. Scheurer, and S. S., arXiv:2302.07885

The Tameness of Quantum Field Theories
2023-04-24 11:00 - 12:00 | Jefferson 368
Swampland Seminar
Speaker: Thomas Grimm (Utrecht U.)

Title: The Tameness of Quantum Field Theories
Abstract: Tameness is a generalized notion of finiteness that is restricting the geometric complexity of sets and functions. The underlying mathematical foundation lies in tame geometry, which is built from o-minimal structures introduced in mathematical logic. In this talk I formalize the connection between quantum field theories and logical structures and argue that the tameness of a quantum field theory relies on its UV definition. I quantify our expectations on the tameness of effective theories that can be coupled to quantum gravity and on CFTs. In particular, I present tameness conjectures about CFT observables and propose universal constraints that render spaces of CFTs to be tame sets. I then highlight the relation of these conjectures to other swampland conjectures, e.g., by arguing that the tameness of CFT observables restricts having parametrical gaps in the operator spectrum.

Recent advances in scalar curvature and positive mass theorems
2023-04-24 13:30 - 14:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
General Relativity Seminar
Speaker: Tin Yau Tsang, University of California Irvine
Title: Recent advances in scalar curvature and positive mass theorems

Abstract: First, we have a review of classical tools for studying scalar curvature and positive mass theorem. Then we are going to discuss some advances and new perspectives on these tools which lead to a deeper understanding of geometry and initial data sets.

On complete Calabi-Yau metrics and some related Monge-Ampere equations
2023-04-25 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Freid Tong
Title: On complete Calabi-Yau metrics and some related Monge-Ampere equations
Abstract: We will give a basic introduction to constructions for complete Calabi-Yau metrics. A systematic approach to construct such metrics using PDE methods was proposed in the work of Tian-Yau in the 90s and have attracted a lot of attention in recent years. I will discuss some joint work with B. Guo and T. Collins on a singular version of such a construction, as well as some ongoing work with Prof. Yau on some related boundary value problems.

Boundary behavior at classical and quantum phase transitions
2023-04-26 12:30 - 13:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Max Metlitski (MIT)
Title: Boundary behavior at classical and quantum phase transitions
Abstract: There has been a lot of recent interest in the boundary behavior of materials. This interest is driven in part by the field of topological states of quantum matter, where exotic protected boundary states are ubiquitous. In this talk, I’ll ask: what happens at a boundary of a system, when the bulk goes through a phase transition. While this question was studied in the context of classical statistical mechanics in the 70s and 80s, basic aspects of the boundary phase diagram for the simplest classical phase transitions have been missed until recently. I’ll describe progress in this field, as well as some extensions to quantum phase transitions.

Toolformer: Language Models Can Teach Themselves to Use Tools
2023-04-26 14:00 - 15:00 | Virtual
New Technologies in Mathematics Seminar
Speaker: Timo Schick, Meta AI
Title: Toolformer: Language Models Can Teach Themselves to Use Tools
Abstract: Language models exhibit remarkable abilities to solve new tasks from just a few examples or textual instructions, especially at scale. They also, paradoxically, struggle with basic functionality, such as arithmetic or factual lookup, where much simpler and smaller models excel. In this talk, we show how these limitations can be overcome by letting language models teach themselves to use external tools via simple APIs. We discuss Toolformer, a model trained to independently decide which APIs to call, when to call them, what arguments to pass, and how to best incorporate the results into future token prediction. Through this, it achieves substantially improved zero-shot performance across a variety of downstream tasks without sacrificing its core language modeling abilities.

Boundary current fluctuations for the half space ASEP
2023-04-26 15:30 - 16:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Probability Seminar
Speaker: Jimmy He (MIT)
Title: Boundary current fluctuations for the half space ASEP
Abstract: The half space asymmetric simple exclusion process (ASEP) is an interacting particle system on the half line, with particles allowed to enter/exit at the boundary. I will discuss recent work on understanding fluctuations for the number of particles in the half space ASEP started with no particles, which exhibits the Baik-Rains phase transition between GSE, GOE, and Gaussian fluctuations as the boundary rates vary. As part of the proof, we find new distributional identities relating this system to two other models, the half space Hall-Littlewood process, and the free boundary Schur process, which allows exact formulas to be computed.

The localized seed-to-solution method for the Einstein constraints
2023-04-27 10:30 - 11:30 | Virtual
General Relativity Seminar
Speaker: Philippe G. LeFloch, Sorbonne University and CNRS
Title: The localized seed-to-solution method for the Einstein constraints
Abstract: I will discuss advances on asymptotically Euclidian initial data sets and the variational method introduced by J. Corvino and R. Schoen. This talk is based on joint papers with The-Cang Nguyen (Montpellier) and Bruno Le Floch (Sorbonne Univ. and CNRS). In the vicinity of any given reference data set, we define a “localized seed-to-solution” map, which allows us to parametrize the initial data sets satisfying the Einstein constraints (possibly with matter fields). The parametrization is defined over classes of data sets understood modulo the image of the dual linearized constraints. Our main contribution concerns the sharp behavior of solutions at infinity, which we can arbitrarily localize in asymptotic cones in the sense of A. Carlotto and R. Schoen. Most importantly, as we prove it, the solutions enjoy sharp decay estimates at the harmonic and super-harmonic levels. In the course of this analysis, we discover the notion of ‘asymptotic modulators’, as we call them, or “correctors” to the standard ADM invariants.

Competition at the front of expanding populations
2023-04-27 13:00 - 14:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Active Matter Seminar

Speaker: Mehran Kardar, MIT
Title: Competition at the front of expanding populations
Abstract: When competing species grow into new territory, the population is dominated by descendants of successful ancestors at the expansion front. Successful ancestry depends on the reproductive advantage (fitness), as well as ability and opportunity to colonize new domains. (1) Based on symmetry considerations, we present a model that integrates both elements by coupling the classic description of one-dimensional competition (Fisher equation) to the minimal model of front shape (KPZ equation). Macroscopic manifestations of these equations on growth morphology are explored, providing a framework to study spatial competition, fixation, and differentiation, In particular, we find that ability to expand in space may overcome reproductive advantage in colonizing new territory. (2) Variations of fitness, as well as fixation time upon differentiation, are shown to belong to distinct universality classes depending on limits to gain of fitness.

Fracton Self-Statistics
2023-04-28 10:00 - 11:30 | Virtual
Quantum Matter Seminar
Title: Fracton Self-Statistics
Speaker: Hao Song (ITP-CAS)
Abstract: Fracton order describes novel quantum phases of matter that host quasiparticles with restricted mobility, and thus lies beyond the existing paradigm of topological order. In particular, excitations that cannot move without creating other excitations are called fractons. Here we address a fundamental open question — can the notion of self-exchange statistics be naturally defined for fractons, given their complete immobility as isolated excitations? Surprisingly, we demonstrate how fractons can be exchanged, and show their self-statistics is a key part of the characterization of fracton orders. We derive general constraints satisfied by the fracton self-statistics in a large class of abelian fracton orders. Finally, we show the existence of semionic or fermionic fracton self-statistics in some twisted variants of the checkerboard model and Haah’s code, establishing that these models are in distinct quantum phases as compared to their untwisted cousins.
References: H Song, N Tantivasadakarn, W Shirley, M Hermele, arXiv:2304.00028.
