CMSA

Moduli Space of Metric SUSY Graphs
2022-03-18 09:30 - 10:30 (America/New_York) | Virtual
Member Seminar
Speaker: Yingying Wu
Title: Moduli Space of Metric SUSY Graphs
Abstract: SUSY curves are algebraic curves with additional supersymmetric or supergeometric structures. In this talk, I will present the construction of dual graphs of SUSY curves with Neveu–Schwarz and Ramond punctures. Then, I will introduce the concept of the metrized SUSY graph and the moduli space of the metric SUSY graphs. I will outline its geometric and topological properties, followed by a discussion on the connection with the classical case.

Fluctuation scaling or Taylor’s law of heavy-tailed data, illustrated by U.S. COVID-19 cases and deaths
2022-03-23 09:30 - 10:30 | MA
Speaker: Joel E. Cohen (Rockefeller University and Columbia University)
Title: Fluctuation scaling or Taylor’s law of heavy-tailed data, illustrated by U.S. COVID-19 cases and deaths
Abstract: Over the last century, ecologists, statisticians, physicists, financial quants, and other scientists discovered that, in many examples, the sample variance approximates a power of the sample mean of each of a set of samples of nonnegative quantities. This power-law relationship of variance to mean is known as a power variance function in statistics, as Taylor’s law in ecology, and as fluctuation scaling in physics and financial mathematics. This survey talk will emphasize ideas, motivations, recent theoretical results, and applications rather than detailed proofs. Many models intended to explain Taylor’s law assume the probability distribution underlying each sample has finite mean and variance. Recently, colleagues and I generalized Taylor’s law to samples from probability distributions with infinite mean or infinite variance and higher moments. For such heavy-tailed distributions, we extended Taylor’s law to higher moments than the mean and variance and to upper and lower semivariances (measures of upside and downside portfolio risk). In unpublished work, we suggest that U.S. COVID-19 cases and deaths illustrate Taylor’s law arising from a distribution with finite mean and infinite variance. This model has practical implications. Collaborators in this work are Mark Brown, Richard A. Davis, Victor de la Peña, Gennady Samorodnitsky, Chuan-Fa Tang, and Sheung Chi Phillip Yam.

Formal Mathematics Statement Curriculum Learning
2022-03-23 14:00 - 15:00 | MA
Speaker: Stanislas Polu, OpenAI
Title: Formal Mathematics Statement Curriculum Learning
Abstract: We explore the use of expert iteration in the context of language modeling applied to formal mathematics. We show that at same compute budget, expert iteration, by which we mean proof search interleaved with learning, dramatically outperforms proof search only. We also observe that when applied to a collection of formal statements of sufficiently varied difficulty, expert iteration is capable of finding and solving a curriculum of increasingly difficult problems, without the need for associated ground-truth proofs. Finally, by applying this expert iteration to a manually curated set of problem statements, we achieve state-of-the-art on the miniF2F benchmark, automatically solving multiple challenging problems drawn from high school olympiads.

Periods for singular CY families and Riemann–Hilbert correspondence
2022-03-25 09:30 - 10:30 | MA
Member Seminar
Speaker: Tsung-Ju Lee
Title: Periods for singular CY families and Riemann–Hilbert correspondence
Abstract: A GKZ system, introduced by Gelfand, Kapranov, and Zelevinsky, is a system of partial differential equations generalizing the hypergeometric structure studied by Euler and Gauss. The solutions to GKZ systems have been found applications in various branches of mathematics including number theory, algebraic geometry and mirror symmetry. In this talk, I will explain the details and demonstrate how to find the Riemann–Hilbert partner of the GKZ system with a fractional parameter which arises from the B model of singular CY varieties. This is a joint work with Dingxin Zhang.

Edge Modes and Gravity
2022-03-30 09:30 - 10:30 | MA
Speaker: Rob Leigh, UIUC
Title: Edge Modes and Gravity
Abstract: In this talk I first review some of the many appearances of localized degrees of freedom — edge modes — in a variety of physical systems. Edge modes are implicated for example in quantum entanglement and in various topological and holographic dualities. I then review recent work in which it has been realized that a careful treatment of such modes, paying attention to relevant symmetries, is required in order to properly understand such basic physical quantities as Noether charges. From many points of view, it is conjectured that this physics may be pointing at basic properties of quantum spacetimes and gravity.

Memorizing Transformers
2022-03-30 14:00 - 15:00 | Virtual
Speaker: Yuhuai Wu, Stanford and Google
Title: Memorizing Transformers
Abstract: Language models typically need to be trained or fine-tuned in order to acquire new knowledge, which involves updating their weights. We instead envision language models that can simply read and memorize new data at inference time, thus acquiring new knowledge immediately. In this talk, I will discuss how we extend language models with the ability to memorize the internal representations of past inputs. We demonstrate that an approximate NN lookup into a non-differentiable memory of recent (key, value) pairs improves language modeling across various benchmarks and tasks, including generic webtext (C4), math papers (arXiv), books (PG-19), code (Github), as well as formal theorems (Isabelle). We show that the performance steadily improves when we increase the size of memory up to 262K tokens. We also find that the model is capable of making use of newly defined functions and theorems during test time.

Diffusive growth sourced by topological defects
2022-04-01 09:00 - 10:30 | Virtual
Member Seminar
Speaker: Farzan Vafa
Title: Diffusive growth sourced by topological defects
Abstract: In this talk, we develop a minimal model of morphogenesis of a surface where the dynamics of the intrinsic geometry is diffusive growth sourced by topological defects. We show that a positive (negative) defect can dynamically generate a cone (hyperbolic cone). We analytically explain features of the growth profile as a function of position and time, and predict that in the presence of a positive defect, a bump forms with height profile h(t) ~ t^(1/2) for early times t. To incorporate the effect of the mean curvature, we exploit the fact that for axisymmetric surfaces, the extrinsic geometry can be deduced entirely by the intrinsic geometry. We find that the resulting stationary geometry, for polar order and small bending modulus, is a deformed football.
We apply our framework to various biological systems. In an ex-vivo setting of cultured murine neural progenitor cells, we show that our framework is consistent with the observed cell accumulation at positive defects and depletion at negative defects. In an in-vivo setting, we show that the defect configuration consisting of a bound +1 defect state, which is stabilized by activity, surrounded by two -1/2 defects can create a stationary ring configuration of tentacles, consistent with observations of a basal marine invertebrate Hydra

What is Mathematical Consciousness Science?
2022-04-06 09:30 - 10:30 | MA
Speaker: Johannes Kleiner, LMU München
Title: What is Mathematical Consciousness Science?
Abstract: In the last three decades, the problem of consciousness – how and why physical systems such as the brain have conscious experiences – has received increasing attention among neuroscientists, psychologists, and philosophers. Recently, a decidedly mathematical perspective has emerged as well, which is now called Mathematical Consciousness Science. In this talk, I will give an introduction and overview of Mathematical Consciousness Science for mathematicians, including a bottom-up introduction to the problem of consciousness and how it is amenable to mathematical tools and methods.

Synthetic Regression Discontinuity: Estimating Treatment Effects using Machine Learning
2022-04-08 08:45 - 10:15 | Virtual
Speaker: Jörn Boehnke
Title: Synthetic Regression Discontinuity: Estimating Treatment Effects using Machine Learning
Abstract: In the standard regression discontinuity setting, treatment assignment is based on whether a unit’s observable score (running variable) crosses a known threshold. We propose a two-stage method to estimate the treatment effect when the score is unobservable to the econometrician while the treatment status is known for all units. In the first stage, we use a statistical model to predict a unit’s treatment status based on a continuous synthetic score. In the second stage, we apply a regression discontinuity design using the predicted synthetic score as the running variable to estimate the treatment effect on an outcome of interest. We establish conditions under which the method identifies the local treatment effect for a unit at the threshold of the unobservable score, the same parameter that a standard regression discontinuity design with known score would identify. We also examine the properties of the estimator using simulations, and propose the use machine learning algorithms to achieve high prediction accuracy. Finally, we apply the method to measure the effect of an investment grade rating on corporate bond prices by any of the three largest credit ratings agencies. We find an average 1% increase in the prices of corporate bonds that received an investment grade as opposed to a non-investment grade rating.

Quantisation in monoidal categories and quantum operads
2022-04-13 09:30 - 10:30 | MA
Abstract: The standard definition of symmetries of a structure given on a set S (in the sense of Bourbaki) is the group of bijective maps S to S, compatible with this structure. But in fact, symmetries of various structures related to storing and transmitting information (such as information spaces) are naturally embodied in various classes of loops such as Moufang loops, – nonassociative analogs of groups. The idea of symmetry as a group is closely related to classical physics, in a very definite sense, going back at least to Archimedes. When quantum physics started to replace classical, it turned out that classical symmetries must also be replaced by their quantum versions, e.g. quantum groups.
In this talk we explain how to define and study quantum versions of symmetries, relevant to information theory and other contexts.

Long common subsequences between bit-strings and the zero-rate threshold of deletion-correcting codes
2022-04-27 09:30 - 10:30 | MA
Speaker: Venkatesan Guruswami, UC Berkeley
Title: Long common subsequences between bit-strings and the zero-rate threshold of deletion-correcting codes
Abstract: Suppose we transmit n bits on a noisy channel that deletes some fraction of the bits arbitrarily. What’s the supremum p* of deletion fractions that can be corrected with a binary code of non-vanishing rate? Evidently p* is at most 1/2 as the adversary can delete all occurrences of the minority bit. It was unknown whether this simple upper bound could be improved, or one could in fact correct deletion fractions approaching 1/2.
We show that there exist absolute constants A and delta > 0 such that any subset of n-bit strings of size exp((log n)^A) must contain two strings with a common subsequence of length (1/2+delta)n. This immediately implies that the zero-rate threshold p* of worst-case bit deletions is bounded away from 1/2.
Our techniques include string regularity arguments and a structural lemma that classifies bit-strings by their oscillation patterns. Leveraging these tools, we find in any large code two strings with similar oscillation patterns, which is exploited to find a long common subsequence.
This is joint work with Xiaoyu He and Ray Li.

Machine Learning the Gravity Equation for International Trade
2022-04-29 09:30 - 11:00 | Virtual
Member Seminar
Speaker: Sergiy Verstyuk
Title: Machine Learning the Gravity Equation for International Trade
Abstract: We will go through modern deep learning methods and existing approaches to their interpretation. Next, I will describe a graph neural network framework. You will also be introduced to an economic analog of gravity. Finally, we will see how these tools can help understand observed trade flows between 181 countries over 68 years. [Joint work with Michael R. Douglas.]

Cobordism and Deformation Class of the Standard Model and Beyond: Proton Stability and Neutrino Mass
2022-05-13 09:30 - 11:00 | MA
Member Seminar
Speaker: Juven Wang
Title: Cobordism and Deformation Class of the Standard Model and Beyond: Proton Stability and Neutrino Mass
Abstract: ‘t Hooft anomalies of quantum field theories (QFTs) with an invertible global symmetry G (including spacetime and internal symmetries) in a d-dim spacetime are known to be classified by a d+1-dim cobordism group TPd+1(G), whose group generator is a d+1-dim cobordism invariant written as a d+1-dim invertible topological field theory. Deformation class of QFT is recently proposed to be specified by its symmetry G and a d+1-dim invertible topological field theory. Seemly different QFTs of the same deformation class can be deformed to each other via quantum phase transitions. We ask which deformation class controls the 4d ungauged or gauged (SU(3)×SU(2)×U(1))/Zq Standard Model (SM) for q=1,2,3,6 with a continuous or discrete (B−L) symmetry and with also a compatible discrete baryon plus lepton Z_{2Nf} B+L symmetry. (The Z_{2Nf} B+L is discrete due to the ABJ anomaly under the BPST instanton.) We explore a systematic classification of candidate perturbative local and nonperturbative global anomalies of the 4d SM, including all these gauge and gravitational backgrounds, via a cobordism theory, which controls the SM’s deformation class. While many Grand Unified Theories violating the discrete B+L symmetry suffer from the proton decay, the SM and some versions of Ultra Unification (constrained by Z_{16} class global anomaly that replaces sterile neutrinos with new exotic gapped/gapless topological or conformal sectors) can have a stable proton. Dictated by a Z_2 class global mixed gauge-gravitational anomaly, there can be a gapless deconfined quantum critical region between Georgi-Glashow and Pati-Salam models — the Standard Model and beyond occur as neighbor phases. We will also comment on a new mechanism to give the neutrino mass via topological field theories and topological defects. Work based on arXiv:2112.14765, arXiv:2204.08393, arXiv:2202.13498 and references therein.

Statistical Mechanics of Mutilated Sheets and Shells
2022-05-18 09:30 - 10:30 | MA
Speaker: David Nelson, Harvard University
Title: Statistical Mechanics of Mutilated Sheets and Shells
Abstract: Understanding deformations of macroscopic thin plates and shells has a long and rich history, culminating with the Foeppl-von Karman equations in 1904, a precursor of general relativity characterized by a dimensionless coupling constant (the “Foeppl-von Karman number”) that can easily reach vK = 10^7 in an ordinary sheet of writing paper. However, thermal fluctuations in thin elastic membranes fundamentally alter the long wavelength physics, as exemplified by experiments that twist and bend individual atomically-thin free-standing graphene sheets (with vK = 10^13!) A crumpling transition out of the flat phase for thermalized elastic membranes has been predicted when kT is large compared to the microscopic bending stiffness, which could have interesting consequences for Dirac cones of electrons embedded in graphene. It may be possible to lower the crumpling temperature for graphene to more readily accessible range by inserting a regular lattice of laser-cut perforations, an expectation an confirmed by extensive molecular dynamics simulations. We then move on to analyze the physics of sheets mutilated with puckers and stitches. Puckers and stitches lead to Ising-like phase transitions riding on a background of flexural phonons, as well as an anomalous coefficient of thermal expansion. Finally, we argue that thin membranes with a background curvature lead to thermalized spherical shells that must collapse beyond a critical size at room temperature, even in the absence of an external pressure.

7/22/2020 Quantum Matter Seminar
2022-07-22 09:30 - 11:00 | MA

Duality in Einstein’s Gravity
2022-09-09 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Title: Duality in Einstein’s Gravity
Abstract: Electric-Magnetic duality has been a key feature behind our understanding of Quantum Field Theory for over a century. In this talk I will describe a similar property in Einstein’s gravity. The gravitational duality reveals, in turn, a wide range of new IR phenomena, including aspects of the double copy for scattering amplitudes, asymptotic symmetries and more.

Strategyproof-Exposing Mechanisms Descriptions
2022-09-14 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Yannai Gonczarowski (Harvard)

Title: Strategyproof-Exposing Mechanisms Descriptions
Abstract: One of the crowning achievements of the field of Mechanism Design has been the design and usage of the so-called “Deferred Acceptance” matching algorithm. Designed in 1962 and awarded the Nobel Prize in 2012, this algorithm has been used around the world in settings ranging from matching students to schools to matching medical doctors to residencies. A hallmark of this algorithm is that unlike many other matching algorithms, it is “strategy-proof”: participants can never gain by misreporting their preferences (say, over schools) to the algorithm. Alas, this property is far from apparent from the algorithm description. Its mathematical proof is so delicate and complex, that (for example) school districts in which it is implemented do not even attempt to explain to students and parents why this property holds, but rather resort to an appeal to authority: Nobel laureates have proven this property, so one should listen to them. Unsurprisingly perhaps, there is a growing body of evidence that participants in Deferred Acceptance attempt (unsuccessfully) to “game it,” which results in a suboptimal match for themselves and for others.
By developing a novel framework of algorithm description simplicity—grounded at the intersection between Economics and Computer Science—we present a novel, starkly different, yet equivalent, description for the Deferred Acceptance algorithm, which, in a precise sense, makes its strategyproofness far more apparent. Our description does have a downside, though: some other of its most fundamental properties—for instance, that no school exceeds its capacity—are far less apparent than from all traditional descriptions of the algorithm. Using the theoretical framework that we develop, we mathematically address the question of whether and to what extent this downside is unavoidable, providing a possible explanation for why our description of the algorithm has eluded discovery for over half a century. Indeed, it seems that in the design of all traditional descriptions of the algorithm, it was taken for granted that properties such as no capacity getting exceeded should be apparent. Our description emphasizes the property that is important for participants to correctly interact with the algorithm, at the expense of properties that are mostly of interest to policy makers, and thus has the potential of vastly improving access to opportunity for many populations. Our theory provides a principled way of recasting algorithm descriptions in a way that makes certain properties of interest easier to explain and grasp, which we also support with behavioral experiments in the lab.
Joint work with Ori Heffetz and Clayton Thomas.

Breaking the one-mind-barrier in mathematics using formal verification
2022-09-14 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speaker: Johan Commelin, Mathematisches Institut, Albert-Ludwigs-Universität Freiburg
Title: Breaking the one-mind-barrier in mathematics using formal verification
Abstract: In this talk I will argue that formal verification helps break the one-mind-barrier in mathematics. Indeed, formal verification allows a team of mathematicians to collaborate on a project, without one person understanding all parts of the project. At the same time, it also allows a mathematician to rapidly free mental RAM in order to work on a different component of a project. It thus also expands the one-mind-barrier.
I will use the Liquid Tensor Experiment as an example, to illustrate the above two points. This project recently finished the formalization of the main theorem of liquid vector spaces, following up on a challenge by Peter Scholze.
Video

Derivation of AdS/CFT for Vector Models
2022-09-16 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar

Speaker: Shai Chester

Title: Derivation of AdS/CFT for Vector Models
Abstract: We derive an explicit map at finite N between the singlet sector of the free and critical O(N) and U(N) vector models in any spacetime dimension above two, and a bulk higher spin theory in anti-de Sitter space in one higher dimension. For the boundary theory, we use the bilocal formalism of Jevicki et al to restrict to the singlet sector of the vector model. The bulk theory is defined from the boundary theory via our mapping, and is a consistent quantum higher spin theory with a well defined action. Our mapping relates bilocal operators in the boundary theory to higher spin fields in the bulk, while single trace local operators in the boundary theory are related to boundary values of higher spin fields. We also discuss generalizations of the map to gauge theories, and at finite temperature.

Moduli spaces of graphs
2022-09-21 12:30 - 13:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium

Speaker: Melody Chan, Brown

Title: Moduli spaces of graphs

Abstract: A metric graph is a graph—a finite network of vertices and edges—together with a prescription of a positive real length on each edge. I’ll use the term “moduli space of graphs” to refer to certain combinatorial spaces—think simplicial complexes—that furnish parameter spaces for metric graphs. There are different flavors of spaces depending on some additional choices of decorations on the graphs, but roughly, each cell parametrizes all possible metrizations of a fixed combinatorial graph. Many flavors of these moduli spaces have been in circulation for a while, starting with the work of Culler-Vogtmann in the 1980s on Outer Space. They have also recently played an important role in some recent advances using tropical geometry to study the topology of moduli spaces of curves and other related spaces. These advances give me an excuse to give what I hope will be an accessible introduction to moduli spaces of graphs and their connections with geometry.

Random determinants, the elastic manifold, and landscape complexity beyond invariance
2022-09-23 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Ben McKenna
Title: Random determinants, the elastic manifold, and landscape complexity beyond invariance
Abstract: The Kac-Rice formula allows one to study the complexity of high-dimensional Gaussian random functions (meaning asymptotic counts of critical points) via the determinants of large random matrices. We present new results on determinant asymptotics for non-invariant random matrices, and use them to compute the (annealed) complexity for several types of landscapes. We focus especially on the elastic manifold, a classical disordered elastic system studied for example by Fisher (1986) in fixed dimension and by Mézard and Parisi (1992) in the high-dimensional limit. We confirm recent formulas of Fyodorov and Le Doussal (2020) on the model in the Mézard-Parisi setting, identifying the boundary between simple and glassy phases. Joint work with Gérard Ben Arous and Paul Bourgade.

The Tree Property and uncountable cardinals
2022-09-28 12:30 - 13:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Dima Sinapova (Rutgers University)
Title: The Tree Property and uncountable cardinals
Abstract: In the late 19th century Cantor discovered that there are different levels of infinity. More precisely he showed that there is no bijection between the natural numbers and the real numbers, meaning that the reals are uncountable. He then went on to discover a whole hierarchy of infinite cardinal numbers. It is natural to ask if finitary and countably infinite combinatorial objects have uncountable analogues. It turns out that the answer is yes.
We will focus on one such key combinatorial property, the tree property. A classical result from graph theory (König’s infinity lemma) shows the existence of this property for countable trees. We will discuss what happens in the case of uncountable trees.

Statistical mechanics of neural networks: From the geometry of high dimensional error landscapes to beating power law neural scaling
2022-09-28 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics
Speaker: Surya Ganguli, Stanford University

Title: Statistical mechanics of neural networks: From the geometry of high dimensional error landscapes to beating power law neural scaling

Abstract: Statistical mechanics and neural network theory have long enjoyed fruitful interactions. We will review some of our recent work in this area and then focus on two vignettes. First we will analyze the high dimensional geometry of neural network error landscapes that happen to arise as the classical limit of a dissipative many-body quantum optimizer. In particular, we will be able to use the Kac-Rice formula and the replica method to calculate the number, location, energy levels, and Hessian eigenspectra of all critical points of any index. Second we will review recent work on neural power laws, which reveal that the error of many neural networks falls off as a power law with network size or dataset size. Such power laws have motivated significant societal investments in large scale model training and data collection efforts. Inspired by statistical mechanics calculations, we show both in theory and in practice how we can beat neural power law scaling with respect to dataset size, sometimes achieving exponential scaling, by collecting small carefully curated datasets rather than large random ones.

References: Y. Bahri, J. Kadmon, J. Pennington, S. Schoenholz, J. Sohl-Dickstein, and S. Ganguli, Statistical mechanics of deep learning, Annual Reviews of Condensed Matter Physics, 2020.

Sorscher, Ben, Robert Geirhos, Shashank Shekhar, Surya Ganguli, and Ari S. Morcos. 2022. Beyond Neural Scaling Laws: Beating Power Law Scaling via Data Pruning https://arxiv.org/abs/2206.14486 (NeurIPS 2022).

Kahler geometry in twisted materials
2022-09-30 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Jie Wang
Title: Kahler geometry in twisted materials
Abstract: Flatbands are versatile platform for realizing exotic quantum phases due to the enhanced interactions. The canonical example is Landau level where fractional quantum Hall physics exists. Although interaction is strong, the fractional quantum Hall effect is relatively well understood thanks to its model wavefunction, exact parent Hamiltonian, conformal field theory analogous and other exact aspects. In generic flatbands, the interacting physics is controlled by the interplay between the interaction scale and intrinsic quantum geometries, in particular the Berry curvature and the Fubini-Study metric, which are in general spatially non-uniform. It is commonly believed that the non-uniform geometries destroy these exact properties of fractional quantum Hall physics, making many-body states less stable in flatbands.
In this talk, I will disprove this common belief by showing a large family of flatbands (ideal flatbands) where quantum geometries can be highly non-uniform, but still exhibit exact properties such as model wavefunctions, density algebra, exact parent Hamiltonians. I will discuss both the theory of ideal flatband, its experimental realization in Dirac materials as well as implications.

Minerva: Solving Quantitative Reasoning Problems with Language Models
2022-10-05 14:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speaker: Guy Gur-Ari, Google Research
Title: Minerva: Solving Quantitative Reasoning Problems with Language Models
Abstract: Quantitative reasoning tasks which can involve mathematics, science, and programming are often challenging for machine learning models in general and for language models in particular. We show that transformer-based language models obtain significantly better performance on math and science questions when trained in an unsupervised way on a large, math-focused dataset. Performance can be further improved using prompting and sampling techniques including chain-of-thought and majority voting. Minerva, a model that combines these techniques, achieves SOTA on several math and science benchmarks. I will describe the model, its capabilities and limitations.

Quantum statistical mechanics of charged black holes and strange metals
2022-10-05 16:00 - 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Please note this colloquium will be held at a special time: 4:00-5:00 pm.
Speaker: Subir Sachdev (Harvard)
Title: Quantum statistical mechanics of charged black holes and strange metals

Abstract: The Sachdev-Ye-Kitaev model was introduced as a toy model of interacting fermions without any particle-like excitations. I will describe how this toy model yields the universal low energy quantum theory of generic charged black holes in asymptotically 3+1 dimensional Minkowski space. I will also discuss how extensions of the SYK model yield a realistic theory of the strange metal phase of correlated electron systems.

Slides: cmsa22

Principal flow, sub-manifold and boundary
2022-10-07 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Zhigang Yao
Title: Principal flow, sub-manifold and boundary
Abstract: While classical statistics has dealt with observations which are real numbers or elements of a real vector space, nowadays many statistical problems of high interest in the sciences deal with the analysis of data which consist of more complex objects, taking values in spaces which are naturally not (Euclidean) vector spaces but which still feature some geometric structure. I will discuss the problem of finding principal components to the multivariate datasets, that lie on an embedded nonlinear Riemannian manifold within the higher-dimensional space. The aim is to extend the geometric interpretation of PCA, while being able to capture the non-geodesic form of variation in the data. I will introduce the concept of a principal sub-manifold, a manifold passing through the center of the data, and at any point on the manifold extending in the direction of highest variation in the space spanned by the eigenvectors of the local tangent space PCA. We show the principal sub-manifold yields the usual principal components in Euclidean space. We illustrate how to find, use and interpret the principal sub-manifold, by which a principal boundary can be further defined for data sets on manifolds.

Complete disorder is impossible: Some topics in Ramsey theory
2022-10-12 12:30 - 13:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: James Cummings,Carnegie Mellon University
Title: Complete disorder is impossible: Some topics in Ramsey theory
Abstract: The classical infinite Ramsey theorem states that if we colour pairs of natural numbers using two colours, there is an infinite set all of whose pairs get the same colour. This is the beginning of a rich theory, which touches on many areas of mathematics including graph theory, set theory and dynamics. I will give an overview of Ramsey theory, emphasizing the diverse ideas which are at play in this area.

Quantum magnet chains and Kashiwara crystals
2022-10-14 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Leonid Rybnikov, Harvard CMSA/National Research University Higher School of Economics
Title: Quantum magnet chains and Kashiwara crystals
Abstract: Solutions of the algebraic Bethe ansatz for quantum magnet chains are, generally, multivalued functions of the parameters of the integrable system. I will explain how to compute some monodromies of solutions of Bethe ansatz for the Gaudin magnet chain. Namely, the Bethe eigenvectors in the Gaudin model can be regarded as a covering of the Deligne-Mumford moduli space of stable rational curves, which is unramified over the real locus of the Deligne-Mumford space. The monodromy action of the fundamental group of this space (called cactus group) on the eigenlines can be described very explicitly in purely combinatorial terms of Kashiwara crystals — i.e. combinatorial objects modeling the tensor category of finite-dimensional representations of a semisimple Lie algebra g. More specifically, this monodromy action is naturally equivalent to the action of the same group by commutors (i.e. combinatorial analog of a braiding) on a tensor product of Kashiwara crystals. This is joint work with Iva Halacheva, Joel Kamnitzer, and Alex Weekes.

The Mobility Edge of Lévy Matrices
2022-10-19 12:30 - 13:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Patrick Lopatto (Brown)
Title: The Mobility Edge of Lévy Matrices
Abstract: Lévy matrices are symmetric random matrices whose entry distributions lie in the domain of attraction of an alpha-stable law; such distributions have infinite variance when alpha is less than 2. Due to the ubiquity of heavy-tailed randomness, these models have been broadly applied in physics, finance, and statistics. When the entries have infinite mean, Lévy matrices are predicted to exhibit a phase transition separating a region of delocalized eigenvectors from one with localized eigenvectors. We will discuss the physical context for this conjecture, and describe a result establishing it for values of alpha close to zero and one. This is joint work with Amol Aggarwal and Charles Bordenave.
