CMSA

Cobordism and Deformation Class of the Standard Model and Beyond: Proton Stability and Neutrino Mass
2022-05-13 09:30 - 11:00 (America/New_York) | 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.

Towards Faithful Reasoning Using Language Models
2022-10-19 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speaker: Antonia Creswell, DeepMind
Title: Towards Faithful Reasoning Using Language Models
Abstract: Language models are showing impressive performance on many natural language tasks, including question-answering. However, language models – like most deep learning models – are black boxes. We cannot be sure how they obtain their answers. Do they reason over relevant knowledge to construct an answer or do they rely on prior knowledge – baked into their weights – which may be biased? An alternative approach is to develop models whose output is a human interpretable, faithful reasoning trace leading to an answer. In this talk we will characterise faithful reasoning in terms of logically valid reasoning and demonstrate where current reasoning models fall short. Following this, we will introduce Selection-Inference, a faithful reasoning model, whose causal structure mirrors the requirements for valid reasoning. We will show that our model not only produces more accurate reasoning traces but also improves final answer accuracy.

Explicit Ramsey Graphs and Two Source Extractors
2022-10-21 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: David Zuckerman, Harvard CMSA/University of Texas at Austin
Title: Explicit Ramsey Graphs and Two Source Extractors
Abstract: Ramsey showed that any graph on N nodes contains a clique or independent set of size (log N)/2. Erdos showed that there exist graphs on N nodes with no clique or independent set of size 2 log N, and asked for an explicit construction of such graphs. This turns out to relate to the question of extracting high-quality randomness from two independent low-quality sources. I’ll explain this connection and our recent exponential improvement in constructing two-source extractors.

Clique listing algorithms
2022-10-26 12:30 - 13:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Virginia Vassilevska Williams (MIT)
Title: Clique listing algorithms
Abstract: A k-clique in a graph G is a subgraph of G on k vertices in which every pair of vertices is linked by an edge. Cliques are a natural notion of social network cohesiveness with a long history.
A fundamental question, with many applications, is “How fast can one list all k-cliques in a given graph?”.
Even just detecting whether an n-vertex graph contains a k-Clique has long been known to be NP-complete when k can depend on n (and hence no efficient algorithm is likely to exist for it). If k is a small constant, such as 3 or 4 (independent of n), even the brute-force algorithm runs in polynomial time, O(n^k), and can list all k-cliques in the graph; though O(n^k) time is far from practical. As the number of k-cliques in an n-vertex graph can be Omega(n^k), the brute-force algorithm is in some sense optimal, but only if there are Omega(n^k) k-cliques. In this talk we will show how to list k-cliques faster when the input graph has few k-cliques, with running times depending on the number of vertices n, the number of edges m, the number of k-cliques T and more. We will focus on the case when k=3, but we will note some extensions.
(Based on joint work with Andreas Bjorklund, Rasmus Pagh, Uri Zwick, Mina Dalirrooyfard, Surya Mathialagan and Yinzhan Xu)

From Engine to Auto
2022-10-26 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speakers: João Araújo, Mathematics Department, Universidade Nova de Lisboa and Michael Kinyon, Department of Mathematics, University of Denver

Title: From Engine to Auto

Abstract: Bill McCune produced the program EQP that deals with first order logic formulas and in 1996 managed to solve Robbins’ Conjecture. This very powerful tool reduces to triviality any result that can be obtained by encoding the assumptions and the goals. The next step was to turn the program into a genuine assistant for the working mathematician: find ways to help the prover with proofs; reduce the lengths of the automatic proofs to better crack them; solve problems in higher order logic; devise tools that autonomously prove results of a given type, etc.

In this talk we are going to show some of the tools and strategies we have been producing. There will be real illustrations of theorems obtained for groups, loops, semigroups, logic algebras, lattices and generalizations, quandles, and many more.

Some non-concave dynamic optimization problems in finance
2022-10-28 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Shuaijie Qian (Harvard CMSA)
Title: Some non-concave dynamic optimization problems in finance
Abstract: Non-concave dynamic optimization problems appear in many areas of finance and economics. Most of existing literature solves these problems using the concavification principle, and derives equivalent, concave optimization problems whose value functions are still concave. In this talk, I will present our recent work on some non-concave dynamic optimization problems, where the concavification principle may not hold and the resulting value function is indeed non-concave.
The first work is about the portfolio selection model with capital gains tax and a bitcoin mining model with exit options and technology innovation, where the average tax basis and the average mining cost serves as an approximation, respectively. This approximation results in a non-concave value function, and the associated HJB equation problem turns out to admit infinitely many solutions. We show that the value function is the minimal (viscosity) solution of the HJB equation problem. Moreover, the penalty method still works and converges to the value function.
The second work is about a non-concave utility maximization problem with portfolio constraints. We find that adding bounded portfolio constraints, which makes the concavification principle invalid, can significantly affect economic insights in the existing literature. As the resulting value function is likely discontinuous, we introduce a new definition of viscosity solution, prove the corresponding comparison principle, and show that a monotone, stable, and consistent finite difference scheme converges to the solution of the utility maximization problem.

Doping and inverting Mott insulators on semiconductor moire superlattices
2022-11-02 12:45 - 13:45 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Liang Fu (MIT)

Title: Doping and inverting Mott insulators on semiconductor moire superlattices
Abstract: Semiconductor bilayer heterostructures provide a remarkable platform for simulating Hubbard models on an emergent lattice defined by moire potential minima. As a hallmark of Hubbard model physics, the Mott insulator state with local magnetic moments has been observed at half filling of moire band. In this talk, I will describe new phases of matter that grow out of the canonical 120-degree antiferromagnetic Mott insulator on the triangular lattice. First, in an intermediate range of magnetic fields, doping this Mott insulator gives rise to a dilute gas of spin polarons, which form a pseudogap metal. Second, the application of an electric field between the two layers can invert the many-body gap of a charge-transfer Mott insulator, resulting in a continuous phase transition to a quantum anomalous Hall insulator with a chiral spin structure. Experimental results will be discussed and compared with theoretical predictions.

Quantum trace and length conjecture for hyperbolic knot
2022-11-11 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Mauricio Romo
Title: Quantum trace and length conjecture for hyperbolic knot
Abstract: I will define the quantum trace map for an ideally triangulated hyperbolic knot complement on S^3. This map assigns an operator to each element L of the Kauffman Skein module of knot complement. Motivated by an interpretation of this operator in the context of SL(2,C) Chern-Simons theory, one can formulate a ‘length conjecture’ for the hyperbolic length of L.

Noether’s Learning Dynamics: Role of Symmetry Breaking in Neural Networks
2022-11-16 12:30 - 13:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Hidenori Tanaka (NTT Research at Harvard)
Title: Noether’s Learning Dynamics: Role of Symmetry Breaking in Neural Networks
Abstract: In nature, symmetry governs regularities, while symmetry breaking brings texture. In artificial neural networks, symmetry has been a central design principle, but the role of symmetry breaking is not well understood. Here, we develop a Lagrangian formulation to study the geometry of learning dynamics in neural networks and reveal a key mechanism of explicit symmetry breaking behind the efficiency and stability of modern neural networks. Then, we generalize Noether’s theorem known in physics to describe a unique symmetry breaking mechanism in learning and derive the resulting motion of the Noether charge: Noether’s Learning Dynamics (NLD). Finally, we apply NLD to neural networks with normalization layers and discuss practical insights. Overall, through the lens of Lagrangian mechanics, we have established a theoretical foundation to discover geometric design principles for the learning dynamics of neural networks.

Light states in the interior of CY moduli spaces
2022-11-18 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Damian van de Heisteeg
Title: Light states in the interior of CY moduli spaces
Abstract: In string theory one finds that states become massless as one approaches boundaries in Calabi-Yau moduli spaces. In this talk we look in the opposite direction, that is, we search for points where the mass gap for these light states is maximized — the so-called desert. In explicit examples we identify these desert points, and find that they correspond to special points in the moduli space of the CY, such as orbifold points and rank two attractors.

Compactness and Anticompactness Principles in Set Theory
2022-12-02 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
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.

How do Transformers reason? First principles via automata, semigroups, and circuits
2022-12-07 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
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.

AI and Theorem Proving
2023-01-31 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Mike Douglas
Title: AI and Theorem Proving
Abstract: We survey interactive theorem proving and the Lean theorem prover, and the use of AI and large language models to improve this technology. We hope to start a discussion on projects we can do at the CMSA.
