CMSA

Minerva: Solving Quantitative Reasoning Problems with Language Models
2022-10-05 14:00 - 16:00 (America/New_York) | 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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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.

Neural Optimal Stopping Boundary
2023-02-02 12:30 - 13:30
Speaker: Max Reppen (Boston University)
Title: Neural Optimal Stopping Boundary
Abstract: A method based on deep artificial neural networks and empirical risk minimization is developed to calculate the boundary separating the stopping and continuation regions in optimal stopping. The algorithm parameterizes the stopping boundary as the graph of a function and introduces relaxed stopping rules based on fuzzy boundaries to facilitate efficient optimization. Several financial instruments, some in high dimensions, are analyzed through this method, demonstrating its effectiveness. The existence of the stopping boundary is also proved under natural structural assumptions.

Motivic Geometry of Two-Loop Feynman Integrals
2023-02-07 12:00 - 13:00
Member Seminar
Speaker: Chuck Doran
Title: Motivic Geometry of Two-Loop Feynman Integrals
Abstract: We study the geometry and Hodge theory of the cubic hypersurfaces attached to two-loop Feynman integrals for generic physical parameters. We show that the Hodge structure attached to planar two-loop Feynman graphs decomposes into a mixed Tate piece and a variation of Hodge structure from families of hyperelliptic curves, elliptic curves, or rational curves depending on the space-time dimension. We give more precise results for two-loop graphs with a small number of edges. In particular, we recover a result of Spencer Bloch that in the well-known double box example there is an underlying family of elliptic curves, and we give a concrete description of these elliptic curves. We show that the motive for the “non-planar” two-loop tardigrade graph is that of a family of K3 surfaces of generic Picard number 11. Lastly, we show that generic members of the multi-scoop ice cream cone family of graph hypersurfaces correspond to pairs of multi-loop sunset Calabi-Yau varieties. Our geometric realization of these motives permits us in many cases to derive in full the homogeneous differential operators for the corresponding Feynman integrals. This is joint work with Andrew Harder and Pierre Vanhove.

From spin glasses to Boolean circuits lower bounds - Algorithmic barriers from the overlap gap property
2023-02-08 12:30 - 13:30
Speaker: David Gamarnik (MIT)
Title: From spin glasses to Boolean circuits lower bounds. Algorithmic barriers from the overlap gap property
Abstract: Many decision and optimization problems over random structures exhibit an apparent gap between the existentially optimal values and algorithmically achievable values. Examples include the problem of finding a largest independent set in a random graph, the problem of finding a near ground state in a spin glass model, the problem of finding a satisfying assignment in a random constraint satisfaction problem, and many many more. Unfortunately, at the same time no formal computational hardness results exist which explains this persistent algorithmic gap.
In the talk we will describe a new approach for establishing an algorithmic intractability for these problems called the overlap gap property. Originating in statistical physics theory of spin glasses, this is a simple to describe property which a) emerges in most models known to exhibit an apparent algorithmic hardness; b) is consistent with the hardness/tractability phase transition for many models analyzed to the day; and, importantly, c) allows to mathematically rigorously rule out a large class of algorithms as potential contenders, specifically the algorithms which exhibit a form of stability/noise insensitivity.
We will specifically show how to use this property to obtain stronger (stretched exponential) than the state of the art (quasi-polynomial) lower bounds on the size of constant depth Boolean circuits for solving the two of the aforementioned problems: the problem of finding a large independent set in a sparse random graph, and the problem of finding a near ground state of a p-spin model.
Joint work with Aukosh Jagannath and Alex Wein

Complete Calabi-Yau metrics: Recent progress and open problems
2023-02-13 12:30 - 13:30
Speaker: Tristan Collins, MIT
Title: Complete Calabi-Yau metrics: Recent progress and open problems
Abstract: Complete Calabi-Yau metrics are fundamental objects in Kahler geometry arising as singularity models or “bubbles” in degenerations of compact Calabi-Yau manifolds. The existence of these metrics and their relationship with algebraic geometry are the subjects of several long standing conjectures due to Yau and Tian-Yau. I will describe some recent progress towards the question of existence, and explain some future directions, highlighting connections with notions of algebro-geometric stability.

Dynamics of active nematic defects on cones
2023-02-14 12:00 - 13:00
Member Seminar
Speaker: Farzan Vafa
Title: Dynamics of active nematic defects on cones
Abstract: In the first part of the talk, we investigate the ground-state configurations of two-dimensional liquid crystals with p-fold rotational symmetry (p-atics) on cones. The cone apex develops an effective topological charge, which in analogy to electrostatics, leads to defect absorption and emission at the cone apex as the deficit angle of the cone is varied. We find three types of ground-state configurations as a function of cone angle, which is determined by charged defects screening the effective apex charge: (i) for sharp cones, all of the +1/p defects are absorbed by the apex; (ii) at intermediate cone angles, some of the +1/p defects are absorbed by the apex and the rest lie equally spaced along a concentric ring on the flank; and (iii) for nearly flat cones, all of the +1/p defects lie equally spaced along a concentric ring on the flank. We check these results with numerical simulations for a set of commensurate cone angles and find excellent agreement. In the second part of the talk, we investigate the dynamics of an active nematic on a cone, and via simulations find long-time circular orbits of either one or two flank defects, with transitions between these states mediated by the apex via defect absorption, emission, or defect pair creation.

Hints of Flat Space Holography
2023-02-21 12:00 - 13:00
Member Seminar
Speaker: Dan Kapec
Title: Hints of Flat Space Holography
Abstract: Despite our detailed understanding of holography in Anti-de Sitter space, flat space holography remains somewhat mysterious. “Celestial CFT” is a formalism which attempts to recast quantum gravity in (d+2)-dimensional asymptotically flat spacetimes in terms of a d-dimensional Euclidean conformal field theory residing at the conformal boundary. I will discuss certain universal aspects of this correspondence. As in Anti-de Sitter space, bulk gravitons produce a boundary stress tensor, and bulk gluons furnish boundary-conserved currents. I will also show that continuous spaces of vacua in the bulk map directly onto the conformal manifold of the boundary CFT. This correspondence provides a new perspective on the role of the BMS group in flat space holography and offers a new interpretation of the antisymmetric double-soft gluon theorem in terms of the curvature of an infinite-dimensional vacuum manifold.

The Black Hole Information Paradox: A Resolution on the Horizon?
2023-02-22 12:30 - 13:30
Speaker: Netta Engelhardt (MIT)
Title: The Black Hole Information Paradox: A Resolution on the Horizon?
Abstract: The black hole information paradox — whether information escapes an evaporating black hole or not — remains one of the most longstanding mysteries of theoretical physics. The apparent conflict between validity of semiclassical gravity at low energies and unitarity of quantum mechanics has long been expected to find its resolution in a complete quantum theory of gravity. Recent developments in the holographic dictionary, and in particular its application to entanglement and complexity, however, have shown that a semiclassical analysis of gravitational physics can reproduce a hallmark feature of unitary evolution. I will describe this recent progress and discuss some promising indications of a full resolution of the information paradox.

New bounds on lattice covering volumes, and nearly uniform covers
2023-03-02 12:00 - 13:00
Member Seminar
Speaker: Barak Weiss
Title: New bounds on lattice covering volumes, and nearly uniform covers
Abstract: Let L be a lattice in R^n and let K be a convex body. The covering volume of L with respect to K is the minimal volume of a dilate rK, such that L+rK = R^n, normalized by the covolume of L. Pairs (L,K) with small covering volume correspond to efficient coverings of space by translates of K, where the translates lie in a lattice. Finding upper bounds on the covering volume as the dimension n grows is a well studied problem, with connections to practical questions arising in computer science and electrical engineering. In a recent paper with Or Ordentlich (EE, Hebrew University) and Oded Regev (CS, NYU) we obtain substantial improvements to bounds of Rogers from the 1950s. In another recent paper, we obtain bounds on the minimal volume of nearly uniform covers (to be defined in the talk). The key to these results are recent breakthroughs by Dvir and others regarding the discrete Kakeya problem. I will give an overview of the questions and results.

The string/black hole transition in anti de Sitter space
2023-03-02 16:00 - 17:00
Speaker: Erez Urbach, Weizmann Institute of Science
Title: The string/black hole transition in anti de Sitter space
Abstract: String stars, or Horowitz-Polchinski solutions, are string theory saddles with normalizable condensates of thermal-winding strings. In the past, string stars were offered as a possible description of stringy (Euclidean) black holes in asymptotically flat spacetime, close to the Hagedorn temperature. I will discuss the thermodynamic properties of string stars in asymptotically (thermal) anti-de Sitter background (including AdS3 with NS-NS flux), their possible connection to small black holes in AdS, and their implications for holography. I will also present new “winding-string gas” saddles for confining holographic backgrounds such as the Witten model, and their relation to the deconfined phase of 3+1 pure Yang-Mills.

Categorical Symmetry of the Standard Model from Gravitational Anomaly
2023-03-07 12:00 - 13:00
Member Seminar
Speaker: Juven Wang
Title: Categorical Symmetry of the Standard Model from Gravitational Anomaly
Abstract: In the Standard Model, the total “sterile right-handed” neutrino number n_{νR} is not equal to the family number Nf. The anomaly index (-Nf+n_{νR}) had been advocated to play an important role in our previous work on Cobordism and Deformation Class of the Standard Model [2112.14765, 2204.08393] and Ultra Unification [2012.15860] in order to predict new highly entangled sectors beyond the Standard Model. Moreover, the invertible baryon minus lepton number B−L symmetry current conservation can be violated quantum mechanically by gravitational backgrounds such as gravitational instantons. In specific, we show that a noninvertible categorical counterpart of the B−L symmetry still survives in gravitational backgrounds. In general, we propose a construction of noninvertible symmetry charge operators as topological defects derived from invertible anomalous symmetries that suffer from mixed gravitational anomalies. Examples include the perturbative local and nonperturbative global anomalies classified by ℤ and ℤ16 respectively. For this construction, we utilize the anomaly inflow concept and the 3d Witten-Reshetikhin-Turaev-type topological quantum field theory corresponding to a 2d rational conformal field theory with an appropriate chiral central charge, or the 3d boundary topological order of 4d ℤTF4-time-reversal symmetric topological superconductor [2302.14862].

Conformal symmetry, Optimization algorithms and the Critical Phenomena
2023-03-08 12:30 - 13:30
Speaker: Ning Su, University of Pisa
Title: Conformal symmetry, Optimization algorithms and the Critical Phenomena
Abstract: In the phase diagram of many substances, the critical points have emergent conformal symmetry and are described by conformal field theories. Traditionally, physical quantities near the critical point can be computed by perturbative field theory method, where conformal symmetry is not fully utilized. In this talk, I will explain how conformal symmetry can be used to determine certain physical quantities, without even knowing the fine details of the microscopic structure. To compute the observables precisely, one needs to develop powerful numerical techniques. In the last few years, we have invented many computational tools and algorithms, and predicted critical exponents of Helium-4 superfluid phase transition and Heisenberg magnet to very high precision.

How to steer foundation models?
2023-03-08 14:00 - 15:00
New Technologies in Mathematics Seminar
Speaker: Jimmy Ba, University of Toronto
Title: How to steer foundation models?
Abstract: By conditioning on natural language instructions, foundation models and large language models (LLMs) have displayed impressive capabilities as general-purpose computers. However, task performance depends significantly on the quality of the prompt used to steer the model. Due to the lack of knowledge of how foundation models work, most effective prompts have been handcrafted by humans through a demanding trial-and-error process. To reduce the human effort in this alignment process, I will discuss a few approaches to steer these powerful models to excel in various downstream language and image tasks.
