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In this talk, I will discuss non-invertible symmetries in familiar 3+1d quantum field theories describing our Nature. In massless QED, the classical U(1) axial symmetry is not completely broken by the ABJ anomaly. Instead, it turns into a discrete, non-invertible symmetry. The non-invertible symmetry operator is obtained by dressing the naïve U(1) axial symmetry operator with a fractional quantum Hall state. We also find a similar non-invertible symmetry in the massless limit of QCD, which provides an alternative explanation for the neutral pion decay. In the latter part of the talk, I will discuss non-invertible time-reversal symmetries in 3+1d gauge theories. In particular, I will show that in free Maxwell theory, there exists a non-invertible time-reversal symmetry at every rational value of the theta angle.

Based on https://arxiv.org/abs/2205.05086 and https://arxiv.org/abs/2208.04331.

For more information on how to join, please see: https://cmsa.fas.harvard.edu/event_category/quantum-matter-seminar/

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Moduli spaces of degree d dynamical systems on projective space are fundamental in algebraic dynamics. When the degree d is at least 2, these moduli spaces can be defined via geometric invariant theory (GIT), but when d = 1, there are no GIT stable linear maps. Inspired by the case of genus 0 curves, we show how to recover a nice moduli space by including marked points. Linear maps are the simplest dynamical systems, but with marked points, the moduli space becomes quite subtle. We construct the moduli space of linear maps with marked points, prove its rationality, and show that GIT stability is characterized by subtle dynamical conditions on the marked map related to Hessenberg varieties. The proof is a combinatorial analysis of polytopes generated by root vectors of the A_N lattice from Lie theory.

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.

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.

For more information on how to join, please see: https://cmsa.fas.harvard.edu/tech-in-math/

Many of the quantities appearing in the conjecture of Birch and Swinnerton-Dyer look suspiciously like squares. Motivated by this and related examples, we may ask if the central value of an L-function “of symplectic type” admits a preferred square root.

The answer is no: there’s an interesting cohomological obstruction. More formally, in the everywhere unramified situation over a function field, I will describe an explicit cohomological formula for the L-function modulo squares. This is based on a purely topological result about 3-manifolds. If time permits I’ll speculate on generalizations. This is based on joint work with Amina Abdurrahman.

https://people.math.harvard.edu/~sli/hnts/

This seminar will be held in Science Center 530 at 4:00pm on Wednesday, September 15th.

Please see the seminar page for more details: https://math.harvard.edu/~ctm/sem/.

The answer I shall argue, is both yes and no. Along the way I will discuss an entirely new approach to the Godel Incompleteness Theorems which has emerged over the last 15 years. I will also introduce some very large finite numbers which arise naturally from finite combinatorics, and indicate how by invoking these large finite numbers, any number theoretic problem of modern interest can be converted to a finitistic statement.

It is unclear which would be more amazing for these problems of modern interest: This conversion does not always produce an equivalent problem, or that this conversion always does.

In this talk I will discuss our recent work that reproduces and extends the famous work of Lehner and Pretorius on the end point of the Gregory-Laflamme instability of black strings. We consider black strings of different thicknesses and our numerics allow us to get closer to the singularity than ever before. In particular, while our results support the picture of the formation of a naked singularity in finite asymptotic time, the process is more complex than previously thought. In addition, we obtain some hints about the nature of the singularity that controls the pinch off of the string.

This seminar will be held in CMSA GR Seminar room and will be broadcast on Zoom.

For more information on how to join, please see: https://cmsa.fas.harvard.edu/event_category/general-relativity/

Instanton homology was first introduced by Floer in 1980s. It has many important applications in the study of 3-manifolds and knots, especially in studying the SU(2) representations of fundamental groups. It is conjectured by Kronheimer and Mrowka that the framed instanton Floer homology of a 3-manifold is isomorphic to the hat version of Heegaard Floer homology, but not much is known beyond families of computational examples. In this talk, I will present some surgery formulae in instanton theory, which helps us understand the framed instanton Floer homology of Dehn surgeries on knots. These surgery formulae provide some important structural properties for instanton theory and enable us to compute the framed instanton Floer homology of many new families of 3-manifolds that come from Dehn surgeries and splicings. This is a join work with Fan Ye.

No additional detail for this event.

This will be more of a mini-course than a traditional seminar.  I will provide details about the (core) entropy algorithms for quadratic polynomials from my work with Tiozzo and Wu.

For more information, please see: https://people.math.harvard.edu/~demarco/AlgebraicDynamics/