## news

##### First Year Undergraduate Math Information

Click the Headline ↑↑↑↑↑ above to go directly to information about Math Department courses and your Math Placement Test.

##### Math Community Web Page

Following up on the #shutdownSTEM discussions, the Department of Mathematics has launched a community web page, with evolving content to be created through community effort.

##### Commendations for Extraordinary Teaching

Dean of Undergraduate Education Amanda Claybaugh sent letters of Special Commendation for Extraordinary Teaching in Extraordinary Times to Wes Cain Janet Chen Voula Collins David...

##### Three New Professors Join Department

Harvard Mathematics welcomes three new Professors to our Department in July: Laura DeMarco in Dynamical Systems, Mihnea Popa in Algebraic Geometry, and Melanie Matchett Wood...

##### #shutdownSTEM–10 June 2020

In partnership with the Harvard Division of Science, the Mathematics Department used the call to #shutdownSTEM on 10 June to initiate a discussion of actions...

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## upcoming events

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• CMSA Geometry and Physics Seminar: Collective integrable systems and global action-angle coordinates

##### CMSA Geometry and Physics Seminar: Collective integrable systems and global action-angle coordinates

9:30 AM-10:30 AM
July 7, 2020

A “collective integrable system” on a symplectic manifold is a commutative integrable system constructed from a Hamiltonian action of a non-commutative Lie group. Motivated by the example of Gelfand-Zeitlin systems, we give a construction of collective integrable systems that generate a Hamiltonian torus action on a dense subset of any Hamiltonian K-manifold, where K is any compact connected Lie group. In the case where the Hamiltonian K-manifold is compact and multiplicity free, the resulting Hamiltonian torus action is completely integrable and yields global action angle coordinates. Moreover, the image of the moment map is a (non-simple) convex polytope.

• Decay estimates and complete Bakry-Emry theory

##### Decay estimates and complete Bakry-Emry theory

10:00 AM-11:00 AM
July 7, 2020

The connection between decay estimates for entropy and logarithmic Sobolev inequalities is well-established for dynamical systems on commutative systems. I will explain how to extend this to matrix-valued functions, and then apply these techniques to Lindbladians on quantum systems interacting with an environment. In fact, some Lindbladian on small quantum systems seems to contain all the relevant information of dynamical systems on groups. This is joint work with Haojian Li and Nick LaRacuente.

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• CMSA Condensed Matter/Math Seminar: Deconfined metallic quantum criticality-I

##### CMSA Condensed Matter/Math Seminar: Deconfined metallic quantum criticality-I

9:00 AM-10:30 AM
July 9, 2020

A number of strongly correlated electronic materials exhibit quantum criticality that does not fit into the conventional Landau-Ginzburg-Wilson paradigm of continuous phase transitions. Inspired by these experimental examples, I will discuss a new class of quantum phase transitions that describe a continuous transition between a Fermi liquid metal with a generic electronic Fermi surface and electrical insulators without Fermi surface of neutral excitations. Such phase transitions are described in terms of a finite density of fractionalized excitations coupled to emergent gauge fields. I will discuss various concrete examples of such gauge theories and describe their associated phase transitions using a renormalization group framework.  Remarkably, we find examples of continuous phase transitions between Landau Fermi liquid metals and insulators, where the quantum critical point hosts a non-Fermi liquid with a sharp Fermi surface but no long-lived quasiparticles. I will comment on the relevance of this new theoretical framework for some of the most pressing questions in the field of quantum matter.

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• CMSA Social Science Applications Forum: Convergence of Large Population Games to Mean Field Games with Interaction Through the Controls

##### CMSA Social Science Applications Forum: Convergence of Large Population Games to Mean Field Games with Interaction Through the Controls

10:00 AM-11:00 AM
July 13, 2020

This work considers stochastic differential games with a large number of players, whose costs and dynamics interact through the empirical distribution of both their states and their controls. We develop a framework to prove convergence of finite-player games to the asymptotic mean field game. Our approach is based on the concept of propagation of chaos for forward and backward weakly interacting particles which we investigate by fully probabilistic methods, and which appear to be of independent interest. These propagation of chaos arguments allow to derive moment and concentration bounds for the convergence of both Nash equilibria and social optima in non-cooperative and cooperative games, respectively. Incidentally, we also obtain convergence of a system of second order parabolic partial differential equations on finite dimensional spaces to a second order parabolic partial differential equation on the Wasserstein space.

For security reasons, you will have to show your full name to join the meeting.

• CMSA Geometry and Physics Seminar: Berry phase in quantum field theory

##### CMSA Geometry and Physics Seminar: Berry phase in quantum field theory

9:00 PM-10:00 PM
July 13, 2020

We will discuss Berry phase in family of quantum field theories using effective field theory. The family is labelled by parameters which we promote to be spacetime-dependent sigma model background fields. The Berry phase is equivalent to Wess-Zumino-Witten action for the sigma model. We use Berry phase to study diabolic points in the phase diagram of the quantum field theory and discuss applications to deconfined quantum criticality and new tests for boson/fermion dualities in (2+1)d.

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• Integrability and Braided Tensor Categories

##### Integrability and Braided Tensor Categories

10:00 AM-11:00 AM
July 14, 2020

Many integrable critical classical statistical mechanical models and the corresponding quantum spin chains possess a fractional-spin conserved current. Such currents have been constructed by utilizing quantum-group algebras, fermionic and parafermionic operators, and ideas from “discrete holomorphicity”. I define them generally and naturally using a braided tensor category, a topological structure familiar from the study of knot invariants, anyons and conformal field theory.  I derive simple constraints on the Boltzmann weights necessary and sufficient for such a current to exist, generalizing those found using quantum-group algebras. I find many solutions, in both geometric and local models. In all cases, the resulting weights are those of an integrable lattice model, giving a linear construction for “Baxterising”, i.e. building a solution of the Yang-Baxter equation out of topological data.

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• CMSA Quantum Matter/Quantum Field Theory Seminar: Interplay between two boundary effects

##### CMSA Quantum Matter/Quantum Field Theory Seminar: Interplay between two boundary effects

10:30 AM-12:00 PM
July 15, 2020

We study the interplay between two nontrivial boundary effects: (1) the d-1 dimensional edge states of d-dimensional strongly interacting symmetry protected topological states, and (2) the boundary fluctuations of d-dimensional bulk quantum criticality. We also discuss states localized at an interface in a higher dimensional bulk, when the bulk undergoes a quantum phase transition. Using controlled analytical methods, we demonstrate that the interplay between the two different boundary effects leads to rich physics at the d-1 dimensional boundary, including new stable fixed points, and also an exotic quantum phase transition which cannot happen in a local d-1 dimensional system alone. Our analytical calculation is qualitatively consistent with recent numerical works on nonlocal quantum many body systems.

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• CMSA Condensed Matter/Math Seminar: Deconfined metallic quantum criticality – II

##### CMSA Condensed Matter/Math Seminar: Deconfined metallic quantum criticality – II

10:30 AM-12:00 PM
July 16, 2020

The main goal of this talk is to discuss in detail a concrete setup for deconfined metallic quantum criticality. In particular, we propose that certain quantum Hall bilayers can host examples of a deconfined metal-insulator transition (DMIT), where a Fermi liquid (FL) metal with a generic electronic Fermi surface evolves into a gapped insulator (or, an insulator with Goldstone modes) through a continuous quantum phase transition. The transition can be accessed by tuning a single parameter, and its universal critical properties can be understood using a controlled framework. At the transition, the two layers are effectively decoupled, where each layer undergoes a continuous transition from a FL to a generalized composite Fermi liquid (gCFL). The thermodynamic and transport properties of the gCFL are similar to the usual CFL, while its spectral properties are qualitatively different. The FL-gCFL quantum critical point hosts a sharply defined Fermi surface without long-lived electronic quasiparticles. Immediately across the transition, the two layers of gCFL are unstable to forming an insulating phase. We discuss the topological properties of the insulator and various observable signatures associated with the DMIT. Some key ingredients of this proposal include Dirac-Chern-Simons theory, color superconductivity, dimensional decoupling, etc.

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• CMSA Geometry and Physics Seminar: A geometric construction of orbifold Jacobian algebras

##### CMSA Geometry and Physics Seminar: A geometric construction of orbifold Jacobian algebras

9:30 PM-10:30 PM
July 20, 2020

We review the definition of a twisted Jacobian algebra of a Landau-Ginzburg orbifold due to Kaufmann et al. Then we construct an A-infinity algebra of a weakly unobstructed Lagrangian submanifold in a symplectic orbifold. We work on an elliptic orbifold sphere and see that above two algebras are isomorphic, and furthermore their structure constants are related by a modular identity which was used to prove the mirror symmetry of closed string pairings. This is a joint work with Cheol-Hyun Cho.

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• Applied von Neumann Algebra

##### Applied von Neumann Algebra

10:00 AM-11:00 AM
July 21, 2020

We are interested in mathematical results which are stated entirely without reference to von Neumann algebras but whose proofs use von Neumann algebras in an essential way. The first stunning example is the Kaplansky result that ab=1 iff ba=1 in a group algebra over a field of characteristic zero. Connes’ noncommutative integration theory yields other examples. We will concentrate on a new example in the theory of zero sets of Bergman spaces where we are able to calculate a certain density of orbits of Fuchsian groups.

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• CMSA Quantum Matter/Quantum Field Theory Seminar: Domain Wall Decorations, Anomalies, and Fermionic SPT

##### CMSA Quantum Matter/Quantum Field Theory Seminar: Domain Wall Decorations, Anomalies, and Fermionic SPT

9:30 AM-11:00 AM
July 22, 2020

In the constructions of symmetry-protected topological (SPT) states, we usually decorate lower-dimensional states to higher codimensional domain walls of the system. In this talk, we will argue that domain wall decorations are basically equivalent to spectral sequences in algebraic topology. I will first illustrate this idea in bosonic systems, with explicit formulas for all differentials on all pages in the Lyndon-Hochschild-Serre spectral sequence. These results are useful in bosonic systems with Lieb-Schultz-Mattis (LSM) theorems, SPT-LSM theorems, and symmetry-enriched gauge theories. The second part of the talk will focus on fermionic SPT states. Using domain wall decorations, we will give a systematic construction and classification of fermionic SPT states in 3+1 or lower dimensions. We can obtain the full classifications for arbitrary finite unitary Abelian symmetries and interacting 10-fold way. All the classifications are consistent with known results from other approaches such as point/loop braiding statistics and spin cobordisms.

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• CMSA Quantum Matter/Quantum Field Theory Seminar: Nonperturbative anomaly inflow for fermions and p-form fields

##### CMSA Quantum Matter/Quantum Field Theory Seminar: Nonperturbative anomaly inflow for fermions and p-form fields

9:30 AM-11:00 AM
July 23, 2020

I will talk about a nonperturbative formulation of chiral fermions and chiral (self-dual) p-form fields, in terms of massive theories in one-higher dimensions. The introduction of the higher dimensional bulk is unavoidable due to the existence of anomalies, and I discuss the modern understanding of anomalies, which appears very naturally in our formulation of chiral theories.

(please use full real name to join in, the Admin/moderator[s] may examine the attendees for zoombombing repeating)

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• CMSA Geometry and Physics Seminar: Parabolic de Rham bundles: motivic vs periodic

##### CMSA Geometry and Physics Seminar: Parabolic de Rham bundles: motivic vs periodic

9:30 PM-10:30 PM
July 27, 2020

Let $C$ be a complex smooth projective curve. We consider the set of parabolic de Rham bundles over $C$ (with rational weights in parabolic structure). Many examples arise from geometry: let $f: X\to U$ be a smooth projective morphism over some nonempty Zariski open subset $U\subset C$. Then the Deligne–Iyer–Simpson canonical parabolic extension of the Gauss–Manin systems associated to $f$ provides such examples. We call a parabolic de Rham bundle \emph{motivic}, if it appears as a direct summand of such an example of geometric origin. It is a deep question in the theory of linear ordinary differential equations and in Hodge theory, to get a characterization of motivic parabolic de Rham bundles. In this talk, I introduce another subcategory of parabolic de Rham bundles, the so-called \emph{periodic} parabolic de Rham bundles. It is based on the work of Lan–Sheng–Zuo on Higgs-de Rham flows, with aim towards linking the Simpson correspondence over the field of complex numbers and the Ogus–Vologodsky correspondence over the finite fields. We show that motivic parabolic de Rham bundles are periodic, and conjecture that they are all periodic parabolic de Rham bundles. The conjecture for rank one case follows from the solution of Grothendieck–Katz p-curvature conjecture, and for some versions of rigid cases should follow from Katz’s work on rigid local systems. The conjecture implies that in a spread-out of any complex elliptic curve, there will be infinitely many supersingular primes, a result of N. Elkies for rational elliptic curves. Among other implications of the conjecture, we would like to single out the conjectural arithmetic Simpson correspondence, which asserts that the grading functor is an equivalence of categories from the category of periodic parabolic de Rham bundles to the category of periodic parabolic Higgs bundles. This is a joint work in progress with R. Krishnamoorthy.

• CMSA Social Science Applications Forum: Measuring Intangible Capital with Market Prices

##### CMSA Social Science Applications Forum: Measuring Intangible Capital with Market Prices

10:00 PM-11:00 PM
July 27, 2020

Despite the importance of intangibles in today’s economy, current standards prohibit the capitalization of internally created knowledge and organizational capital, resulting in a downward bias of reported assets. As a result, researchers estimate this value by capitalizing prior flows of R&D and SG&A. In doing so, a set of capitalization parameters, i.e. the R&D depreciation rate and the fraction of SG&A that represents a long-lived asset, must be assumed. Parameters now in use are derived from models with strong assumptions or are ad hoc. We develop a capitalization model that motivates the use of market prices of intangibles to estimate these parameters. Two settings provide intangible asset values: (1) publicly traded equity prices and (2) acquisition prices. We use these parameters to estimate intangible capital stocks and subject them to an extensive set of diagnostic analyses that compare them with stocks estimated using existing parameters. Intangible stocks developed from exit price parameters outperform both stocks developed by publicly traded parameters and those stocks developed with existing estimates. (Joint work with Ryan Peters and Sean Wang.)

For security reasons, you will have to show your full name to join the meeting.

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• The representation theory of the Clifford group, with applications in quantum information

##### The representation theory of the Clifford group, with applications in quantum information

10:00 AM-11:00 AM
July 28, 2020

The Clifford group is the most important set of quantum circuits that allow for an efficient classical description. It therefore forms an integral part of many protocols in quantum information, and it turns out that their properties can often be described in terms of representation-theoretic data. Motivated by these connections, we start our analysis by establishing an analogue of Schur-Weyl duality. As is the case for SW-duality between U(d) and  Sthe resulting description of the tth tensor power of the n-qubit Clifford group is independent of the number n of qubits! This uniform theory implies that, maybe surprisingly, several tasks in quantum information theory can be performed with a system-size independent amount of resources. Examples include: Testing whether an unknown state is a stabilizer state, and constructing unitary designs with few non-Clifford gates.

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• CMSA Condensed Matter/Math Seminar: Topological qauntum field theory in 3+1D and a potential origin of dark matter

##### CMSA Condensed Matter/Math Seminar: Topological qauntum field theory in 3+1D and a potential origin of dark matter

9:30 AM-11:00 AM
July 30, 2020

Topological qauntum field theory(TQFT) is a very powerful theoretical tool to study topological phases and phase transitions. In 2 + 1D, it is well known that the Chern-Simons theory captures all the universal topological data of topological phases, e.g., quasi-particle braiding statistics, chiral central charge and even provides us a deep insight for the nature of topological phase transitions. Recently, topological phases of quantum matter are also intensively studied in 3 + 1D and it has been shown that loop like excitation obeys the so-called three-loop-braiding statistics. In this talk, I will discuss a TQFT framework to understand the quantum statistics of loop like excitation in 3 + 1D. Most surprisingly, this new class of 3+1D TQFT even provides us a new route towards understanding quantum gravity. I will also discuss a generalized Einstein equation which might naturally include dark matter sector.

Please use full real name to join in, the Admin/moderator[s] may examine the attendees to block zoombombing.

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• CMSA Quantum Matter/Quantum Field Theory Seminar: Fermion-fermion dualities in 3+1 dimensions

##### CMSA Quantum Matter/Quantum Field Theory Seminar: Fermion-fermion dualities in 3+1 dimensions

3:00 PM-4:30 PM
July 31, 2020

Dualities play an important role in both quantum field theories and condensed matter systems. They can map hard-to-solve, interacting theories to free, non-interacting ones often trigger a deeper understanding of the systems to which they apply. Recently, a web of (non-supersymmetric) dualities has been discovered in 2+1 dimensions inspired by novel developments in topological phases of matter.

In this talk, I will present some extensions of the original 2+1-dimensional fermion-fermion duality in 3+1 dimensions and in presence of axial gauge fields. By employing the slave-rotor approach in the lattice, I will show the central role of the Kalb-Ramond field and chiral anomaly in their formulation. Finally, I will present some applications of these novel dualities in topological systems such as Weyl and Dirac semimetals and non-symmorphic topological insulators.

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