Non-Fermi liquid phenomena arise naturally near Landau ordering transitions in metallic systems. Here, we leverage quantum anomalies as a powerful nonperturbative tool to calculate optical transport in these models in the infrared limit. While the simplest such models with a single boson flavor (N=1) have zero incoherent conductivity, a recently proposed large N deformation involving flavor-random Yukawa couplings between N flavors of bosons and fermions admits a nontrivial incoherent conductivity   (z is the boson dynamical exponent) when the order parameter is odd under inversion. The presence of incoherent conductivity in the random flavor model is a consequence of its unusual anomaly structure. From this we conclude that the large N deformation does not share important nonperturbative features with the physical N = 1 model, though it remains an interesting theory in its own right. Going beyond the IR fixed point, we also consider the effects of irrelevant operators and show, within the scope of the RPA expansion, that the old result   due to Kim et al. is incorrect for inversion-odd order parameters.

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

For more information, please see: https://math.harvard.edu/~ctm/sem

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/

Just as moments of a distribution of real numbers are a powerful tool that (if they are not too large) determine the distribution uniquely, we can consider certain averages as moments of a distribution of groups (or more general algebraic structures).  We discuss how these moments determine a distribution of groups uniquely, and how that can be applied to find distributions of fundamental groups of random 3-manifolds, fundamental groups of curves over finite fields, and to give conjectures for distributions of important groups arising in number theory such as class groups.

Zoom:   https://harvard.zoom.us/j/779283357?pwd=MitXVm1pYUlJVzZqT3lwV2pCT1ZUQT09

We prove that the categories of coherent (or, equivalently, of quasi-coherent) sheaves over two noetherian non-commutative schemes X and Y are equivalent if and only if there centers C(X) and C(Y) are isomorphic and there is a local progenerator in the category of coherent sheaves over X whose sheaf of endomorphisms is anti-isomorphic to the inverse image of the structure sheaf of Y under the isomorphism X->Y. To prove it, we combine the classical Morita theorem with the Gabriel’s theory of locally noetherian categories.

We generalize the flux insertion argument due to Laughlin, Niu-Thouless-Tao-Wu, and Avron-Seiler-Zograf to the case of fractional quantum Hall states on a higher-genus surface. We propose this setting as a test to characterise the robustness, or topologicity, of the quantum state of matter and apply our test to the Laughlin states. Laughlin states form a vector bundle, the Laughlin bundle, over the Jacobian – the space of Aharonov-Bohm fluxes through the holes of the surface. The rank of the Laughlin bundle is the degeneracy of Laughlin states or, in presence of quasiholes, the dimension of the corresponding full many-body Hilbert space; its slope, which is the first Chern class divided by the rank, is the Hall conductance. We compute the rank and all the Chern classes of Laughlin bundles for any genus and any number of quasiholes, settling, in particular, the Wen-Niu conjecture. Then we show that Laughlin bundles with non-localized quasiholes are not projectively flat and that the Hall current is precisely quantized only for the states with localized quasiholes. Hence our test distinguishes these states from the full many-body Hilbert space.Based on joint work with Dimitri Zvonkine (CNRS, University of Paris-Versaille)

The adaptive immune system is able to learn from past experiences to better fit an unforeseen future. This is made possible by a diverse and dynamic repertoire of cells expressing unique antigen receptors and capable of rapid Darwinian evolution within an individual. However, naturally occurring immune responses exhibit limits in efficacy, speed and capacity to adapt to novel challenges. In this talk, I will discuss theoretical frameworks we developed to (1) explore functional impacts of non-equilibrium antigen recognition, and (2) identify conditions under which natural selection acting local in time can find adaptable solutions favorable in the long run, through exploiting environmental variations and functional constraints.

This seminar will be held at CMSA, 20 Garden St, seminar room G-10 and on Zoom. For more information on how to join the Zoom, please see: https://cmsa.fas.harvard.edu/event/active-matter-seminar-6/

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.

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

This seminar will be held in Science Center 530 at 4:00pm on Wednesday, September 21st.

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

Few notions within the realm of mathematical physics succeed in capturing the imagination and inspiring awe as well as that of a black hole. First encountered in the Schwarzschild solution, discovered a few months after the presentation of the Field Equations of General Relativity at the Prussian Academy of Sciences, the black hole as a mathematical phenomenon accompanies and prominently features within the history of General Relativity since its inception. In this talk we will lay out a brief history of the question of dynamical black hole formation in General Relativity and discuss a result, in collaboration with Xinliang An, on a scale-critical trapped surface formation criterion for the Einstein-Maxwell system.

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

Understanding the geometry of The Mandelbrot set, which records dynamical information about every quadratic polynomial, has been a central task in holomorphic dynamics over the past forty years. Near parabolic parameters, the structure of the Mandelbrot set is asymptotically self-similar and resembles a parade of elephants. Near parabolic parameters on these “elephants”, the Mandelbrot set is again self-similar and resembles another parade of elephants. This phenomenon repeats infinitely, and we see different parades of elephants at each scale. In this talk, we will explore the implications of controlling the geometry of these elephants. In particular, we will partially answer Milnor’s conjecture on the optimality of the Yoccoz inequality, and see potential connections to the local connectivity of the Mandelbrot set.

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

This talk is on work in preparation with Ali Daemi.

I will explain why Kim Froyshov’s mod 2 instanton invariant q_3 gives information about indefinite 4-manifolds: if H_1(W;Z/2) = 0 and W has boundary Y, then -b^+(W) <= q_3(Y) <= b^-(W). This is the first invariant known to enjoy comparable bounds for indefinite manifolds W.

The key observation is that even when b^+(W) > 0, one can define a Donaldson invariant in the (tilde) instanton homology of boundary(W) — not in the usual instanton tilde complex, but rather a “suspension”. This suspension process accounts for the role of obstructed gluing theory, and does not destroy information about q_3 (but does destroy information about all other types of h-invariant).

As a corollary, we show that there exist integer homology spheres with arbitrary integral surgery number S(Y_n) = n. This answers a question of Dave Auckly. Previously, the state of the art was n=2, and further progress was obstructed by the possibility that the linking matrix be indefinite.

CMSA Fall Gathering will be held on September 23rd from 4:30-6:00pm. All CMSA and Math Affiliates Invited! Hot dogs, popcorn, and hot cider courtesy of Dylan and Pete’s.

In the talk I will discuss the possibility and the obstructions of finding non-supersymmetric dualities for 4d gauge theories. I will review consistency conditions based on Weingarten inequalities, anomalies and large N, and clarify some subtle points and misconceptions about them. Later I will go over some old and new examples of candidates for non-supersymmetric dualities. The will be based on 2208.07842

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

Arithmetic quotients of Type IV Hermitian symmetric domains have cohomology-valued modular forms whose coefficients are special cycles, by work of Borcherds. These can be interpreted as non-compact period spaces for K3-type Hodge structures. I will describe recent results (joint with P. Engel and S. Tayou) that give mock modular forms whose coefficients are compactified special cycles in a simplicial toroidal compactification. Next, I will discuss an application to the geometry of Severi curves associated to a rational elliptic surface.

One question in the study of topological phases is to identify the topological data from the ground state wavefunction without accessing the Hamiltonian. Since local measurement is not enough, entanglement becomes an indispensable tool. Here, we use modular Hamiltonian (entanglement Hamiltonian) and modular flow to rephrase previous studies on topological entanglement entropy and motivate a natural generalization, which we call the entanglement linear response. We will show how it embraces a previous work by Kim&Shi et al on the chiral central charge, and furthermore, inspires a new formula for the quantum Hall conductance.

This seminar will be held in person, for more information, please see: https://cmsa.fas.harvard.edu/tech-in-math/

This seminar will be held in Science Center 530 at 4:00pm on Wednesday, September 28th.

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

The discovery of the quark-gluon plasma that forms in heavy-ion collision experiments provides a unique opportunity to study the properties of matter under extreme conditions, as the quark-gluon plasma is the hottest, smallest, and densest fluid known to humanity. Studying the quark-gluon plasma also provides a window into the earliest moments of the universe, since microseconds after the Big Bang the universe was filled with matter in the form of the quark-gluon plasma. For more than two decades, the community has intensely studied the quark-gluon plasma with the help of a rich interaction between experiments, theory, phenomenology, and numerical simulations. From these investigations, a coherent picture has emerged, indicating that the quark-gluon plasma behaves essentially like a relativistic liquid with viscosity. More recently, state-of-the-art numerical relativity simulations strongly suggested that viscous and dissipative effects can also have non-negligible effects on gravitational waves produced by binary neutron star mergers. But despite the importance of viscous effects for the study of such systems, a robust and comprehensive theory of relativistic fluids with viscosity is still lacking. This is due, in part, to difficulties to preserve causality upon the inclusion of viscous and dissipative effects into theories of relativistic fluids. In this talk, we will survey the history of the problem and report on a new approach to relativistic viscous fluids that addresses these issues.

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

Kuznetsov’s Homological projective duality (HPD) in algebraic geometry is a powerful theorem that allows to extract information about semiorthogonal decompositions of derived categories of certain varieties. I will give a GLSMs perspective based on categories of B-branes. I will focus mostly on the case of Fano (hypersurfaces) manifolds. In general, for such cases the HPD can be interpreted as a non-commutative (nc) resolution of a compact variety. I will give a physical interpretation of this fact and present some conjectures.

For information on how to join, please see: https://cmsa.fas.harvard.edu/event_category/algebraic-geometry-in-string-theory/

Teachers change lives – and you can be part of that change by becoming a teacher at the Harvard Graduate School of Education. HGSE’s new Teaching and Teacher Leadership master’s program will prepare you to lead transformative learning experiences that expand opportunity, fuel student success, and make a positive deep impact on young people and their communities. TTL builds on the successes of HGSE’s former teacher education programs, including the Harvard Teacher Fellows, and is committed to welcoming Harvard College undergraduates into meaningful careers in education. The program offers two pathways: a residency model, where you’ll jump into the classroom right away as a teacher of record, and an internship model for those seeking to ramp up teaching responsibility more gradually. TTL is committed to supporting new teachers – with extensive funding to minimize student debt, mentors to learn from, and field experiences that will fuel your professional growth.

 

Learn more about becoming a math teacher through TTL at our Math Info Session Friday, September 30 at 3pm in SC105. Register here.  

In this talk, we will discuss bordered aspects of the Heegaard Floer surgery formulas of  Ozsvath–Szabo and Manolescu–Ozsvath. In particular, we will explain how their theories naturally define bordered invariants for manifolds with toroidal boundary components. Time permitting, we will discuss applications of the theory. One application is a proof of the equivalence of lattice homology and Heegaard Floer homology. Another application is a description of the link Floer complexes of algebraic links, which is parallel to Ozsvath and Szabo’s description of the knot Floer complex of an L-space link. The latter application is joint with M. Borodzik and B. Liu.