Calendar

I discuss the impact of the positive cosmological constant on the interplay between the equivalence principle in general relativity, and the rules of quantum mechanics. There is an ambiguity in the definition of a phase of a wave function measured by inertial and accelerating observes which is a non-relativistic analogue of the Unruh effect. This will be put in the framework of a non-relativistic limit of twistor space.

We prove a conjecture by Cameron and Aydinian in three dimensions, showing that the density of the largest sum-free subset of [n]^3 approaches (10 + √15)/20 as n approaches infinity. The resolution of the two-dimensional problem was accomplished by Elsholtz and Rackham in 2017. In this work, we introduce a different approach that involves considering a relaxed linear program on the projection [n]^2 and then constructing a suitable dual solution. We also explore potential approaches for addressing the higher-dimensional version of this problem.

For information about the Combinatorics Seminar, please visit http://math.mit.edu/seminars/combin/

In the setting of lattice gauge theories with finite (possibly non-Abelian) gauge groups at weak coupling, we prove exponential decay of correlations for a wide class of gauge invariant functions, which in particular includes arbitrary functions of Wilson loop observables. Based on joint work with Sky Cao.

Large ensembles of points with Coulomb interactions arise in various settings of condensed matter physics, classical and quantum mechanics, statistical mechanics, random matrices and even approximation theory, and they give rise to a variety of questions pertaining to analysis, Partial Differential Equations and probability.

We will first review these motivations, then present the ”mean-field” derivation of effective models and equations describing the system at the macroscopic scale. We then explain how to analyze the next order behavior, giving information on the configurations at the microscopic level and connecting with crystallization questions, and finish with the description of the effect of temperature.

Talk will be followed by Tea in the Math Common Room – Science Center, 4th Floor

The flag variety of rank r=(r_1,…,r_k) has points corresponding to collections of subspaces (V_1,…, V_k) with V_i of dimension r_i such that V_i is contained in V_{i+1}. It can be embedded into a multi-projective space, where it is cut out by the incidence Plücker relations. We explore two extensions of this variety: First, we study the nonnegative flag variety, which corresponds to a subset of the flag variety consisting of flags that can be represented by totally positive matrices. Second, we study the tropicalization of the flag variety and, more specifically, its nonnegative part. In both cases, we provide equivalent descriptions of these spaces for flag varieties of rank r=(a,a+1,…,b), where r consists of consecutive integers. We also explore descriptions of the nonnegative tropical flag variety in terms of polytopal subdivisions. Finally, based on explicit computations, we highlight a number of possible fan structures on the totally nonnegative tropical flag variety. This talk is based on joint work with Chris Eur and Lauren Williams.

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For more info, see https://math.mit.edu/combin/

I will begin the talk with an overview of the topic of quantitative homogenization for elliptic and parabolic equations. Homogenization refers to the procedure of replacing a very “noisy” equation– one with rapidly oscillating coefficients– with a nicer, “effective” equation in a large-scale limit. There is a very abstract theory of (qualitative) homogenization, which is classical. We will discuss the more concrete theory of quantitative homogenization, which has been developed recently. A central role in the story concerns certain “coarse-graining” arguments, which can be seen as constituting a rigorous renormalization group-type approach, formulated in the language of analysis. These methods have surprising applications in mathematical physics and probability, which are still emerging. In the second part of the talk, I will discuss one such application (in a recent joint work with V. Vicol) to turbulence theory: namely, a proof of anomalous diffusion for an advection-diffusion equation.

Talk will be followed by Tea in the Math Common Room – Science Center, 4th Floor

A chaotic quantum system can be studied using the out-of-time-order correlator (OTOC). I will tell you about pole skipping — a recently discovered feature of the retarded Green’s function — that seems to also know things: things like the Lyapunov exponent and the butterfly velocity, which are important quantifiers of the OTOC. Then I will talk about a systematic way of deriving pole-skipping conditions for general holographic CFTs dual to classical bulk theories and how to use this framework to derive a few interesting statements including: (1) theories with higher spins generally violate the chaos bound; (2) the butterfly velocity calculated using pole skipping agrees with that calculated using shockwaves for arbitrary higher-derivative gravity coupled to ordinary matter; (3) shockwaves are related to a special type of quasinormal modes. As we will see, the techniques are entirely classically gravitational, which I will go through with a certain level of details.

Zoom Link: https://harvard.zoom.us/j/99317037992
Password: cmsa

Given a moduli space of either sheaves on a smooth projective variety or a moduli space of representations of a quiver, there are several invariants that we can extract. One of the ways to get numbers out of a moduli space is to integrate (possibly against a virtual fundamental class) certain tautological classes. Such numbers often have interesting structures behind, and I will talk about two: how they change when one changes a stability condition (wall-crossing formulas) and some universal and explicit linear relations that those invariants always seem to satisfy (Virasoro constraints). Both of these phenomena are related to a vertex algebra found by D. Joyce. For simplicity I will mostly focus on the case of representations of a quiver. The talk is based on joint work with A. Bojko and W. Lim.

We study Clifford locality-preserving unitaries and stabilizer
Hamiltonians by means of Hermitian K-theory. We demonstrate how
the notion of algebraic homotopy of modules over Laurent polynomial
rings translates into the connectedness of two short-range entangled
stabilizer Hamiltonians by a shallow Clifford circuit. We apply this
observation to a classification of homotopy classes of loops of Clifford
unitaries. The talk is based on a work in collaboration with Yichen Hu
https://arxiv.org/abs/2306.09903.

This seminar will be held in person and on Zoom: https://harvard.zoom.us/j/97514733653?pwd=Q05XN3oxSnYvaXlnS0dsRnVyMXZMUT09

Password: 353114

Motivated by the recent experimental breakthroughs in observing Fractional Quantum Anomalous Hall
(FQAH) states in moir\’e Transition Metal Dichalcogenide (TMD) bilayers, we propose and study various
unconventional phase transitions between quantum Hall phases and Fermi liquids or charge ordered
phases upon tuning the bandwidth. At filling -2/3, we describe a direct transition between the FQAH state
and a Charge Density Wave (CDW) insulator. The critical theory resembles that of the familiar deconfined
quantum critical point (DQCP) but with an additional Chern-Simons term. At filling -1/2, we study the
possibility of a continuous transition between the composite Fermi liquid (CFL) and the Fermi liquid (FL)
building on and refining previous work by Barkeshli and McGreevy. Crucially we show that translation
symmetry alone is enough to enable a second order CFL-FL transition. We argue that there must be critical
CDW fluctuations though neither phase has long range CDW order. A striking signature is a universal jump
of resistivities at the critical point. With disorder, we argue that the CDW order gets pinned and the CFL-FL
evolution happens through an intermediate electrically insulating phase with mobile neutral fermions. A
clean analog of this insulating phase with long range CDW order and a neutral fermi surface can potentially
also exist. We also present a critical theory for the CFL to FL transition at filling -3/4. Our work opens up a
new avenue to realize deconfined criticality and fractionalized phases beyond familiar Landau level physics
in the moire Chern band system.

This seminar will be held in person and on Zoom: https://harvard.zoom.us/j/97514733653?pwd=Q05XN3oxSnYvaXlnS0dsRnVyMXZMUT09

Password: 353114

Scattering amplitudes in strong background fields provide an arena where perturbative and non-perturbative physics meet, with important applications ranging from laser physics to black holes, but their study is hampered by the cumbersome nature of QFT in the background field formalism. In this talk, I will try to convince you that strong-field scattering amplitudes contain a wealth of physical information which cannot be obtained with standard perturbative techniques, ranging from all-order classical observables to constraints on exact solutions. Furthermore, I will discuss how amplitudes in certain chiral strong fields can be obtained to all-multiplicity twistor and string methods.

Cell fate decisions are made by allowing external signals to govern the steady-state pattern adopted by networks of interacting regulatory factors governing transcription and translation. One of these decisions, of importance for both developmental processes and for cancer metastasis, is the epithelial-mesenchymal transition (EMT). In this talk, we will argue that these biological networks have highly non-generic interaction structures such that they allow for phenotypic states with very low frustration, i.e. where most interactions are satisfied. This property has important consequences for the allowed dynamics of these systems.

This seminar will be held in person and on Zoom. For more information on how to join, please see: https://harvard.zoom.us/j/96657833341

Since Floer’s work in 1988, various Floer homologies have been constructed for closed 3-manifolds, knots, and sutured manifolds.  In 2008, Kronheimer-Mrowka proposed a conjecture about isomorphisms among Floer homologies. In this talk, I will introduce an approach to proving the isomorphisms and mention some partial results, based on combinatorial version of Floer homology. This work is joint with Baldwin, Li, and Sivek.

Zoom: https://harvard.zoom.us/j/7855806609

Password: cmsa

I spent most of my time here cycling (or is it biking?) and thinking about algebraic cycles from a homotopical viewpoint. I will speak about the latter. In joint work with Matthew Morrow, we developed a theory of motivic cohomology of schemes beyond the case of smooth schemes over a field. I will explain the cycle-theoretic aspects of this construction, focusing on the case of surfaces, revisiting older results of Krishna and Srinivas.

Flat bands, like those in the kagome lattice or twisted bilayer graphene, are a
natural setting for studying strongly coupled physics since the interaction
strength is the only energy scale in the problem. They can exhibit
unconventional behavior in the multi-orbital case: the mean-field theory of
flat band attractive Hubbard models shows the possibility of
superconductivity even though the Fermi velocity of the bands is strictly zero.
However, it is not necessary to resort to this approximation. We demonstrate
that the groundstates and low-energy excitations of a large class of attractive
Hubbard models are exactly solvable, offering a rare, microscopic view of
their physics. The solution reveals the importance of quantum geometry in
escaping (some of) BCS phenomenology within a tractable and nontrivial
strong coupling theory.

This seminar will be held in person and on Zoom: https://harvard.zoom.us/j/97514733653?pwd=Q05XN3oxSnYvaXlnS0dsRnVyMXZMUT09

Password: 353114

While generative language models exhibit powerful capabilities at large scale, when either the model
or the number of training steps is too small, they struggle to produce coherent and fluent text:
Existing models whose size is below a few billion parameters often do not generate coherent text
beyond a few sentences. Hypothesizing that one of the main reasons for the strong reliance on size is
the vast breadth and abundance of patterns in the datasets used to train those models, this
motivates the following question: Can we design a dataset that preserves the essential elements of
natural language, such as grammar, vocabulary, facts, and reasoning, but that is much smaller and
more refined in terms of its breadth and diversity?
In this talk, we introduce TinyStories, a synthetic dataset of short stories that only contain words
that 3 to 4-year-olds typically understand, generated by GPT-3.5/4. We show that TinyStories can
be used to train and analyze language models that are much smaller than the state-of-the-art models
(below 10 million parameters), or have much simpler architectures (with only one transformer
block), yet still produce fluent and consistent stories with several paragraphs that are diverse and
have almost perfect grammar, and demonstrate certain reasoning capabilities. We also show that
the trained models are substantially more interpretable than larger ones, as we can visualize and
analyze the attention and activation patterns of the models, and show how they relate to the
generation process and the story content. We hope that TinyStories can facilitate the development,
analysis and research of language models, especially for low-resource or specialized domains, and
shed light on the emergence of language capabilities in LMs.

This seminar will be held in person and on Zoom: https://harvard.zoom.us/j/95706757940?pwd=dHhMeXBtd1BhN0RuTWNQR0xEVzJkdz09

The class number formula describes the behavior of the Dedekind zeta function at $s=0$ and $s=1$. The Stark conjecture extends the class number formula, describing the behavior of Artin $L$-functions and $p$-adic $L$-functions at $s=0$ and $s=1$ in terms of units. The Harris–Venkatesh conjecture describes the residue of Stark units modulo $p$, giving a modular analogue to the Stark and Gross conjectures while also serving as the first verifiable part of the broader conjectures of Venkatesh, Prasanna, and Galatius. In this talk, I will draw an introductory picture, formulate a unified conjecture combining Harris–Venkatesh and Stark for weight one modular forms, and describe the proof of this in the imaginary dihedral case.

The replica method, together with Parisi’s symmetry breaking mechanism, is an extremely powerful tool to compute the limiting free energy of virtually any mean field disordered system. Unfortunately, the tool is dramatically flawed from a mathematical point of view. I will discuss a truly elementary procedure which allows to rigorously implement two (out of three) steps of the replica method, and conclude with some remarks on the relation between this new point of view and old work by Mezard and Virasoro on the microstructure of ultrametricity, the latter being the fundamental yet unjustified Ansatz in the celebrated Parisi solution. We are still far from a clear understanding of the issues, but quite astonishingly, evidence is mounting that Parisi’s ultrametricity assumption, the onset of scales and the universal hierarchical self-organisation of random systems in the infinite volume limit, is intimately linked to hidden geometrical properties of large random matrices which satisfy rules reminiscent of the popular SUDOKU game.

The SL(3) web basis is a special basis of certain spaces of tensor invariants developed in the late 90’s by Khovanov and Kuperberg as a tool for computing quantum link invariants. Since then this basis has found connections and applications to cluster algebras, canonical bases, dimer models, quantum topology, and tableau combinatorics. A main open problem has remained: how to find a basis replicating the desirable properties of this basis for SL(4) and beyond? I will describe joint work with Oliver Pechenik, Stephan Pfannerer, Jessica Striker, and Josh Swanson in which we construct such a basis for SL(4). Modified versions of plabic graphs and the six-vertex model and new tableau combinatorics will appear along the way.

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For more info, see https://math.mit.edu/combin/

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https://people.math.harvard.edu/~gammage/ons/

The theme of the Thursday Seminar this term is “topological aspects of quantum field theory.”   The first lecture will be a brief overview, followed by an introduction to topological quantum field theories.

We start with an introduction to Szpiro’s conjecture and its equivalent formulation, the abc conjecture. Szpiro’s goal in the 1980s was to prove the Mordell conjecture (effectively too!); it was soon observed that the implications of this conjecture and its alternate versions also include landmarks such as Fermat’s Last Theorem, Baker’s theorem, Roth’s theorem, and the non-existence of Siegel zeroes for certain Dirichlet L-functions. We will then highlight recent work by Nicole Looper on new implications in arithmetic dynamics, in particular that the abcd conjecture implies uniform boundedness for preperiodic points of polynomials as well as a weak dynamical Lang’s conjecture for points of small height.

http://people.math.harvard.edu/~demarco/AlgebraicDynamics/

In this talk, I will introduce a new kind of measurement-based quantum computation inspired by Floquet codes. In this model, the quantum logical gates are implemented by short sequences of low-weight measurements which simultaneously encode logical information and enable error correction. We introduce a new class of quantum error-correcting codes generalizing Floquet codes that achieve this, which we call dynamic automorphism (DA) codes.
As in Floquet codes, the instantaneous codespace of a DA code at any fixed point in time is that of a topological code. In this case, the quantum computation can be viewed as a sequence of time-like domain walls implementing automorphisms of the topological order, which can be understood in terms of reversible anyon condensation paths in a particular parent model. This talk will introduce all of these concepts as well as provide a new perspective for thinking about Floquet codes.
The explicit examples that we construct, which we call DA color codes, can implement the full Clifford group of logical gates in 2+1d by two- and, rarely three-body measurements. Using adaptive two-body measurements, we can achieve a non-Clifford gate in 3+1d, making the first step towards universal quantum computation in this model.
The talk is based on recent work with Nathanan Tantivasadakarn, Shankar Balasubramanian, and David Aasen [arxiv: 2307.10353].

This seminar will be held in person and on Zoom: https://harvard.zoom.us/j/977347126

Password: cmsa

The Ramsey number r_k(G, H) of two k-uniform hypergraphs G and H is the smallest n such that any edge-coloring of the complete k-uniform hypergraph on n vertices contains either a red copy of G or a blue copy of H. Hypergraph Ramsey theory is concerned with the growth rates of such Ramsey numbers, particularly when one or both of {G, H} is a clique of size tending to infinity. Classical arguments of Erdős-Rado and Erdős-Hajnal reduce most major problems in this area from all higher uniformities to uniformity k=3, but fail to bridge the gap from the graph case k=2 to the hypergraph case k=3.

In this talk, we survey the known approaches for lifting Ramsey graphs to 3-uniform hypergraphs, most famously the stepping-up construction of Erdős-Hajnal. We collect these constructions under the umbrella term “pair constructions” and present new variations proving new lower bounds. In particular, we describe an explicit family of 3-uniform hypergraphs H (including links of odd cycles and tight cycles of length not divisible by 3) which satisfy r_3(H, K_n) > 2^{c n log n}. We also prove the existence of a linear 3-uniform hypergraph H for which r_3(H, K_n) grows superpolynomially.

Based on joint work with David Conlon, Benjamin Gunby, Jacob Fox, Dhruv Mubayi, Andrew Suk, and Jacques Verstraete.

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For more info, see https://math.mit.edu/combin/

A matrix M of real numbers is called totally positiveif every minor of $M$ is nonnegative. Gantmakher and Krein showedin 1937 that a Hankel matrix $H = (a_{i+j})_{i,j \ge 0}$of real numbers is totally positive if and only if the underlyingsequence $(a_n)_{n \ge 0}$ is a Stieltjes moment sequence,i.e.\ the moments of a positive measure on $[0,\infty)$.Here I will introduce a generalization: a matrix $M$ of polynomials(in some set of indeterminates) will be called{\em coefficientwise totally positive}\/ if every minor of $M$is a polynomial with nonnegative coefficients. And a sequence$(a_n)_{n \ge 0}$ of polynomials will be called{\em coefficientwise Hankel-totally positive}\/ if the Hankel matrix$H = (a_{i+j})_{i,j \ge 0}$ associated to $(a_n)$ is coefficientwisetotally positive.It turns out that many sequences of polynomials arising naturallyin enumerative combinatorics are (empirically) coefficientwiseHankel-totally positive. I will discuss some methods for proving this.But these proofs fall far short of what appears to be true.

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For more info, see https://math.mit.edu/combin/

String theories feature a wide array of moduli spaces. We propose that the energy cutoff scale of these theories – the so-called species scale – can be determined through higher-curvature corrections. This species scale varies with the moduli; we use it both asymptotically to bound the diameter of the field space, as well as in the interior to determine a “desert point” where it is maximized.

Please note that there will be a pre-talk aimed at graduate students by David Wu from 10:00-10:30am

One of the strangest consequences of the Axiom of Choice is the following Hardin-Taylor 2008 result:  there is a “predictor” such that for every function $f$ from the reals to the reals—even nowhere continuous $f$—the predictor applied to $f \restriction (-\infty,t)$ correctly predicts $f(t)$ for *almost every* $t \in R$.  They asked how robust such a predictor could be, with respect to distortions in the time (input) axis; more precisely, for which subgroups $H$ of Homeo^+(R) do there exist $H$-invariant predictors?  Bajpai-Velleman proved an affirmative answer when H=Affine^+(R), and a negative answer when H is (the subgroup generated by) C^\infty(R).  They asked about the intermediate region; in particular, do there exist analytic-invariant predictors?  We have partially answered that question:  assuming the Continuum Hypothesis (CH), the answer is “no”. Regarding other subgroups of Homeo^+(R), we have affirmative answers that rely solely on topological group-theoretic properties of the subgroup.  But these properties are very restrictive; e.g., all known positive examples are metabelian.  So there remain many open questions. This is joint work with Aldi, Buffkin, Cline, Cody, Elpers, and Lee.

Crepant resolutions have inspired connections between birational geometry, derived categories, representation theory, and motivic integration. In this talk, we prove that every variety with log-terminal singularities admits a crepant resolution by a smooth stack. We additionally prove a motivic McKay correspondence for stack-theoretic resolutions. Finally, we show how our work naturally leads to a generalization of twisted mapping spaces. No prior knowledge of stacks will be assumed. This is joint work with Jeremy Usatine.

https://sites.google.com/view/harvardmitag

Transformers can be trained to solve problems of mathematics. I present two recent applications, in mathematics and physics: predicting integer sequences, and discovering the properties of scattering amplitudes in a close relative of Quantum ChromoDynamics.
Problems of mathematics can also help understand transformers. Using two examples from linear algebra and integer arithmetic, I show that model predictions can be explained, that trained models do not confabulate, and that carefully choosing the training distributions can help achieve better, and more robust, performance.

This seminar will be held in person and on Zoom: https://harvard.zoom.us/j/95706757940?pwd=dHhMeXBtd1BhN0RuTWNQR0xEVzJkdz09

We prove, for any prime ∫ greater than or equal to 5, that a density one set of rational elliptic curves are ∫-Diophantine stable in the sense of Mazur and Rubin. This is joint work with Anwesh Ray.

A parking function is a sequence (a_1,…,a_n) of positive integers such that if b_1<=…<=b_n is the increasing rearrangement of a_1,…,a_n, then b_i<=i for 1<=i<=n. Parking functions of length n are in bijection with labelled forests on the vertex set [n]={1,2,…,n} (or rooted trees on [n]_0={0,1,…,n} with root 0). We obtain some new results on the enumeration of parking functions and labelled forests, concentrating in particular on the joint distribution of several sets of statistics on parking functions. The distribution of most of these individual statistics is known, but the joint distributions are new. Extensions of our techniques are discussed. Joint work with Richard Stanley and with Stephan Wagner.

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For more info, see https://math.mit.edu/combin/

The more prosaic cousin of active matter, driven inactive matter, is still full of unexpected phenomena. I will discuss two projects involving two seemingly mundane systems, a phase-separating colloidal mixture and a lipid membrane, which demonstrate counterintuitive properties when driven out of equilibrium. We will see that the phase separating mixture, when driven by a uniform force, develops (in simulations) an intriguing pattern with a characteristic length scale set by the magnitude of the drive. We will look at some theoretical approaches to understanding the pattern formation and possible experimental realizations. The membrane, when driven by an oscillatory electric field, develops (in experiments) a long-lived metastable state with a decreased capacitance and increased dissipation. This state may have implications for neuronal processing and memory formation.

This seminar will be held in person and on Zoom. For more information on how to join, please see: https://harvard.zoom.us/j/96657833341

Random surfaces are central paradigms in equilibrium statistical mechanics. As these surfaces become larger, their statistical behaviors become strongly dependent on how their boundaries are pinned down. This can lead to phase transitions, such as facet edges separating a flat region of the surface from the places where it curves. How do these surfaces look and fluctuate near such interfaces? The purpose of this talk is to explain some of these aspects, and to touch on some of the diverse mathematical concepts that go into its analysis.

Talk will be followed by Tea in the Math Common Room – Science Center, 4th Floor

In this talk I will explain my works with Y. Tachikawa to study anomaly in heterotic string theory via homotopy theory, especially the theory of Topological Modular Forms (TMF). TMF is an E-infinity ring spectrum which is conjectured by Stolz-Teichner to classify two-dimensional supersymmetric quantum field theories in physics. In the previous work (https://arxiv.org/abs/2108.13542), we proved the vanishing of anomalies in heterotic string theory mathematically by using TMF.

Furthermore, we have a recent update (https://arxiv.org/abs/2305.06196) on the previous work. Because of the vanishing result, we can consider a secondary transformation of spectra, which is shown to coincide with the Anderson self-duality morphism of TMF. This allows us to detect subtle torsion phenomena in TMF by differential-geometric ways, and leads us to new conjectures on the relation between VOAs and TMF.

This seminar will be held in person and on Zoom: https://harvard.zoom.us/j/977347126

Password: cmsa

Oriented matroids are matroids with extra sign data, and they are useful in the tropical study of real algebraic geometry. In order to study the topology of real algebraic hypersurfaces constructed from patchworking, Renaudineau and Shaw introduced an algebraically defined filtration of the tope space of an oriented matroid based on Quillen filtration. We will prove the equality between their filtration (together with the induced maps), the topologically defined Kalinin filtration, and the combinatorially defined Varchenko-Gelfand dual degree filtration over Z/2Z. We will also explain how the dual degree filtration can serve as a Z-coefficient version of the other two in this setting. This is joint work with Kris Shaw.

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For more info, see https://math.mit.edu/combin/

No additional detail for this event.