Quantum Field Theory and Entanglement - Edward Witten @videosfromIAS

Quantum Field Theory and Entanglement - Edward Witten

Institute for Advanced Study 2018-08-15 | Prospects in Theoretical Physics 2018: From Qubits to Spacetime

Topic: Quantum Field Theory and Entanglement
Speaker: Edward Witten
Date: July 16, 2018

For more video please visit http://video.ias.edu

The Spin SYK Model - Michael Winer

Institute for Advanced Study 2024-10-18 | IAS High Energy Theory Seminar

Topic: The Spin SYK Model
Speaker: Michael Winer
Affiliation: Institute for Advanced Study
Date: October 18, 2024

We analyze a model of qubits which we argue has an emergent quantum gravitational description similar to the fermionic Sachdev-Ye-Kitaev (SYK) model. The model we consider is known as the quantum q
-spin model because it features q
-local interactions between qubits. It was previously studied as a model of a quantum spin glass, and while we find that the model is glassy for q=2
, q=3
, and likely q=4
, we also find evidence for previously unexpected SYK-like behavior for the quenched free energy down to the lowest temperatures for q≥5
. This SYK-like physics includes power-law correlation functions and an extensive low temperature entropy, so we refer to the model as \name{}. The model is generic in that it includes all possible q
-body couplings, lacks most symmetries, and has no spatial structure, so our results can be construed as establishing a certain ubiquity of quantum holography in systems dominated by many-body interactions. Furthermore, we discuss a generalized family of models which includes \name{} and which provably exhibit SYK-like physics in the solvable limit of large local Hilbert space dimension. We also comment on implications of a bosonic system with SYK-like properties for the study of holography, Hamiltonian complexity, and related topics.

Equivariant Symplectic Homology - Jean Gutt

Institute for Advanced Study 2024-10-18 | Joint IAS/Princeton/Montreal/Paris/Tel-Aviv Symplectic Geometry Zoominar

Topic: Ekeland-Hofer Capacities as Coming from Positive S1
Equivariant Symplectic Homology
Speaker: Jean Gutt
Affiliation: Institute for Advanced Study
Date: October 18, 2024

Associated to a star-shaped domain in ℝ2nR2n are two increasing sequences of capacities: the Ekeland-Hofer capacities and the so-called Gutt-Hutchings capacities. I shall recall both constructions and then present the main theorem that they are the same. This is joint work with Vinicius Ramos.

First Explicit Reciprocity Law for Unitary Friedberg—Jacquet PeriodsMurilo Zanarella

Institute for Advanced Study 2024-10-18 | Joint PU/IAS Number Theory

Topic: First Explicit Reciprocity Law for Unitary Friedberg—Jacquet Periods
Speaker: Murilo Zanarella
Affiliation: John Hopkins University
Date: October 17, 2024

In the early 2000's, Bertolini and Darmon introduced a new technique to bound Selmer groups of elliptic curves via level raising congruences. This was the first example of what is now termed a "bipartite Euler system", and over the last decade we have seen many breakthroughs on constructing such systems for other Galois representations, including settings such as twisted and cubic triple product, symmetric cube, and Rankin—Selberg, with applications to the Bloch—Kato conjecture and to Iwasawa theory.

For this talk, I'll consider Galois representations attached to automorphic forms on a totally definite unitary group U(2r) over a CM field which are distinguished by the subgroup U(r) x U(r). I'll discuss a new "first explicit reciprocity law" in this setting and its application to the corresponding Bloch—Kato conjecture, focusing on new obstacles which arise from the lack of local multiplicity one.

Scattering Amplitudes, Multi-variate Residues and Valuated Matroids - Nick Early

Institute for Advanced Study 2024-10-17 | Special Year Seminar II

Topic: Scattering Amplitudes, Multi-variate Residues and Valuated Matroids
Speaker: Nick Early
Affiliation: Institute for Advanced Study
Date: October 17, 2024

Multi-variate residues on Grassmannians G(k,n) and moduli spaces M0,n are ubiquitous in the study of scattering amplitudes; they provide a powerful and essential tool. Amenable theories include the biadjoint scalar, NLSM, Yang-Mills, gravity and N=4 Super Yang-Mills and more. I will review a relation between such multi-variate residues and the real tropical Grassmannian, i.e. the set of valuated matroids which are realizable over the real numbers. In this way, a physical problem becomes a question in geometric combinatorics.

Discrete and Continuous Duality Algebras - Leonid Monin

Institute for Advanced Study 2024-10-17 | Special Year Seminar I

Topic: Discrete and Continuous Duality Algebras
Speaker: Leonid Monin
Affiliation: École Polytechnique Fédérale de Lausanne
Date: October 16, 2024

A classical construction associates a Poincare duality algebra to a homogeneous polynomial on a vector space. This construction was used to give a presentation for cohomology rings of complete smooth toric varieties by Khovanskii and Pukhlikov and of some spherical varieties (including full flag varieties) by Kaveh.

In my talk, I will recall the above results and will give two recent generalizations of the construction of duality algebras. The first one replaces the homogeneous polynomial by weighted homogeneous polynomial (and more general functions). In contrast to the classical construction,this allows us to construct Poincare duality algebras which are not necessarily generated in degree 1. The second extension is the discrete analogue of the classical construction, which associates an algebra with Gorenstein duality to a polynomial on a lattice (free abelian group). As a corollary, this provides a presentation for K-ring of smooth complete toric varieties as well as full flag varieties.

Decoupling Identities and Oriented Matroids and Beyond - Nick Early

Institute for Advanced Study 2024-10-16 | IAS Physics Group Meeting

Topic: Decoupling Identities and Oriented Matroids and Beyond
Speaker: Nick Early
Affiliation: Institute for Advanced Study
Date: October 16, 2024

Closure of Orbits of the Pure Mapping Class Group on the Character Variety- Alireza Salehi Golsefidy

Institute for Advanced Study 2024-10-16 | Joint IAS/PU Groups and Dynamics Seminar
5:00pm|Simonyi 101
Topic: Closure of Orbits of the Pure Mapping Class Group on the Character Variety
Speaker: Alireza Salehi Golsefidy
Affiliation: University of California San Diego
Date: October 15, 2024

For every surface S, the pure mapping class group G_S acts on the (SL_2)-character variety Ch_S of a fundamental group P of S. The character variety Ch_S is a scheme over the ring of integers. Classically this action on the real points Ch_S(R) of the character variety has been studied in the context of the Teichmuller theory and SL(2,R)-representations of P. In a seminal work, Goldman studied this action on a subset of Ch_S(R) which comes from SU(2)-representations of P. In this case, Goldman showed that if S is of genus g greater than 1 and zero punctures, then the action of G_S is ergodic. Previte and Xia studied this question from topological point of view, and when g greater than 0, proved that the orbit closure is as large as algebraically possible. Bourgain, Gamburd, and Sarnak studied this action on the F_p-points Ch_S(F_p) of the character variety where S is a puncture torus. They conjectured that in this case, this action has only two orbits, where one of the orbits has only one point. Recently, this conjecture was proved for large enough primes by Chen. When S is an n-punture sphere, the finite orbits of this action on Ch_S(C) are connected to the algebraic solutions of Painleve differential equations. In this talk, I will report on our recent contributions to this theory. Here are some sample results:

An almost complete description of the Zariski-closure of infinite G_S-orbits in Ch_S(F) where F is a characteristic zero field. Answering a question of Goldman-Previte-Xia by understanding the orbit closure of G_S on SU(2)-representation part of Ch_S(R) where S is an n-puncture sphere. Show that the original result of Previte and Xia is not accurate and give a description of the cases where it fails. Proving that in most cases the closure of G_S-orbits in the p-adic integer points Ch_S(Z_p) are open within given polynomial constrains. We give precise description of exceptional cases. (This is a joint work with Natallie Tamam.)

The Ramsey Numbers – New Results and New Perspectives - Julian Sahasrabudhe

Institute for Advanced Study 2024-10-15 | Bourgain Lecture

Topic: The Ramsey Numbers – New Results and New Perspectives
Speaker: Julian Sahasrabudhe
Affiliation: Institute for Advanced Study
Date: October 15, 2024

In this talk I will try to motivate the interest and some of the mystery in the Ramsey numbers R(k), which are fundamental quantities in combinatorics. I will go on to discuss some recent progress on our understanding of these numbers and make some connections to problems about the geometry of random variables in high dimensions.

Serrin’s Overtermined Problem In Rough Domains - Alessio Figalli

Institute for Advanced Study 2024-10-15 | Analysis and Mathematical Physics

Topic: Serrin’s Overtermined Problem In Rough Domains
Speaker: Alessio Figalli
Affiliation: Institute for Advanced Study
Date: October 15, 2024

The classical Serrin’s overdetermined theorem states that a C^2 bounded domain, which admits a function with constant Laplacian that satisfies both constant Dirichlet and Neumann boundary conditions, must necessarily be a ball. While extensions of this theorem to non-smooth domains have been explored since the 1990s, the applicability of Serrin’s theorem to Lipschitz domains remained unresolved.

In this talk, I will discuss a recent answer to this open question, which was given in a joint paper with Yi Zhang.

As we shall see, our approach shows that the result holds for domains that are sets of finite perimeter with a uniform upper bound on the density, and it also allows for slit discontinuities.

Relative Symplectic Cohomology of Pairs - Adi Dickstein

Institute for Advanced Study 2024-10-15 | Symplectic Geometry Seminar

Topic: Relative Symplectic Cohomology of Pairs
Speaker: Adi Dickstein
Affiliation: Tel Aviv University
Date: October 15, 2024

Relative symplectic cohomology, an invariant of subsets in a symplectic manifold, was recently introduced by Varolgunes. In this talk, I will present a generalization of this invariant to pairs of subsets, which shares similar properties with the singular cohomology of pairs, such as excision and a product structure. Using this new invariant, I will demonstrate new symplectic rigidity phenomena. Joint with Yaniv Ganor, Leonid Polterovich and Frol Zapolsky.

Analytic Insights into the Zig-Zag Product and Its Friends: Part II - Gil Cohen

Institute for Advanced Study 2024-10-15 | Computer Science/Discrete Mathematics Seminar II

Topic: Analytic Insights into the Zig-Zag Product and Its Friends: Part II
Speaker: Gil Cohen
Affiliation: Tel Aviv University
Date: October 15, 2024

The well-known Zig-Zag product and related graph operators, like derandomized squaring, are fundamentally combinatorial in nature. Classical bounds on their behavior often rely on a mix of combinatorics and linear algebra. However, these traditional bounds are not tight and frequently fail to align with experimental results.

In these talks, we will present a more refined analysis that utilizes the full spectrum of the graph, rather than relying solely on its spectral expansion. This approach produces results that both match experimental observations and, in a sense, are proved to be optimal. Our technique is analytic, diverging from classical methods: for the upper bound, we apply finite free probability, while for the lower bound, we draw on results from analytic combinatorics.

Based on joint works with Itay Cohen, Gal Maor, and Yuval Peled.

No prior knowledge is required.

Relevant papers:

Random Walks on Rotating Expanders (STOC 2023)
Tight Bounds for the Zig-Zag Product (FOCS 2024)
Derandomized Squaring: An Analytic Insight into Its True Behavior (manuscript)

Subgroup Tests and the Aldous Lyons Conjecture - Michael Chapman

Institute for Advanced Study 2024-10-15 | Computer Science/Discrete Mathematics Seminar I

Topic: Subgroup Tests and the Aldous--Lyons Conjecture
Speaker: Michael Chapman
Affiliation: New York University
Date: October 07, 2024

A common theme in mathematics is that limits of finite objects are well behaved. This allows one to prove many theorems about finitely approximable objects, while leaving the general case open --- examples for this are Gottschalk's conjecture, Kaplansky's direct finiteness conjecture, and many more. When the topology of the space is somewhat coarse, it becomes very hard to decide whether every object is approximable by finite ones, or whether there exist non-approximable objects. Some of the more famous problems in various fields of mathematics can be framed this way; this includes Connes' embedding problem (CEP) in functional analysis, the Aldous--Lyons conjecture in probability theory, the soficity problem of Gromov--Weiss in group theory, and more.

In 2019, Ji--Natarajan--Vidick--Wright--Yuen resolved CEP in the negative, namely, they showed that there are non-approximable characters of the free group. The amazing thing about their result is that it is deduced from complexity theory, and specifically from undecidability in certain (quantum) interactive proof systems. Inspired by their work, we suggest a novel interactive proof system which is related to the Aldous--Lyons conjecture in the following way: If the Aldous--Lyons conjecture was true, then every language in this interactive proof system is decidable. A key concept we introduce for this purpose is that of a Subgroup Test, which is our analogue of a Non-local Game. By providing a reduction from the Halting Problem to this new proof system, we refute the Aldous-Lyons conjecture.

In this first talk we introduce the main concepts, motivate the conjecture and give a high level description of the proof strategy. No special mathematical background will be assumed.

This talk is based on a joint work with Lewis Bowen, Alex Lubotzky and Thomas Vidick.

On the converse to Eisensteins last theorem - Daniel Litt

Institute for Advanced Study 2024-10-15 | Joint IAS/PU Arithmetic Geometry

Topic: On the converse to Eisenstein's last theorem
Speaker: Daniel Litt
Affiliation: Institute for Advanced Study
Date: October 14, 2024

Eisenstein proved, in 1852, that if a function f(z) is algebraic, then its Taylor expansion at a point has coefficients lying in some finitely-generated Z-algebra. I will explain ongoing joint work with Josh Lam which studies the extent to which the converse of this theorem holds. Namely, we conjecture that if f(z) satisfies a (possibly non-linear!) algebraic ODE, non-singular at 0, and its Taylor expansion has coefficients lying in a finitely-generated Z-algebra, then f is algebraic. For linear ODE, we prove this conjecture when f(z) satisfies a Picard-Fuchs equation, with initial conditions the class of an algebraic cycle, and in some other cases. For non-linear ODE, we prove it when f(z) satisfies an "isomonodromy" ODE with "Picard-Fuchs" initial conditions. I'll also discuss some motivic corollaries of these results, and explain their relationship to the Grothendieck-Katz p-curvature conjecture.

Candidate de Sitter Vacua - Richard Nally

Institute for Advanced Study 2024-10-14 | IAS High Energy Theory Seminar

Topic: Candidate de Sitter Vacua
Speaker: Richard Nally
Affiliation: Stanford University
Date: October 14, 2024

We construct compactifications of type IIB string theory that yield, at leading order in the α ′ α ′ and g s g s expansions, de Sitter vacua of the form envisioned by Kachru, Kallosh, Linde, and Trivedi. We specify explicit Calabi-Yau orientifolds and quantized fluxes for which we derive the four-dimensional effective supergravity theories, incorporating the exact flux superpotential, the nonperturbative superpotential from Euclidean D3-branes, and the Kähler potential at tree level in the string loop expansion but to all orders in α ′ α ′ . Each example includes a Klebanov-Strassler throat region containing a single anti-D3-brane, whose supersymmetry-breaking energy, computed at leading order in α ′ α ′ , causes an uplift to a metastable de Sitter vacuum in which all moduli are stabilized. Finding vacua that demonstrably survive subleading corrections, and in which the quantization conditions are completely understood, is an important open problem for which this work has prepared the foundations. Based on arxiv.org/abs/2406.13751.

Reading Alan Turing - Avi Wigderson

Institute for Advanced Study 2024-10-14 | Members' Colloquium

Topic: Reading Alan Turing
Speaker: Avi Wigderson
Affiliation: Institute for Advanced Study
Date: October 14, 2024

I will discuss some well-known and less-known papers of Turing, exemplify the scope of deep, prescient ideas he put forth, and mention follow-up work on these by the Theoretical CS community.

No special background will be assumed.

Analytic Insights into the Zig-Zag Product and Its Friends: Part I - Gil Cohen

Institute for Advanced Study 2024-10-14 | Computer Science/Discrete Mathematics Seminar I

Topic: Analytic Insights into the Zig-Zag Product and Its Friends: Part I
Speaker: Gil Cohen
Affiliation: Tel Aviv University
Date: October 14, 2024

The well-known Zig-Zag product and related graph operators, like derandomized squaring, are fundamentally combinatorial in nature. Classical bounds on their behavior often rely on a mix of combinatorics and linear algebra. However, these traditional bounds are not tight and frequently fail to align with experimental results.

In these talks, we will present a more refined analysis that utilizes the full spectrum of the graph, rather than relying solely on its spectral expansion. This approach produces results that both match experimental observations and, in a sense, are proved to be optimal. Our technique is analytic, diverging from classical methods: for the upper bound, we apply finite free probability, while for the lower bound, we draw on results from analytic combinatorics.

Based on joint works with Itay Cohen, Gal Maor, and Yuval Peled.

No prior knowledge is required.

Relevant papers:

Random Walks on Rotating Expanders (STOC 2023)
Tight Bounds for the Zig-Zag Product (FOCS 2024)
Derandomized Squaring: An Analytic Insight into Its True Behavior (manuscript)

Floer Homology with DG Coefficients. Applications to Cotangent Bundles - Alexandru Oancea

Institute for Advanced Study 2024-10-11 | Joint IAS/Princeton/Montreal/Paris/Tel-Aviv Symplectic Geometry Zoominar

Topic: Floer Homology with DG Coefficients. Applications to Cotangent Bundles
Speaker: Alexandru Oancea
Affiliation: University of Strasbourg
Date: October 11, 2024

Given a path-connected topological space X, a differential graded (DG) local system (or derived local system) is a module over the DGA of chains on the based loop space of X. I will explain how to define in the symplectically aspherical case Hamiltonian Floer homology with coefficients in a DG local system, how this homology fits into a filtered homological toolbox, and will present a number of dynamical applications to cotangent bundles. This is joint work with Jean-François Barraud, Mihai Damian and Vincent Humilière. The construction of Floer homology with enriched coefficients was originally discovered by Barraud-Cornea, and it was revisited over the years in different settings by Abouzaid, Charette, Zhou, and Rezchikov.

Non-escape of Mass for Arithmetic Quantum Limits on Hyperbolic 4-Manifolds - Alexandre de Faveri

Institute for Advanced Study 2024-10-11 | Joint IAS/PU Groups and Dynamics Seminar

Topic: Non-escape of Mass for Arithmetic Quantum Limits on Hyperbolic 4-Manifolds
Speaker: Alexandre de Faveri
Affiliation: Stanford University
Date: October 08, 2024

The Arithmetic Quantum Unique Ergodicity (AQUE) conjecture predicts that the L2 mass of Hecke-Maass cusp forms on an arithmetic hyperbolic manifold becomes equidistributed as the Laplace eigenvalue grows. If the underlying manifold is non-compact, mass could “escape to infinity”. This possibility was ruled out by Soundararajan for arithmetic surfaces, which when combined with celebrated work of Lindenstrauss completed the proof of AQUE for surfaces.

We establish non-escape of mass for Hecke-Maass cusp forms on a congruence quotient of hyperbolic 4-space. Unlike in the setting of hyperbolic 2- or 3-manifolds (for which AQUE has been proved), the number of terms in the Hecke relations is unbounded, which prevents us from naively applying Cauchy-Schwarz. We instead view the isometry group as a group of quaternionic matrices, and rely on non-commutative unique factorization along with certain structural features of the Hecke action. Joint work with Zvi Shem-Tov.

A Mathematical Vocabulary for Textures and its Astronomical Applications - Sihao Cheng

Institute for Advanced Study 2024-10-11 | Friends Lunch with a Member

Topic: A Mathematical Vocabulary for Textures and its Astronomical Applications
Speaker: Sihao Cheng
Affiliation: Institute for Advanced Study, School of Natural Scinces
Date: October 10, 2024

Entanglement Implications of the Pythons Lunch Conjecture - Alex May

Institute for Advanced Study 2024-10-11 | IAS Quantum Aspects of Black Holes Group Meeting

Topic: Entanglement Implications of the Pythons Lunch Conjecture
Speaker: Alex May
Affiliation: Perimeter Institute
Date: October 10, 2024

Second Moment of the GL_3 Standard L-function on the Critical LineMatthew Young

Institute for Advanced Study 2024-10-11 | Joint PU/IAS Number Theory

Topic: Second Moment of the GL_3 Standard L-function on the Critical Line
Speaker: Matthew Young
Affiliation: Rutgers University
Date: October 10, 2024

The second and fourth moments of the Riemann zeta function have been known for about a century, but the sixth moment remains elusive.

The sixth moment of zeta can be thought of as the second moment of a GL_3 Eisenstein series, and it is natural to consider variants of the problem

where the Eisenstein series is replaced by a cusp form. I will discuss recent work with Agniva Dasgupta and Wing Hong Leung where we obtain a

nontrivial bound on this second moment. I will also discuss some applications, including an improvement on the Rankin-Selberg problem.

Gravitational atoms and black hole binaries - Giovanni Maria Tomaselli

Institute for Advanced Study 2024-10-10 | Institute for Advanced Study Astrophysics Seminar

Topic: Gravitational atoms and black hole binaries
Speaker: Giovanni Maria Tomaselli
Affiliation: Institute for Advanced Study
Date: October 10, 2024

Superradiant instabilities may create clouds of ultralight bosons around rotating black holes, forming so-called "gravitational atoms". In this talk, I will review a series of papers that study the effects of a binary companion’s presence. The companion can induce transitions between bound states of the cloud (resonances), as well as transitions from bound to unbound states (ionization). These processes backreact on the binary’s dynamics and leave characteristic imprints on the emitted gravitational waves (GWs), providing direct information about the mass of the boson and the state of the cloud. However, some of the resonances may destroy the cloud before the binary enters the frequency band of future gravitational wave detectors. This destruction leaves a mark on the binary’s eccentricity and inclination, which can be identified through a statistical analysis of a population of binary black holes.

Combinatorial Inequalities and Combinatorial Interpretations: Part III - Igor Pak

Institute for Advanced Study 2024-10-10 | Special Year Seminar I

Topic: Combinatorial Inequalities and Combinatorial Interpretations: Part III
Speaker: Igor Pak
Affiliation: Institute for Advanced Study
Date: October 09, 2024

In the third talk, I will concentrate on inequalities for linear extensionsof finite posets. I will start with several inequalities which do have a combinatorial proof. I will then turn to Stanley's inequality and outline the proof why its defect does not have a combinatorialinterpretation. Time permitting I will discuss what does that sayabout the equality cases and stability of the Alexandrov-Fenchelinequality.

Joint work with Swee Hong Chan (STOC 2024).

Higher Dimensional Fourier Quasicrystals from Lee-Yang Varieties - Pavel Kurasov

Institute for Advanced Study 2024-10-08 | Analysis and Mathematical Physics

Topic: Higher Dimensional Fourier Quasicrystals from Lee-Yang Varieties
Speaker: Pavel Kurasov
Affiliation: Stockholm University
Date: October 08, 2024

Fourier Quasicrystals (FQ) are defined as crystalline measures
μ=∑λ∈Λaλδλ,μ̂ =∑s∈Sbsδs, so that not only μ (and hence μ̂ ) are tempered distributions, but also |μ|:=∑λ∈Λ|aλ|δλand|μ̂ |:=∑s∈S|bs|δs,
are tempered.

One-dimensional FQs with positive integer weights (that is aλ∈ℕ) can be described using stable Lee-Yang polynomials, as was proven in a joint work with Peter Sarnak. Multidimensional Fourier quasicrystals are discussed in the current talk. It is shown that a rather general family of FQs in ℝd
with positive integer weights can be constructed using co-dimension d Lee-Yang varieties in ℂn,n greater than d.
These complex algebraic varieties are symmetric and avoid certain regions in ℂn, thus generalising zero sets of Lee-Yang polynomials.

It is shown that such FQs can be supported by Delaunay almost periodic sets and are genuinely multidimensional in the sense that their restriction to any one-dimensional subspace is not given by a one-dimensional FQ. Connections to alternative recent approaches by Yves Meyer, Lawton-Tsikh and de Courcy-Ireland-K. are clarified. Is it possible that our construction gives all multidimensional FQs with positive integer masses?

This is joint work with L. Alon, M. Kummer, and C. Vinzant (arXiv:2407.11184).

Combinatorial Inequalities and Combinatorial Interpretations: Part II - Igor Pak

Institute for Advanced Study 2024-10-08 | Special Year Seminar

Topic: Combinatorial Inequalities and Combinatorial Interpretations: Part II
Speaker: Igor Pak
Affiliation: Institute for Advanced Study
Date: October 08, 2024

In the second talk, I will concentrate on polynomial inequalities and whether the defect (the difference of two sides) has a combinatorial interpretation. For example, does the inequality x2+y2≥2xy
have a combinatorial proof and what does that mean? There is a surprisingly clean answer of the most general questions of this kind which can be presented in purely algebraic terms.

Joint work with Christian Ikenmeyer (FOCS 2022).

Subgroup Tests and Tailored Non-local Games - Michael Chapman

Institute for Advanced Study 2024-10-08 | Computer Science/Discrete Mathematics Seminar II

Topic: Subgroup Tests and Tailored Non-local Games
Speaker: Michael Chapman
Affiliation: New York University
Date: October 08, 2024

In the previous talk, we defined Subgroup Tests and the interactive proof system induced by them. In addition, we showed that if the Aldous--Lyons conjecture was true, then this interactive proof system contains only decidable languages. In this talk, we describe why the Halting Problem can be decided in our interactive proof system, which in turn refutes the Aldous--Lyons conjecture. This is done in two steps: The first relates Subgroup Test to a new subclass of non-local games which we term Tailored Games. The second shows that the techniques of MIP*=RE can be refined so that all the games in it are tailored, or in acronym fashion, that Tailored-MIP*=RE.

This talk is based on a joint work with Lewis Bowen, Alex Lubotzky and Thomas Vidick.

Imagining Other Worlds: How Planet-Forming Disks Can Shed Light on What Is Possible

Institute for Advanced Study 2024-10-08 | Institute for Advanced Study / Princeton University Joint Astrophysics Colloquium

Topic: Imagining Other Worlds: How Planet-Forming Disks Can Shed Light on What Is Possible
Speaker: Ilse Cleeves
Affiliation: University of Virginia
Date: October 08, 2024

Hodge theory for non-Archimedean analytic spaces - Vladimir Berkovich

Institute for Advanced Study 2024-10-08 | Joint IAS/PU Arithmetic Geometry

Topic: Hodge theory for non-Archimedean analytic spaces
Speaker: Vladimir Berkovich
Affiliation: Institute for Advanced Study
Date: October 07, 2024

By Deligne's Hodge theory, the integral cohomology groups H^n(X^h, Z) of the C-analytification of a separated scheme X of finite type over C are provided with a mixed Hodge structure, functorial in X. Given a non-Archimedean field K isomorphic to the field of Laurent power series C((z)), there is a functor X \mapsto X^an_K that takes X to the non-Archimedean K-analytification of X_K = X \otimes_C K. I'll describe a Hodge theory for the full subcategory of the category of K-analytic spaces that consists of the strictly K-analytic spaces with the property that each compact analytic subdomain is isomorphic to an analytic domain in a boundaryless space. It extends Deligne's Hodge theory through the above functor and generalizes previously known complex analytic constructions.

Combinatorial Inequalities and Combinatorial Interpretations: Part I - Igor Pak

Institute for Advanced Study 2024-10-08 | Special Year Seminar

Topic: Combinatorial Inequalities and Combinatorial Interpretations: Part I
Speaker: Igor Pak
Affiliation: Institute for Advanced Study
Date: October 07, 2024

In the first talk, I will give a broad survey of classical inequalities that arise in enumerative and algebraic combinatorics. I will discuss how these inequalities lead to questions about combinatorial interpretations, and how these questions influence the area in return. I will conclude with a complexity theoretic setup which allows one to formulate negative results. This talk is aimed at the general audience and absolutely no prior knowledge of computational complexity is assumed.

Non-Invertible Symmetries, Boundary Conditions, and Topological Field Theory - Yichul Choi

Institute for Advanced Study 2024-10-04 | IAS High Energy Theory Seminar

Topic: Non-Invertible Symmetries, Boundary Conditions, and Topological Field Theory
Speaker: Yichul Choi
Affiliation: Institute for Advanced Study
Date: October 04, 2024

In recent years, there have been various activities trying to better understand the interplay between non-invertible global symmetries and boundary conditions in quantum field theories, with applications ranging from scattering of solitons to symmetry-resolved entanglement entropies. In this talk, I will focus on 1+1d conformal field theories, and start by reviewing some of the fundamental properties of topological line defects that generate global symmetries. I will then discuss how the algebra of symmetry operators is modified in the presence of conformal boundary conditions. Universal kinematic aspects of this symmetry algebra, such as its representation theory, are neatly captured by a 2+1d topological field theory defined on manifolds with corners, which I will explain.
Based on arxiv.org/pdf/2409.02159 and arxiv.org/pdf/2409.02806.

AGN HBT OMG - Neal Dalal

Institute for Advanced Study 2024-10-03 | Institute for Advanced Study Astrophysics Seminar

Topic: AGN HBT OMG
Speaker: Neal Dalal
Affiliation: Perimeter Institute
Date: October 3, 2024

I’ll discuss an old technique called intensity interferometry, pioneered by Hanbury Brown and Twiss (HBT) that allows us to achieve sub-microarcsecond angular resolution using ordinary ground-based optical telescopes observing visible light. This technique is best suited for compact, bright sources, and I will discuss applications of intensity interferometry observations of bright, supermassive black holes called active galactic nuclei (AGN). I’ll describe a few possible applications, like measuring the Hubble constant H0 and probing the innermost structure of AGN accretion disks.

The Cosmic Quest for Neutrino Mass - Marilena LoVerde

Institute for Advanced Study 2024-10-03 | Institute for Advanced Study / Princeton University Joint Astrophysics Colloquium

Topic: The Cosmic Quest for Neutrino Mass
Speaker: Marilena LoVerde
Affiliation: University of Washington
Date: September 24, 2024

Neutrinos are the second most abundant particle in the known Universe yet they remain mysterious. While they played an important role in the early Universe, today they contribute at most 1-2% of the energy budget and leave only faint signatures. A major goal of the current decade of experiments is to measure the neutrino contribution to the energy budget at high significance, thereby determining the neutrino mass scale and perhaps even the ordering of masses. I will describe the physical effects of neutrino mass, highlighting the distinct roles played geometry and structure on constraints, and how some dataset combinations give a peculiar preference for “negative neutrino mass,” while others hint at a detection. I will also discuss novel astrophysical probes of neutrino mass that may ultimately help resolve current conundrums and contribute to validating any detection of relic neutrinos in the late Universe.

Derived Categories of Permutahedral Varieties Through Matroids - Mario Sanchez

Institute for Advanced Study 2024-10-03 | Short Talks by Postdoctoral Members

Topic: Derived Categories of Permutahedral Varieties Through Matroids
Speaker: Mario Sanchez
Affiliation: Institute for Advanced Study
Date: October 2, 2024

The Mysterious Kronecker Coefficients - Greta Panova

Institute for Advanced Study 2024-10-03 | Special Year Seminar I
11:00am|Rubenstein Commons | Meeting Room 5
Topic: The Mysterious Kronecker Coefficients
Speaker: Greta Panova
Affiliation: Institute for Advanced Study
Date: October 1, 2024

The Kronecker coefficients of the Symmetric group Sn
are the multiplicities of an irreducible Sn representation in the tensor product of two other irreducibles. They were introduced in 1938 by Murnaghan and generalize the beloved Littlewood-Richardson coefficients of the General Linear group. 86 years later still very little is known about them and they are subject of some of the major open problems in Algebraic Combinatorics. They also appear within Geometric Complexity theory, in the quest for multiplicity obstructions to show computational lower bounds for VP versus VNP. In this talk, we will give an overview of the problems and discuss some recent developments on positivity, asymptotics and computational complexity of the Kronecker coefficients and the reduced Kronecker coefficients.

We will feature results from joint works with Christian Ikenmeyer, Igor Pak and Damir Yeliussizov.

Characterizations of PAC learnability - Nataly Brukhim

Institute for Advanced Study 2024-10-03 | Short Talks by Postdoctoral Members

Topic: Characterizations of PAC learnability
Speaker: Nataly Brukhim
Affiliation: Institute for Advanced Study
Date: October 2, 2024

Elliptic Regularity and Singularity - Federico Franceschini

Institute for Advanced Study 2024-10-02 | Short Talks by Postdoctoral Members

Topic: Elliptic Regularity and Singularity
Speaker: Federico Franceschini
Affiliation: Institute for Advanced Study
Date: October 2, 2024

Energy Dissipation of Navier-Stokes Equation with Non-Characteristic Boundary - Jincheng Yang

Institute for Advanced Study 2024-10-02 | Short Talks by Postdoctoral Members

Topic: Energy Dissipation of Navier-Stokes Equation with Non-Characteristic Boundary
Speaker: Jincheng Yang
Affiliation: Institute for Advanced Study
Date: October 2, 2024

Stability of Geometric Inequalities: Continuous and Discrete Perspectives - Peter van Hintum

Institute for Advanced Study 2024-10-02 | Short Talks by Postdoctoral Members

Topic: Stability of Geometric Inequalities: Continuous and Discrete Perspectives
Speaker: Peter van Hintum
Affiliation: Institute for Advanced Study
Date: October 2, 2024

Trees, Self Dual Gravity, Gravity MHV Amplitudes, Lw1+infinity - Noah Miller

Institute for Advanced Study 2024-10-02 | IAS Physics Group Meeting

Topic: Trees, Self Dual Gravity, Gravity MHV Amplitudes, Lw1+infinity
Speaker: Noah Miller
Affiliation: Institute for Advanced Study
Date: October 2, 2024

Abstract: Based on arxiv.org/abs/2408.11139 . In this talk we'll explain how self-dual metrics (which correspond to spacetimes containing only positive helicity gravitons) can be written as a sum over connected "tree" diagrams, which are different from Feynman diagrams. If we add two anti-self-dual perturbations to the spacetime and evaluate the on-shell action of said spacetime, we can give a first principles proof of the NSVW tree formula for graviton MHV scattering in Einstein gravity. Time permitting we'll also discuss how the tree formula in SDG can be used to express the action of the so-called Lw1+infinity algebra on metrics in self dual gravity.

How Efficiently Can We Solve Unique Games? - Yotam Dikstein

Institute for Advanced Study 2024-10-02 | Short Talks by Postdoctoral Members

Topic: How Efficiently Can We Solve Unique Games?
Speaker: Yotam Dikstein
Affiliation: Institute for Advanced Study
Date: October 1, 2024

Singularities of Constructible Sheaves - Tong Zhou

Institute for Advanced Study 2024-10-02 | Short Talks by Postdoctoral Members

Topic: Singularities of Constructible Sheaves
Speaker: Tong Zhou
Affiliation: Institute for Advanced Study
Date: October 1, 2024

On Recent Advances in Dynamic Graph Algorithms - Yang Liu

Institute for Advanced Study 2024-10-02 | Short Talks by Postdoctoral Members

Topic: On Recent Advances in Dynamic Graph Algorithms
Speaker: Yang Liu
Affiliation: Institute for Advanced Study
Date: October 1, 2024

The Orlik-Terao Algebra and a Surprising Duality - Colin William Crowley

Institute for Advanced Study 2024-10-01 | Short Talks by Postdoctoral Members

The Orlik-Terao Algebra and a Surprising Duality
Colin William Crowley
Affiliation: Institute for Advanced Study
Date: October 1, 2024

Expanders and Property Testing - Siqi Liu

Institute for Advanced Study 2024-10-01 | Short Talks by Postdoctoral Members

Topic: Expanders and Property Testing
Speaker: Siqi Liu
Affiliation: Institute for Advanced Study
Date: October 1, 2024

Swallowed Kashiwara Crystals by Accident - Xiao (Griffin) Wang

Institute for Advanced Study 2024-10-01 | Short Talks by Postdoctoral Members

Topic: Swallowed Kashiwara Crystals by Accident
Speaker: Xiao (Griffin) Wang
Affiliation: Institute for Advanced Study
Date: October 1, 2024

A New Approach to Strong Convergence - Ramon Van Handel

Institute for Advanced Study 2024-10-01 | Computer Science/Discrete Mathematics Seminar II

Topic: A New Approach to Strong Convergence
Speaker: Ramon Van Handel
Affiliation: Institute for Advanced Study
Date: October 1, 2024

It was conjectured by Alon in the 1980s that random d-regular graphs have the largest possible spectral gap (up to negligible error) among all d-regular graphs. This conjecture was proved by Friedman in 2004 in major tour de force. In recent years, deep generalizations of Friedman's theorem, such as strong convergence of random permutation matrices due to Bordenave and Collins, have played a central role in a series of breakthrough results on random graphs, geometry, and operator algebras.

In this talk, I will discuss a surprisingly simple new approach to such results that is almost entirely based on soft arguments. This approach makes it possible to address previously inaccessible questions: for example, it enables a sharp understanding of the large deviation probabilities in Friedman's theorem, and establishes strong convergence of very high-dimensional representations of the symmetric and classical groups. I will aim to explain some of these results and the basic ideas on which they are based.

Joint work with Chen, Garza-Vargas, and Tropp.

Some Disassembly Required - Eugene Chiang

Institute for Advanced Study 2024-10-01 | Institute for Advanced Study / Princeton University Joint Astrophysics Colloquium

Topic: Some Disassembly Required
Speaker: Eugene Chiang
Affiliation: University of California, Berkeley
Date: October 01, 2024

Planetary systems are shaped as much by destructive processes --- N-body instabilities, catastrophic impacts, and atmospheric loss --- as by accretionary ones. We examine the histories of violence written in: (a) the orbital architectures of super-Earths and sub-Neptunes; (b) the scattered light morphologies of debris disks; (c) chondritic meteorites; and (d) the chaotically variable light curves of disintegrating planets.

Total Positivity and Determinantal Inequalities - Daniel Soskin

Institute for Advanced Study 2024-09-30 | Short Talks by Postdoctoral Members

Topic: Total Positivity and Determinantal Inequalities
Speaker: Daniel Soskin
Affiliation: Institute for Advanced Study
Date: September 30, 2024

Quasipolynomial Bounds for Some Problems in Extremal Combinatorics - Zander Kelley

Institute for Advanced Study 2024-09-30 | Short Talks by Postdoctoral Members

Topic: Quasipolynomial Bounds for Some Problems in Extremal Combinatorics
Speaker: Zander Kelley
Affiliation: Institute for Advanced Study
Date: September 30, 2024

Determinants of Cohomology for Number Fields - Artane Jeremie Siad

Institute for Advanced Study 2024-09-30 | Short Talks by Postdoctoral Members

Topic: Determinants of Cohomology for Number Fields
Speaker: Artane Jeremie Siad
Affiliation: Institute for Advanced Study
Date: September 30, 2024