Talks

I will answer the question in the title. I will also describe a new quantum algorithm for Boolean formula evaluation and an improved analysis of an existing quantum algorithm for st-connectivity. Joint work with Stacey Jeffery. 

About a decade ago Eynard and Orantin proposed a powerful computation algorithm
known as topological recursion. Starting with a ``spectral curve" and some ``initial data"
(roughly, meromorphic differentials of order one and two) the topological recursion produces by induction
a collection of symmetric meromorphic differentials on the spectral curves parametrized by
pairs of non-negative integers (g,n) (g should be thought of as a genus and n as the number of punctures).
Despite of many applications of the topological recursion (matrix integrals, WKB expansions, TFTs, etc.etc.)
the nature of the recursive relations was not understood.

Recently, in a joint work with Maxim Kontsevich we found a simple underlying structure of the recursive relations of Eynard and Orantin. We call it  ``Airy structure". In this talk I am going to define this notion and explain how the recursive relations of Eynard and Orantin follow from the  quantization of
a quadratic Lagrangian subvariety in a symplectic vector space.

Imagine going beyond treating the symptoms of disease and instead stopping it and reversing it. This is the promise of regenerative medicine.

In her Perimeter Institute public lecture, Prof. Molly Shoichet will tell three compelling stories that are relevant to cancer, blindness and stroke. In each story, the underlying innovation in chemistry, engineering, and biology will be highlighted with the opportunities that lay ahead. 

To make it personal, Shoichet’s lab has figured out how to grow cells in an environment that mimics that of the native environment. Now she has the opportunity to grow a patient’s cancer cells in the lab and figure out which drugs will be most effective for that individual. 

In blindness, the cells at the back of the eye often die. We can slow the progression of disease but we cannot stop it because there is no way to replace those cells. With a newly engineered biomaterial, Shoichet’s lab can now transplant cells to the back of the eye and achieve some functional repair. 

The holy grail of regenerative medicine is stimulation of the stem cells resident in us. The challenge is to figure out how to stimulate those cells to promote repair. Using a drug-infused “band-aid” applied directly on the brain, Shoichet’s team achieved tissue repair. 

These three stories underline the opportunity of collaborative, multi-disciplinary research. It is exciting to think what we will discover as this research continues to unfold.

 Non-Fermi liquids are exotic metallic states which do not support well defined quasiparticles. Due to strong quantum fluctuations and the presence of extensive gapless modes near the Fermi surface, it has been difficult to understand universal low energy properties of non-Fermi liquids reliably.  In this talk, we will discuss recent progress made on field theories for non-Fermi liquids. Based on a dimensional regularization scheme which tunes the number of co-dimensions of Fermi surface, critical exponents that control scaling behaviors of physical observables can be computed in controlled ways. The systematic expansion also provides important insight into strongly interacting metals. This allows us find the non-perturbative solution for the strange metal realized at the antiferromagnetic quantum critical point in 2+1 dimensions and predict the exact critical exponents.

Analyzing characteristics of an unknown quantum system in a device-independent manner, i.e., using only the measurement statistics, is a fundamental task in quantum physics and quantum information theory. For example, device-independence is a very important feature in the study of quantum cryptography where the quantum devices may not be trusted. In this talk, I will discuss the ability to characterize the state that Alice and Bob share in various physical scenarios using only the correlation data. I first give a lower bound on the dimension of the underlying Hilbert spaces required by Alice and Bob to generate a given correlation in the Bell scenario. Also, I give two properties that the Schmidt coefficients of their shared state must satisfy. I’ll provide examples showing that our results can be tight, and examine when the shared pure state is characterized completely. Lastly, I will discuss these ideas in the Prepare-and-Measure scenario. 

This is joint work with Antonios Varvitsiotis and Zhaohui Wei. 

References: 

Phys. Rev. Lett. 117, 060401, 

Phys. Rev. A, to appear. (ArXiv:1606.03878), 

ArXiv:1609.01030. 

3d N=4 theories on the sphere have interesting supersymmetric sectors described by 1d QFTs and defined as the cohomology of a certain supercharge. One can define such a 1d sector for the Higgs branch or for the Coulomb branch. We study the Higgs branch case, meaning that the 1d QFT captures exact correlation functions of the Higgs branch operators of the 3d theory. The OPE of the 1d theory gives a star-product on the Higgs branch which encodes the data of these correlation functions. When the 3d theory is superconformal, the 1d theory is topological and coincides with the known construction in flat space, where the topological 1d theory lives in the cohomology of Q+S. Our construction thus generalizes it away from the conformal point. We then focus on theories constructed from vector and hypermultiplets. Using supersymmetric localization, we explicitly describe their 1d sector as the gauged topological quantum mechanics, or equivalently a gaussian theory coupled to a matrix model. This provides a very simple technique to compute the Higgs branch correlators.

Integral values of zeta functions are important not only for what they say about other values of their respective functions, but also for what they say about transcendence degree questions for appropriate extensions of the rationals or other number fields.  They  also appear in some recent computations relevant to particle physics.
In this talk we will give a quick introduction to the theory of periods and motives, relate said theory to special values of zeta functions, and discuss a graphical definition of the associated category of motives.
Any original work discussed in this talk is joint with Susama Agarwala.

The Sachdev-Ye-Kitaev model exhibits conformal invariance and a maximal Lyapunov exponent in the large-N and low temperature limit, and thus belongs to the same universality class as a two-dimensional anti-de Sitter black hole. Poles corresponding to a tower of operators that are bilinear in the microscopic Majorana fermions can be found in the four-point function of the fermions. We propose a renormalization theory for UV perturbations to the model, and derive an effective action for the soft mode of the model that results from integrating out the lowest resonance and which is dominant in the IR. We show that a two-dimensional dilaton theory with a quadratic term precisely reproduces this effective action on its boundary.

A geometric approach to investigation of quantum entanglement is advocated.
We discuss first the geometry of the (N^2-1)--dimensional convex body
of  mixed quantum states acting  on an N--dimensional Hilbert space
and study projections of this set into 2- and 3-dimensional spaces.
For composed dimensions, N=K^2, one consideres the subset
of separable states and shows that it has a positive measure.
Analyzing its properties contributes to our understanding of
quantum entanglement and its time evolution.

We discuss recent work showing that in type A_n the category of equivariant perverse coherent sheaves on the affine Grassmannian categorifies the cluster algebra associated to the BPS quiver of pure N=2 gauge theory. Physically, this can be understood as a statement about line operators in this theory, following ideas of Gaiotto-Moore-Neitzke, Costello, and Kapustin-Saulina -- in short, coherent IC sheaves are the precise algebro-geometric counterparts of Wilson-'t Hooft line operators. The proof relies on techniques developed by Kang-Kashiwara-Kim-Oh in the setting of KLR algebras. A key moral is that the appearance of cluster structures is in large part forced by the compatibility between chiral and tensor structures on the category in question (i.e. by formal features of holomorphic-topological field theory). This is joint work with Sabin Cautis.

In this talk, I would introduce spontaneous nematicity in the background of fractional quantum Hall fluids where symmetry breaking phenomenon intertwined with topological phase of matter. The resulting nematic FQH state is characterized by an order parameter that represents these quadrupolar fluctuations, which play the role of fluctuations of the local geometry of the quantum fluid. We demonstrate that the low-energy effective theory of the nematic order parameter has z=2 dynamical scaling exponent, due to a Berry phase term of the order parameter, which is related to the nondissipative Hall viscosity. By investigating the spectrum of collective excitations, we demonstrate that the mass gap of the Girvin-MacDonald-Platzman mode collapses at the isotropic-nematic quantum phase transition. An interesting feature of the nematic phase is that it has topological defects carrying nontrivial braiding statistics and fractional charge inherited from the topological fluid nature. In addition, I would also mention the decorated nodal line condensation in pair density wave SC, where the topological phase emerges concurrently with symmetry recovery by decorated defect condensation.