Scientific Series

In this talk, intended for a broader audience, I promise to use techniques only at the level of university calculus. While staying at this level, our goal will be to learn conceptual lessons for categorification of quantum group invariants of knots and 3-manifolds, also known as the Witten-Reshetikhin-Turaev (or WRT) invariants. In particular, we will introduce new q-series invariants of 3-manifolds that have integer powers and integer coefficients and, if time permits, discuss their various constructions and properties. The talk is based on several papers with D. Pei and/or P. Putrov, C. Vafa, as well as ongoing work with C. Manolescu.

When a system interacts with an environment with which it is initially uncorrelated, its evolution is described by a completely positive map.  The common wisdom in the field of quantum information theory, however, is that when the system is initially correlated with the environment, the map describing its evolution may fail to be completely positive. This has motivated many researchers to try and characterize this putatively more general sort of dynamics, and even the textbook of Nielsen and Chuang suggests that "It is an interesting problem for further research to study quantum information processing beyond the quantum operations formalism."  This talk will demonstrate that this common wisdom is mistaken.  The error can be traced to the standard argument for how the evolution map ought to be defined in such circumstances.  One can show that anomolous dynamics would arise even in completely classical examples if one were to follow the prescription of the standard argument.  The framework of classical causal models specifies how dynamics ought to be defined in such circumstances, and

the quantum analogue of this framework provides the correct definition of the quantum evolution map, which is found to be always completely positive.

Joint work with David Schmid

Polyakov’s bootstrap programme aims at solving conformal field theories using 
unitarity and conformal symmetry. Its implementation in two dimensions has been 
highly successful and numerical studies, in particular of the 3-dimensional Ising 
model, have clearly demonstrated the potential for higher dimensional theories. 
Analytical results in higher dimensions, however, require significant insight 
into the conformal group and its representations. Surprisingly little is actually 
known about this important group theory challenge. I  will explain a remarkable 
and unexpected connection with a class of Schroedinger equations that was uncovered 
in recent joint work with M. Isachenkov. The study of the relevant quantum mechanics 
systems has created an entire branch of modern mathematics whose results can now be 
put to use in the conformal bootstrap program. 

Canonical quantization schemes often suggest modifications to classical dynamics, such as in an effective Friedmann equation. However, although often ignored, they also necessarily imply new effects for quantum space-time leading to new (quantum) symmetries. The invariant line-element, corresponding to  new geometrical structures emerging in the presence of holonomy modifications in loop quantum gravity, shall be consistently derived in this talk. We shall use black-hole models to illustrate new features of this quantum space-time, going beyond standard Riemannian manifolds.

The general purpose computer can run any program we can express in
symbolic logic; that makes it the go-to tool for accomplishing any task
that can be reduced to a computable function, and that's why software is
eating the world and cars and colliders and airplanes and pacemakers and
toasters are all just turning into computers in fancy cases.

That also means that every policy problem we can imagine will eventually
involve a computer -- and thus involve a lawmaker insisting that it must
be possible to make a computer that can run every program except for
some program that creates a problem in the world.

We don't know how to make that computer, but we can approximate it by
creating a computer with some spyware-like program running on it that
checks to see whether you're running a "naughty" program and shuts it down.

That's a spider we swallow to catch a fly. Here's the bird we swallow to
catch the spider: laws like Canada's C-11 make it a crime to
investigate, circumvent, or point out defects in systems that have these
spyware-like processes.

Companies have noticed that this offers some real upsides for them: they
can use these spyware-like processes to prevent users their customers
from using their own consumables (e.g. printer ink), parts, service
depots, apps -- anything that the company can command a high margin on,
provided that it's illegal for a customer to choose someone else's products.

What a deadly combination: companies are rapidly expanding the
constellation of devices that are locked in this way, and once a device
is locked in this way, it can't be investigated, its defects can't be
disclosed, and it can't be modified to improve it, mitigate its flaws,
and protects its owner from cybersecurity risk.

Peer review exists out of recognition that there is no way to know if
you're right until you let your enemies try to prove you wrong; in
creating a monopolistic right to control the use of products after they
were sold, Parliament also created the right for companies to enjoin the
most fundamental task of scientific knowledge creation: finding mistakes
and pointing them out.

This is urgent: our world is made of computers, those computers are
designed to treat their owners as their enemies, and security
researchers can't audit those computers without risking civil and
criminal reprisals. That's a catastrophe in the making.

In order to think about the foundations of physics it is important to understand the logical relationships among the physical principles that sustain the building. As part of these axioms of physics there is the core hypothesis that, how the Universe is partitioned into systems and subsystems is a subjective choice of the observer that should not affect the predictions of physics. Other foundational principles are the Postulates of Quantum Mechanics. However, we prove that these are not independent from the “independence of subsystem partitioning” hypothesis described above. In particular, we prove that the mathematical structure of quantum measurements and the formula for assigning outcome probabilities are implied by the mentioned hypothesis and the rest of quantum postulates. 

I will review recent progress in computing the exact planar spectrum of closed strings in
AdS3 backgrounds with 8+8 supercharges, including the derivation of the protected spectrum.
Such theories have a multi-dimensional moduli space, and I will show what effect varying
the moduli has on the exact closed string spectrum, and how the integrable structure changes
as we do so, paying particular attention to the background supported by NS-NS flux.

Ultralight bosons exist in various proposed extensions to the Standard Model, which can form condensates around rapidly rotating black holes through a process called superradiance. These boson clouds have many interesting observational consequences, such as the continuous emission of monochromatic gravitational waves.  In this talk, I will describe the dynamics of the system when it is part of a binary black hole. I will show that the presence of a binary companion greatly enriches the evolution of the boson clouds, most remarkably through the existence of resonant transitions between growing and decaying modes of the clouds. Finally, I will sketch some phenomenological consequences, both for the gravitational waves emitted by the clouds and the finite-size effects imprinted in the waveforms of the binary signal.

The cobordism hypothesis gives a functorial bijection between oriented 

n-dimensional fully local topological field theories, valued in some 

higher category C, and the fully dualizable object of C equipped with 

the structure of SO(n)-fixed point.  In this talk I'll explain recent 

works of Haugseng, Johnson-Freyd and Scheimbauer which construct a 

Morita 4-category of braided tensor categories, and I'll report on joint 

work with Brochier and Snyder which identifies two natural subcategories 

therein which are 3- and 4-dualizable.  These are the rigid braided 

tensor categories with enough compact projectives, and the braided 

fusion categories, respectively.  I'll also explain work in progress by 

us to construct SO(3)- and SO(4)-fixed point structures in each case, 

starting from ribbon and pre-modular categories, respectively.

Applying the cobordism hypothesis, we obtain 3- and 4-dimensional fully 

local TFT's, which extend the 2-dimensional TFT's we constructed with 

Ben-Zvi and Brochier, and which conjecturally relate to a number of 

constructions in the literature, including: skein modules, quantum 

A-polynomials, Crane-Kauffmann-Yetter invariants; hence our construction 

puts these on firm foundational grounds as fully local TFT's.  A key 

feature of our construction in dimension 3 is that we require the input 

braided tensor category neither to be finite, nor semi-simple, so this 

opens up new examples -- such as non-modularized quantum groups at roots 

of unity -- which were not obtainable by earlier methods.

Port-based teleportation (PBT) is a variant of the well-known task of quantum teleportation in which Alice and Bob share multiple entangled states called "ports". While in the standard teleportation protocol using a single entangled state the receiver Bob has to apply a non-trivial correction unitary, in PBT he merely has to pick up the right quantum system at a port specified by the classical message he received from Alice. PBT has applications in instantaneous non-local computation and can be used to attack position-based quantum cryptography. Since perfect PBT protocols are impossible, there is a trade-off between error and entanglement consumption (or the number of ports), which can be analyzed using representation theory of the symmetric and unitary groups. In particular, without loss of generality the resource state can be assumed to have a “purified" Schur-Weyl duality symmetry. I will give an introduction to the task of PBT and its symmetries, and show how the asymptotics of existing formulas for the optimal performance for a given number of ports can be derived using a connection between representation theory and the Gaussian unitary ensemble.

Joint work with M. Christandl, C. Majenz, G. Smith, F. Speelman & M. Walter