Talks

Self-gravitating quantum matter may exist in a wide range of cosmological and astrophysical settings: from the very early universe through to present-day boson stars. Such quantum matter arises in UltraLight Dark Matter (ULDM): an exciting axion-like particle candidate which keeps the successes of CDM on large scales but alleviates tensions on small scales. This small scale behavior is due to characteristic cores in ULDM called solitons, which also correspond to the ground state of the self-gravitating quantum system governing ULDM. We calculate the full spectrum of eigenstates and decompose simulations of ULDM into these states, allowing us to precisely track the evolution of the tell-tale soliton cores and the surrounding halo “skirt”. Using this formalism, we investigate formation of halos through binary soliton collisions and the dependence of the final halo product on initial parameters. We further link characteristic ULDM halo behavior—such as the soliton “breathing mode” and random walk of the center of mass—to the presence of certain modes. Finally, we comment on the relationship between eigenenergies and oscillatory timescales present in the system, as well as future directions for understanding ULDM through the language of its eigenstates.

 

Entropic Dynamics (ED) is a framework in which Quantum Mechanics is derived as an application of entropic methods of inference. In ED the dynamics of the probability distribution is driven by entropy subject to constraints that are codified into a quantity later identified as the phase of the wave function. The challenge is to specify how those constraints are themselves updated.

The important ingredients are two: the cotangent bundle associated to the probability simplex inherits (1) a natural symplectic structure from ED, and (2) a natural metric structure from information geometry.

The requirement that the dynamics preserves both the symplectic structure (a Hamilton flow) and the metric structure (a Killing flow) leads to a Hamiltonian dynamics of probabilities in which the linearity of the Schrödinger equation, the emergence of a complex structure, Hilbert spaces, and the Born rule, are derived rather than postulated.

 

 I will report on my progress, joint with David Reutter, to construct and analyze the algebraic closure of nVec – in other words, the universal n-category of framed nD TQFTs. The invertibles are Pontryagin dual to the stable homotopy groups of spheres. The Galois group is almost, but not quite, the stable PL group. An invertible TQFT can be condensed from the vacuum if and only if it trivializes on (possibly-exceptional) spheres.
 

 

Symmetries play a unifying role in physics and many other sciences. In deep learning, symmetries have been incorporated into neural networks through the concept of equivariance. One of the major benefits is that it will reduce the number parameters through parameter sharing and as such can learn with less data. In this talk I will ask the question, can equivariance also help in RL? Besides the obvious idea of using equivariant value functions, we explore the idea of deep equivariant policies. We make a connection between equivariance and MDP homomorphisms, and generalize to distributed multi-agent settings.   Joint work with Elise van der Pol (main contributor), Herke van Hoof and Frans Oliehoek. 

We report on the geometric structure of symmetry adapted PSD cones and symmetry adapted Gram spectrahedra of symmetric polynomials. We determine the dimension of symmetry adapted PSD cones, describe its extreme rays, and discuss the structure of its matrix representations. We also focus on symmetry adapted Gram spectrahedra of symmetric binary forms, quadrics, ternary quartics and sextics. In particular, we characterize extreme points of these spectrahedra for symmetric binary forms that are of rank two, and we report what we know about the facial structure of the same spectrahedra. The talk will be based on two collaborations, one with Alex Heaton and Isabelle Shankar, and another one with Matthew Heid.

For quantum gravity states associated to open spin network graphs, we study how the boundary (the set of open edges, which carries spin degrees of freedom) is affected by the bulk, specifically by its combinatorial structure and by the quantum correlations among the intertwiners. In particular, we determine under which conditions certain classes of quantum gravity states map bulk degrees of freedom into boundary ones isometrically (which is a necessary condition for holography). We then look at the entanglement entropy of the boundary and recover, for slightly entangled intertwiners, the Ryu-Takayanagi formula with corrections induced by the entanglement entropy of the bulk state. We also show that the presence of a region with highly entangled intertwiners deforms the minimal-area surface, which is prevented from entering that region when the entanglement entropy of the latter exceeds a certain bound, a mechanism which thus leads to the rise of a black hole-like region in the bulk.

Zoom Link: https://pitp.zoom.us/j/96356007543?pwd=U2VrRlhyOThMODdMYllDMnB6VjlZQT09

Tensor network states form a variational ansatz class widely used in the study of quantum many-body systems. Geometrically, these states form an algebraic variety of tensors with rich representation theoretic structure. It is known that tensors on the "boundary" of this variety can provide more efficient representations for states of physical interest, but the pathological geometric properties of the boundary make it difficult to extend the classical optimization methods. In recent work, we introduced a new ansatz class which includes states at the boundary of the tensor network variety. I will present some of the geometric features of this class and explain how it can be used in the variational approach. This is based on joint work with M. Christandl, D. Stilck-Franca and A. Werner.

The era of multi-messenger astronomy is well and truly upon us, with 90 compact binaries observed since the Advanced LIGO detectors saw first light in 2015. Despite our very own cosmic backyard, the Milky Way, being ripe with prospective sources for ground-based gravitational wave detectors, the closest source detected thus far (GW170817, the famed binary neutron star merger) was at a distance of 40 Mpc. In this talk, I will outline a number of prospective Galactic multi-messenger sources, and discuss several ways in which their detection over the next twenty years can be improved through both experimental and analytical techniques. 

Zoom Link: https://pitp.zoom.us/j/93664322641?pwd=VUoxbmFGZE1KczJ1RU1MR09TQ05Ldz09

Geometrically, a linear program is a polyhedron together with an orientation of its graph. A simplex method determines a path from any given starting vertex to the sink. The centerpiece of any simplex algorithm is the pivot rule that successively selects outgoing edges along the path. We introduce normalized-weight pivot rules, a class of pivot rules that are memory-less, that subsume many of the most-used pivot rules, and that can be parametrized in a natural continuous manner. We introduce two polytopes that capture the behavior of normalized-pivot rules on polytopes and linear programs. On the one hand, this gives a new perspective on the performance of pivot rules. On the other hand, our constructions generalize classical constructions (e.g. monotone path polytopes) and objects (e.g. permutahedra, associahedra, multiplihedra) from geometric combinatorics, This is joint work with Alex Black, Jesus De Loera, and Niklas Lütjeharms.

Conformal field theories (CFTs) are ubiquitous in theoretical physics as fixed points of renormalization, descriptions of critical systems and more. In these theories the conformal symmetry is a powerful tool in the computation of correlation functions, especially in 2 dimensions where the conformal algebra is infinite. Discretization of field theories is another powerful tool, where the theory on the lattice is both mathematically well-defined and easy to put on a computer. In this talk I will outline how these are combined using a discrete version of the 2d conformal algebra that acts in lattice models. I will also discuss recent work on convergence of this discretization, as well as on applications to non-unitary CFTs that appear in descriptions of problems of interest in condensed matter physics such as polymers, percolation and disordered systems.

Zoom Link: https://pitp.zoom.us/j/95048143778?pwd=N1hhVHlsZThVYzBWTy9CNlBTUHIydz09

I plan to survey some parts of Strassen's seminal body of work on the theory of asymptotic spectra, in which optimization and symmetry play crucial roles. In particular, I will motivate and explain the notion of asymptotic spectra, Strassen's duality theorem (far reaching generalization of linear programming duality, with connections to Positivstellensatz), and Strassen's connectivity theorem of the spectra of matrix multiplication.

The Kronecker coefficients of the symmetric group are the multiplicities of irreducible representations in the decomposition of the tensor product of two other irreducible representations. Ever since their definition by Murnaghan more than 80 years ago they've presented a major mystery and open problem in Algebraic Combinatorics. Recently they have come to play a crucial role in Geometric Complexity Theory in the quest for separation of VP and VNP. In this talk I'll give a broad overview of this topic with respect to combinatorics, asymptotics and computational complexity.