Colloquium

Applications of physics to geometry have deep historical roots going back at least to Archimedes, while recent decades have seen structures of classical and quantum gauge theory lead to enormous advances in the theory of three and four manifolds. Meanwhile, geometry provides conceptual tools for physics as foundational  as variational principles, the kinematics of spacetime, and the topological  classification of matter. This talk will describe some possibilities for bringing number theory, more specifically, *arithmetic geometry* into this interaction. After giving some examples of the suggestive interplay of  number theory and physics, I will speculate somewhat on how ideas of quantum field theory may elucidate some of the deepest and hardest problems of number theory, especially the effective Mordell conjecture and the Hasse-Weil conjecture.

Many of the rich interactions between mathematics and physics arise using general mathematical frameworks that describe a host of physical phenomena: from differential equations, to algebra, to topology and geometry. On the other hand, mathematics also possesses many examples of "exceptional objects": they constitute the finite set of leftovers that appear in numerous classification problems. For example, groups of symmetries in three dimensions appear in two infinite families (cyclic groups and dihedral groups of n-sided polygons) and the symmetry groups of the five Platonic solids--- the 'exceptional' structures. 

The mathematical subject of moonshine refers to surprising relationships between other kinds of special/exceptional objects that arise from the theory of finite groups and from number theory. Increasingly, string theory has been a source of insights in and explanations for moonshine. It is even the source of new examples of moonshine that further implicate special objects in geometry. We will review moonshine, survey these developments, and highlight some of the (many!) exciting mysteries that remain.

The cosmic microwave background radiation has been an indispensable tool for learning about the origins and evolution of our Observable Universe. Satellites and ground based experiments measuring the temperature and polarization anisotropies with ever increasing angular resolution and sensitivity have established the standard cosmological model, LCDM, and constrained or ruled out a huge variety of theoretical models of the early Universe. Most of this cosmological information has thus far been derived from the primary CMB, anisotropies largely sourced during the epoch of recombination some 380,000 years after the Big Bang. However, experiments are achieving the sensitivity and resolution necessary to access a whole new treasure-trove of cosmological information: secondary temperature and polarization anisotropies induced by photons scattering from mass (CMB lensing) and free electrons (the Sunyaev Zel'dovich effect). Unlike the primary CMB, where cosmological information is contained in the pattern of fluctuations in statistically isotropic temperature/polarization anisotropies, the secondary CMB encodes cosmological information in the statistical anisotropies of non-gaussian temperature/polarization fields on the sky. In this talk, I will describe the possibilities for future CMB experiments, in concert with large galaxy surveys, to reconstruct the cosmological information contained in the statistics of the secondary CMB anisotropies. In particular, I will describe the information that can be reconstructed from kinetic/polarized Sunyaev Zel'dovich effect, CMB lensing, and the moving lens effect. I speculate on what we might learn from future measurements, and conclude by discussing several projects in progress.

Sexual harassment and sexual assault in the workplace is almost always a severe betrayal of trust.  I will describe research and theory that my students and I have developed over the last 25 years regarding interpersonal and institutional betrayals of trust.  My presentation will include an explanation of betrayal trauma theory and information about institutional betrayal.   I will present data from some of our research studies, including results from a study of sexual harassment of graduate students.   Included will be research-based recommendations for how to respond well to disclosures of harassment and sexual violence as well as steps individuals and institutions can take to constructively address sexual violence and promote institutional courage.  

Jennifer J. Freyd, PhD, is a Fellow at the Center for Advanced Study in the Behavioral Sciences at Stanford University and a Professor of Psychology at the University of Oregon.

From earliest infancy, we live in and learn to function in a world of causes and effects. Yet science has had an ambivalent, even hostile attitude toward causation for more than a century. Statistics courses teach us that “correlation is not causation,” yet they are strangely silent about what is causation.

A central reason for this silence is that causation does not reside in data alone, but in the process that generates the data. In order to answer causal questions, like “What would happen if we lowered the price of toothpaste?” or “Should I brake for this object?” we need a model of causes and effects. Judea Pearl has developed a simple calculus for expressing our cause-effect knowledge in a diagram and using that diagram to tell us how to interpret the data we gather from the real world. His methods are already transforming the practice of statistics and could equip future artificial intelligences with causal reasoning abilities they currently lack.

This talk is largely based on Mackenzie’s book co-written with Pearl, The Book of Why.

While it’s undeniably sexy to work with infinite-dimensional categories “model-independently,” we contend there is a categorical imperative to familiarize oneself with at least one concrete model in order to check that proposed model-independent constructions interpret correctly. With this aim in mind, we recount the n-complicial sets model of (∞,n)-categories for 0 ≤ n ≤ ∞, the combinatorics of which are quite similar to its low-dimensional special cases: quasi-categories (n=1) and Kan complexes (n=0). We conclude by reporting on an encounter with 2-complicial sets in the wild, where a suitably-defined fibration of 2-complicial sets enables the comprehension construction introduced in joint work with Verity. Special cases of the comprehension construction can be used to “straighten” a co/cartesian fibration of (∞,1)-categories into a homotopy coherent functor, exhibit a quasi-categorical version of the “unstraightening” construction, and define an internal model of the Yoneda embedding for (∞,1)-categories.

A century ago, Emmy Noether published an article, "Invariante Variationsprobleme", which contained 2 theorems on symmetries and conservation laws in the Gottingen Nachrichten. How original, how modern, how influential were her results? Did they shed light on the question of energy conservation that had arisen in the general theory of relativity? I shall comment on the curious transmission of her theorems and the later recognition of their wide applicability in very numerous fields of physics.

Most machine learning algorithms involve optimizing a single set of parameters to decrease a single cost function. In adversarial machine learning, two or more "players" each adapt their own parameters to decrease their own cost, in competition with the other players. In some adversarial machine learning algorithms, the algorithm designer contrives this competition between two machine learning models in order to produce a beneficial side effect. For example, the generative adversarial networks framework involves a contrived conflict between a generator network and a discriminator network that results in the generator learning to produce realistic data samples. In other contexts, adversarial machine learning models a real conflict, for example, between spam detectors and spammers. In general, moving machine learning from optimization and a single cost to game theory and multiple costs has led to new insights in many application areas.

The strong interaction of quarks and gluons is described theoretically within the framework of Quantum Chromodynamics (QCD). The most promising way to evaluate QCD for all energy ranges is to formulate the theory on a 4-dimensional Euclidean space-time grid, which allows for numerical simulations on state of the art supercomputers. We will review the status of lattice QCD calculations providing examples such as the hadron spectrum and the inner structure of nucleons. We will then point to problems that cannot be solved by conventional Monte Carlo simulation techniques, i.e. non-zero baryon density, the matter-antimatter asymmetry and real time simulations. It will be demonstrated at the example of the 1+1 dimensional Schwinger model that tensor network techniques are able to overcome these problems showing that with this approach --and eventual quantum simulations-- a path for solving QCD is opening up.

Energy and momentum are conserved in Newton's laws of gravitation.
Numerical integration of the equations of motion should comply to
these requirements in order to guarantee the correctness of a
solution, but this turns out to be insufficient.  The steady growth of
numerical errors and the exponential divergence, renders numerical
solutions over more than a dynamical time-scale meaningless.  Even
time reversibility is not a guarantee for finding the definitive
solution to the numerical few-body problem.  As a consequence,
numerical N-body simulations produce questionable results.  Using
brute force integrations to arbitrary numerical precision I will
demonstrate empirically that the statistics of an ensemble of resonant
3-body interactions is independent of the precision of the numerical
integration, and conclude that, although individual solutions using
common integration methods are unreliable, an ensemble of approximate
3-body solutions accurately represent the ensemble of true solutions.

I give an overview of trapped ion quantum information experiments and discuss prospects for implementing multi-valued quantum logic using trapped ions. Qudits (the multi-state generalization of qubits) are attractive for quantum computing because they enable a much larger Hilbert space for the same number of trapped ions, which may allow us to improve the information capacity of a quantum processor. I describe possible advantages and disadvantages of using qudits in place of qubits, and lay out some of the protocols that my lab will test for implementing measurements, single-qudit operations, and two-qudit operations in a trapped ion system.

In this talk I will discuss generalising the mapping from quantum error correcting codes to stat. mech. models (originally due to Dennis, Kitaev, Landahl and Preskill) to the case of correlated Pauli noise. This mapping connects the error correcting threshold to a phase transition, allowing us to use methods for studying stat. mech. models to estimate code thresholds. Using this, I will present numerical estimates of toric code thresholds under correlated noise. Time permitting, I will also discuss how this mapping can be used to give a tensor network-based algorithm for approximate optimal decoding of 2d topological codes, generalising the decoder of Bravyi, Suchara and Vargo.