ACT@UCR Seminar: The Pi Calculus - Christian Williams Applied Category Theory @ UCR 2019-03-29 | Historically, code represents a sequence of instructions for a single machine. Each computer is its own world, and only interacts with others by sending and receiving data through external ports. As society becomes more interconnected, this paradigm becomes more inadequate - these virtually isolated nodes tend to form networks of great bottleneck and opacity. Communication is a fundamental and integral part of computing, and needs to be incorporated in the theory of computation. To describe systems of interacting agents with dynamic interconnection, in 1980 Robin Milner invented the pi calculus: a formal language in which a term represents an open, evolving system of processes (or agents) which communicate over names (or channels). Because a computer is itself such a system, the pi calculus can be seen as a generalization of traditional computing languages; there is an embedding of lambda into pi - but there is an important change in focus: programming is less like controlling a machine and more like designing an ecosystem of autonomous organisms.We review the basics of the pi calculus, and explore a variety of examples which demonstrate this new approach to programming. We will discuss some of the history of these ideas, called “process algebra”, and see exciting modern applications in blockchain and biology. “… as we seriously address the problem of modelling mobile communicating systems we get a sense of completing a model which was previously incomplete; for we can now begin to describe what goes on outside a computer in the same terms as what goes on inside - i.e. in terms of interaction. Turning this observation inside-out, we may say that we inhabit a global computer, an informatic world which demands to be understood just as fundamentally as physicists understand the material world.” — Robin Milnerslides: http://math.ucr.edu/home/baez/mathematical/ACTUCR/Williams_Pi.pdfThis talk is part of the Applied Category Theory at U. C. Riverside Seminar: johncarlosbaez.wordpress.com/2018/12/14/applied-category-theory-seminar
ACT@UCR Seminar: Values and Inclusivity in the ACT Community - Nina Otter Applied Category Theory @ UCR 2020-06-04 | Saddened by the current events, we are taking this opportunity to pause and reflect on what we can do to change the status quo and try to bring about real and long-lasting change. Thus, we are holding a discussion aimed at finding concrete solutions to make the Applied Category Theory community more inclusive, and also to reflect about the values that our community would like to stand for and endorse, in particular, in terms of which sources of funding go against our values. While this discussion is specific to the applied category theory community, we believe that many of the topics will be of interest also to people in other fields, and thus we welcome anybody with an interest to attend. The discussion will consist of two parts: we will have first several people give short talks to discuss common issues that we need to address, as well as present specific plans for initiatives that we could take. We believe that the current pandemic, and the fact that all activities are now taking place remotely, gives us the opportunity to involve people who would otherwise find it difficult to travel, because of disabilities, financial reasons or care-taking responsibilities. Thus, now we have the opportunity to come up with new types of mentoring, collaborations, and many other initiatives that might have been difficult to envision until just a couple of months ago. The second part of the discussion will take place on the category theory community server, and its purpose is to allow for a broader participation in the discussion, and ideally during this part we will be able to flesh out in detail the specific initiatives that have been proposed in the talks. Slides and more: johncarlosbaez.wordpress.com/2020/05/27/values-and-inclusivity-in-actThis talk on 2020 June 4 is part of the Applied Category Theory at U. C. Riverside Seminar: sites.google.com/ucr.edu/actucr
ACT@UCR Seminar: The Legendre Transform: a Category Theoretic Perspective - Simon Willerton Applied Category Theory @ UCR 2020-05-27 | The Legendre-Fenchel transform is a classical piece of mathematics with many applications. In this talk I’ll show how it arises in the context of category theory using categories enriched over the extended real numbers [-∞,+∞]. It turns out that it arises out of nothing more than the pairing between a vector space and its dual in the same way that the many classical dualities (e.g. in Galois theory or algebraic geometry) arise from a relation between sets.Slides and more: johncarlosbaez.wordpress.com/2020/05/26/the-legendre-transform-a-category-theoretic-perspectiveThis talk on 2020 May 27 is part of the Applied Category Theory at U. C. Riverside Seminar: sites.google.com/ucr.edu/actucr
ACT@UCR Seminar: A Complete Axiomatisation of Partial Differentiation - Gordon Plotkin Applied Category Theory @ UCR 2020-05-20 | We formalise the well-known rules of partial differentiation in a version of equational logic with function variables and binding constructs. We prove the resulting theory is complete with respect to polynomial interpretations. The proof makes use of Severi’s theorem that all multivariate Hermite problems are solvable. We also hope to present a number of related results, such as decidability and Hilbert–Post completeness. Slides and more: johncarlosbaez.wordpress.com/2020/05/18/a-complete-axiomatisation-of-partial-differentiationThis talk on 2020 May 20 is part of the Applied Category Theory at U. C. Riverside Seminar: sites.google.com/ucr.edu/actucr
ACT@UCR Seminar: Formal Concepts vs Eigenvectors of Density Operators - Tai-Danae Bradley Applied Category Theory @ UCR 2020-05-13 | In this talk, I’ll show how any probability distribution on a product of finite sets gives rise to a pair of linear maps called density operators, whose eigenvectors capture “concepts” inherent in the original probability distribution. In some cases, the eigenvectors coincide with a simple construction from lattice theory known as a formal concept. In general, the operators recover marginal probabilities on their diagonals, and the information stored in their eigenvectors is akin to conditional probability. This is useful in an applied setting, where the eigenvectors and eigenvalues can be glued together to reconstruct joint probabilities. This naturally leads to a tensor network model of the original distribution. I’ll explain these ideas from the ground up, starting with an introduction to formal concepts. Time permitting, I’ll also share how the same ideas lead to a simple framework for modeling hierarchy in natural language. As an aside, it’s known that formal concepts arise as an enriched version of a generalization of the Isbell completion of a category. Oftentimes, the construction is motivated by drawing an analogy with elementary linear algebra. I like to think of this talk as an application of the linear algebraic side of that analogy. Slides and more: johncarlosbaez.wordpress.com/2020/05/07/formal-conceptsThis talk on 2020 May 13 is part of the Applied Category Theory at U. C. Riverside Seminar: sites.google.com/ucr.edu/actucr
ACT@UCR Seminar: Separation Logic Through a New Lens - Sarah Rovner-Frydman Applied Category Theory @ UCR 2020-05-07 | Separation logic aims to reason compositionally about the behavior of programs that manipulate shared resources. When working with separation logic, it is often necessary to manipulate information about program state in patterns of deconstruction and reconstruction. This achieves a kind of “focusing” effect which is somewhat reminiscent of using optics in a functional programming language. We make this analogy precise by showing that several interrelated techniques in the literature for managing these patterns of manipulation can be derived as instances of the general definition of profunctor optics. In doing so, we specialize the machinery of profunctor optics from categories to posets and to sets. This simplification makes most of this machinery look more familiar, and it reveals that much of it was already hiding in separation logic in plain sight. Slides and more: johncarlosbaez.wordpress.com/2020/05/05/separation-logic-through-a-new-lensThis talk on 2020 May 6 is part of the Applied Category Theory at U. C. Riverside Seminar: sites.google.com/ucr.edu/actucr
ACT@UCR Seminar: A Localic Approach to Dependency, Conflict, and Concurrency - Gershom Bazerman Applied Category Theory @ UCR 2020-04-30 | Petri nets have been of interest to applied category theory for some time. Back in the 1980s, one approach to their semantics was given by algebraic gadgets called “event structures.” We use classical techniques from order theory to study event structures without conflict restrictions (which we term “dependency structures with choice”) by their associated “traces”, which let us establish a one-to-one correspondence between DSCs and a certain class of locales. These locales have an internal logic of reachability, which can be equipped with “versioning” modalities that let us abstract away certain unnecessary detail from an underlying DSC. With this in hand we can give a general notion of what it means to “solve a dependency problem” and combinatorial results bounding the complexity of this. Time permitting, I will sketch work-in-progress which hopes to equip these locales with a notion of conflict, letting us capture the full semantics of general event structures in the form of homological data, thus providing one avenue to the topological semantics of concurrent systems. This is joint work with Raymond Puzio.Slides and more: johncarlosbaez.wordpress.com/2020/04/28/a-localic-approach-to-dependency-conflict-and-concurrencyThis talk on 2020 April 29 is part of the Applied Category Theory at U. C. Riverside Seminar: sites.google.com/ucr.edu/actucr
ACT@UCR Seminar: Star-Autonomous Envelopes - Michael Shulman Applied Category Theory @ UCR 2020-04-22 | Symmetric monoidal categories with duals, a.k.a. compact monoidal categories, have a pleasing string diagram calculus. In particular, any compact monoidal category is closed with [A,B] = (A* ⊗ B), and the transpose of A ⊗ B → C to A → [B,C] is represented by simply bending a string. Unfortunately, a closed symmetric monoidal category cannot even be embedded fully-faithfully into a compact one unless it is traced; and while string diagram calculi for closed monoidal categories have been proposed, they are more complicated, e.g. with “clasps” and “bubbles”. In this talk we obtain a string diagram calculus for closed symmetric monoidal categories that looks almost like the compact case, by fully embedding any such category in a star-autonomous one (via a functor that preserves the closed structure) and using the known string diagram calculus for star-autonomous categories. No knowledge of star-autonomous categories will be assumed. Slides and more: johncarlosbaez.wordpress.com/2020/04/21/star-autonomous-envelopesThis talk on 2020 April 22 is part of the Applied Category Theory at U. C. Riverside Seminar: sites.google.com/ucr.edu/actucr
ACT@UCR Research Meeting: Sheaves and Ringed Spaces - Joe Moeller Applied Category Theory @ UCR 2020-04-16 | A category theorist attempts to read Hartshorne. For the sake of learning the material myself, I'm giving a series of talks to explain the material in chapter 2 of Hartshorne's book Algebraic Geometry. This is the first talk in this series. I go through the definitions and some examples of the following:1. presheaves2. sheaves3. spectrum of a ring4. ringed spaces5. schemes
ACT@UCR Seminar: Open Games: the Long Road to Practical Applications - Jules Hedges Applied Category Theory @ UCR 2020-04-16 | I will talk about open games, and the closely related concepts of lenses/optics and open learners. My goal is to report on the successes and failures of an ongoing effort to try to realise the often-claimed benefits of categories and compositionality in actual practice. I will introduce what little theory is needed along the way. Here are some things I plan to talk about:— Lenses as an abstraction of the chain rule— Comb diagrams— Surprising applications of open games: Bayesian inference, value function iteration— The state of tool support— Open games in their natural habitat: microeconomics— Sociological aspects of working with economics slides: johncarlosbaez.wordpress.com/2020/04/13/open-games-the-long-road-to-practical-applicationsThis talk on 2020 April 15 is part of the Applied Category Theory at U. C. Riverside Seminar: sites.google.com/ucr.edu/actucr
ACT@UCR Research Meeting: Quantifiers as Adjoints - Christian Williams Applied Category Theory @ UCR 2020-04-08 | This is an introduction to the role of adjoints in categorical logic. The logical operations "or" and "and" are left adjoints of the diagonal map on propositions. More generally, existential and universal quantifiers can be seen as left and right adjoints to substitution. In a similar way, composition with a function between sets defines a functor between their slice categories that has both left and right adjoints. This in turn is a special case of how in any locally cartesian closed category, any morphism induces a functor between slice categories that has both left and right adjoints.slides: http://math.ucr.edu/home/baez/mathematical/ACTUCR/Williams_Quantifiers_as_Adjoints.pdfThis talk on 2020 April 8 was part of the Applied Category Theory group meetings at U. C. Riverside.
ACT@UCR Seminar: A Categorical View of Conditional Expectation - Prakash Panagaden Applied Category Theory @ UCR 2020-04-08 | This talk is a fragment from a larger work on approximating Markov processes. I will focus on a functorial definition of conditional expectation without talking about how it was used. We define categories of cones—which are abstract versions of the familiar cones in vector spaces—of measures and related categories cones of Lp functions. We will state a number of dualities and isomorphisms between these categories. Then we will define conditional expectation by exploiting these dualities: it will turn out that we can define conditional expectation with respect to certain morphisms. These generalize the standard notion of conditioning with respect to a sub-sigma algebra. Why did I use the plural? Because it turns out that there are two kinds of conditional expectation, one of which looks like a left adjoint (in the matrix sense not the categorical sense) and the other looks like a right adjoint. I will review concepts like image measure, Radon-Nikodym derivatives and the traditional definition of conditional expectation. This is joint work with Philippe Chaput, Vincent Danos and Gordon Plotkin. slides and more: johncarlosbaez.wordpress.com/2020/04/02/a-categorical-view-of-conditional-expectationThis talk on 2020 April 8 is part of the Applied Category Theory at U. C. Riverside Seminar: sites.google.com/ucr.edu/actucr
ACT@UCR Seminar: Structured Cospans and Double Categories - John Baez Applied Category Theory @ UCR 2020-04-01 | One goal of applied category theory is to better understand networks appearing throughout science and engineering. Here we introduce “structured cospans” as a way to study networks with inputs and outputs. Given a functor L: A → X, a structured cospan is a diagram in X of the form L(a) → x ← L(b). If A and X have finite colimits and L is a left adjoint, we obtain a symmetric monoidal category whose objects are those of A and whose morphisms are certain equivalence classes of structured cospans. However, this arises from a more fundamental structure: a symmetric monoidal double category where the horizontal 1-cells are structured cospans, not equivalence classes thereof. We explain the mathematics and illustrate it with an example from epidemiology. slides and more: http://math.ucr.edu/home/baez/structured/This talk on 2020 April 1 is part of the Applied Category Theory at U. C. Riverside Seminar: sites.google.com/ucr.edu/actucr
ACT@UCR Seminar: Odums Energy Systems Language - Joe Moeller Applied Category Theory @ UCR 2019-07-02 | In this talk, we give a quick overview of some of the elements of Odum's energy systems language. We then give a suggestion of a way to potentially bring these ideas into categorical network theory via network models, a categorical approach to generalized networks. slides: joemathjoe.files.wordpress.com/2019/03/odum_s_energese.pdf This talk is part of the Applied Category Theory at U. C. Riverside Seminar: johncarlosbaez.wordpress.com/2018/12/14/applied-category-theory-seminarEdited by Paola Fernandez
ACT@UCR Seminar: Category Theory for Genetics - Kenny Courser Applied Category Theory @ UCR 2019-05-06 | Rémy Tuyéras has developed a categorical framework aimed at handling various commonly studied subjects from the theory of genetics. Some of these include alignment methods, CRISPR, homologous recombination, haplotypes, and genetic linkage. In this talk, I will introduce the foundations on which this framework is built and give a few examples related to DNA and RNA sequencing which are able to be described in this environment.slides: http://math.ucr.edu/home/baez/mathematical/ACTUCR/Courser_Category_Theory_and_Genetics.pdfThis talk is part of the Applied Category Theory at U. C. Riverside Seminar: johncarlosbaez.wordpress.com/2018/12/14/applied-category-theory-seminarEdited by Paola Fernandez
ACT@UCR Seminar: Mathematics in the 21st Century - John Baez [with slides] Applied Category Theory @ UCR 2019-02-26 | [Same as the other video, but with slides embedded in the video.]Abstract: The global warming crisis is part of a bigger transformation in which humanity realizes that the Earth is a finite system and that our population, energy usage, and the like cannot continue to grow exponentially. If civilization survives this transformation, it will affect mathematics – and be affected by it – just as dramatically as the agricultural revolution or industrial revolution. We should get ready!slides: http://math.ucr.edu/home/baez/planet/planet_ucr.pdf This talk is part of the Applied Category Theory at U. C. Riverside Seminar: johncarlosbaez.wordpress.com/2018/12/14/applied-category-theory-seminarEdited by Paola Fernandez
ACT@UCR Seminar: Social Contagion Modeled On Random Networks - Daniel Cicala Applied Category Theory @ UCR 2019-02-05 | A social contagion may manifest as a cultural trend, a spreading opinion or idea or belief. In this talk, we explore a simple model of social contagion on a random network. We also look at the effect that network connectivity, edge distribution, and heterogeneity has on the diffusion of a contagion. Slides: http://math.ucr.edu/home/baez/mathematical/ACTUCR/Cicala_Social_Contagion.pdf This talk is part of the Applied Category Theory at U. C. Riverside Seminar: johncarlosbaez.wordpress.com/2018/12/14/applied-category-theory-seminarEdited by Paola Fernandez
ACT@UCR Seminar: Systems as Wiring Diagram Algebras - Christina Vasilakopoulou Applied Category Theory @ UCR 2019-01-29 | We will describe the monoidal category of labeled boxes and wiring diagrams and its induced operad. Various kinds of systems such as discrete and continuous dynamical systems have been expressed as algebras for that operad, namely lax monoidal functors into the category of categories. A major advantage of this approach is that systems can be composed to form a system of the same kind, completely determined by the specific way the composite systems are interconnected (‘wired’ together). We will elaborate on the compositional machinery by a few examples of Moore machines and their explicit composition.This talk is part of the Applied Category Theory at U. C. Riverside Seminar: johncarlosbaez.wordpress.com/2018/12/14/applied-category-theory-seminarEdited by Paola Fernandez
ACT@UCR Seminar: Problems in Symplectic Linear Algebra - Jonathan Lorand Applied Category Theory @ UCR 2019-01-20 | In this talk we will look at various examples of classification problems in symplectic linear algebra: conjugacy classes in the symplectic group and its Lie algebra, linear lagrangian relations up to conjugation, tuples of (co)isotropic subspaces. I will explain how many such problems can be encoded using the theory of symplectic poset representations, and will discuss some general results of this theory. Finally, I will recast this discussion from a broader category-theoretic perspective.Slides: http://math.ucr.edu/home/baez/mathematical/ACTUCR/Lorand_Linear_Symplectic_Classifications.pdfThis talk is part of the Applied Category Theory at U. C. Riverside Seminar: johncarlosbaez.wordpress.com/2018/12/14/applied-category-theory-seminarEdited by Paola Fernandez(Sorry for the typo on the title card, "symplectic", not "simplectic".)
ACT@UCR Seminar: Mathematics in the 21st Century - John Baez Applied Category Theory @ UCR 2019-01-13 | The global warming crisis is part of a bigger transformation in which humanity realizes that the Earth is a finite system and that our population, energy usage, and the like cannot continue to grow exponentially. If civilization survives this transformation, it will affect mathematics – and be affected by it – just as dramatically as the agricultural revolution or industrial revolution. We should get ready!slides: http://math.ucr.edu/home/baez/planet/planet_ucr.pdf This talk is part of the Applied Category Theory at U. C. Riverside Seminar: : johncarlosbaez.wordpress.com/2018/12/14/applied-category-theory-seminarEdited by Paola Fernandez
