# MPC Seminar

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==== ''Speaker:'' '''Simon Cho''' ==== | ==== ''Speaker:'' '''Simon Cho''' ==== | ||

− | '' | + | ''Title:'' '''A categorical perspective on persistent and magnitude homology''' |

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+ | ''Abstract:'' We define and explain both persistent homology and magnitude homology, and their respective roles in applications (in both pure and applied settings). We will exhibit how both arise as a "singular complex" of a metric space in essentially the same way, but for different values of a parameter. | ||

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## Revision as of 17:36, 10 February 2020

This is the homepage of the Chapman University **Mathematics, Physics, and Computation Seminars**
(**MPC Seminar**)

*Seminar Organizers:* Roman Buniy and Peter Jipsen

## Spring 2020

The seminar talks are held in **Keck Center for Science and Engineering, KC 370** (Center St. Orange, CA 92866, intersection of Center St. And Sycamore St.), **usually on Wednesday at 4 pm**.
Sometimes there will be a change of time or venue and the announcement will reflect this change.

See [http://www.chapman.edu/about/maps-directions/index.aspx Maps and directions], Keck Center is Building 28 on the Campus Map [https://www.chapman.edu/about/_files/maps-and-directions/current-maps/campus-map.pdf Campus map]

### Thursday, March 12th, 2020, 4 - 5 pm in Keck 370, refreshments at 3:45 pm (same room)

#### *Speaker:* **Yelin Ou, University of Texas at Commerce**

*Title:* **Some study of biharmonic maps and submanifolds since 2000**

*Abstract:* Biharmonic maps are maps between Riemannian manifolds which are critical points of the bi-energy. They are solutions of a system of 4thorder PDEs and they include harmonic maps and biharmonic functions as special cases. Biharmonic submanifolds (which include minimal submanifolds as special cases) are the images of biharmonic isometric immersions. The talk will review some problems, including classification of biharmonic submanifolds in space forms, biharmonic maps into spheres, biharmonic conformal maps, and unique continuation theorems, studied in this field and their progress since 2000. The talk also presents some recent work on equivariant biharmonic maps and the stability and index of biharmonic hypersurfaces in space forms.

### Friday, Feb 21st, 2020, 3 - 4 pm in Keck 370, refreshments at 2:45 pm (same room)

#### *Speaker:* **Christian Williams**

* *

### Wednesday, Feb 19th, 2020, 4 - 5 pm in Keck 370, refreshments at 3:45 pm (same room)

#### *Speaker:* **Simon Cho**

*Title:* **A categorical perspective on persistent and magnitude homology**

*Abstract:* We define and explain both persistent homology and magnitude homology, and their respective roles in applications (in both pure and applied settings). We will exhibit how both arise as a "singular complex" of a metric space in essentially the same way, but for different values of a parameter.

### Tuesday, Jan 14th, 2020, 4 - 5 pm in Keck 370, refreshments at 3:45 pm (same room)

#### *Speaker:* **Pedram Roushan, Google Inc., Santa Barbara, CA**

*Title:* **Quantum supremacy using a programmable superconducting processor**

*Abstract:* The promise of quantum computers is that certain computational tasks might be executed exponentially faster on a quantum processor than on a classical processor1. A fundamental challenge is to build a high-fidelity processor capable of running quantum algorithms in an exponentially large computational space. Here we report the use of a processor with programmable superconducting qubits to create quantum states on 53 qubits, corresponding to a computational state-space of dimension 2^53 (about 10^16). Measurements from repeated experiments sample the resulting probability distribution, which we verify using classical simulations. Our Sycamore processor takes about 200 seconds to sample one instance of a quantum circuit a million times—our benchmarks currently indicate that the equivalent task for a state-of-the-art classical supercomputer would take approximately 10,000 years. This dramatic increase in speed compared to all known classical algorithms is an experimental realization of quantum supremacy for this specific computational task, heralding a much-anticipated computing paradigm. If time permits, I will present some of our more recent measurements.

## Fall 2019

The seminar talks are held in **Keck Center for Science and Engineering, KC 370** (Center St. Orange, CA 92866, intersection of Center St. And Sycamore St.), **usually on Wednesday at 4 pm**.
Sometimes there will be a change of time or venue and the announcement will reflect this change.

See [http://www.chapman.edu/about/maps-directions/index.aspx Maps and directions], Keck Center is Building 28 on the Campus Map [https://www.chapman.edu/about/_files/maps-and-directions/current-maps/campus-map.pdf Campus map]

### Thursday, Dec 12th, 2019, 4 - 5 pm in Keck 370, refreshments at 3:45 pm (same room)

#### *Speaker:* **Alí Guzmán Adán, Universiteit Gent, Belgium**

*Title:* **Pizzetti and Cauchy formulae for higher dimensional surfaces: a**
distributional approach

*Abstract:* In this talk, we study Pizzetti-type formulas for Stiefel
manifolds and Cauchy-type formulas for the tangential Dirac operator
from a distributional perspective. First, we illustrate a general
distributional method for integration over manifolds in R^m defined by
means of k equations. We apply this method to derive an alternative
proof of the Pizzetti formulae for the real Stiefel manifolds
SO(m)/SO(m-k). Besides, a distributional interpretation of invariant
oriented integration is provided. In particular, we obtain a
distributional Cauchy theorem for the tangential Dirac operator on an
embedded (m-k)-dimensional smooth surface.

### Wednesday, Dec 11th, 2019, 4 - 5 pm in Keck 370, refreshments at 3:45 pm (same room)

#### *Speaker:* **Arjendu K Pattanayak, Physics and Astronomy, Carleton College, Northfield, Minnesota**

*Title:* **Quantum entanglement and tunneling oscillations in a (few) many-body nonlinear spin system**

*Abstract:* Quantum tunnelling permits recoherence or refocusing in real or phase space after the initial quantum state has seemingly delocalized, exhibiting so-called classically forbidden dynamics. For a nonlinearly interacting many-body spin system this interim state can be an entangled state. We report on recent results from investigations into this phenomenon for initially spin coherent pure states in the paradigmatic kicked top (vale Haake) system. This has a mixed regualar and chaotic phase space in the classical limit and exhibits the entanglement tunneling described above, including coherent dynamics between phase-space stability islands; here the interim state is maximally entangled. We map the dependence on various parameters including nonlinear spin and number of spins involved (focusing mostly on the remarkable special case N=4) using a metric that attempts to quantify the quality and rate of tunnelling. The calculations of tunneling rates using eigenvalues and eigenstates of the time evolution operator as a function of initial condition compared to classical space structures demonstrates several nontrivial ways in which quantum behavior transitions to classical as the size of the system grows.

### Wednesday, Dec 4th, 2019, 4 - 5 pm in Keck 370, refreshments at 3:45 pm (same room)

#### *Speaker:* **Raphael Drumond, Universidade Federal de Minas Gerais, Brazil**

*Title:* **The basics of Quantum Darwinism (and its relationship with non-Markovianity)**

*Abstract:* Some aspects of physical systems described by quantum
mechanics, most notably the possibility of violating a Bell inequality,
suggests (to some) or unavoidably implies (to others) that certain
properties of microscopic systems, like the precise values of position
and momentum of a fundamental particle, do not have an objective
reality. This strongly contrasts with the objectivity of properties of
everyday macroscopic systems like the (rough estimate of) position and
momentum of a baseball. Now, how can this objectivity of the
macroscopic world emerge from a theory where, apparently, not all
aspects of physical systems are objective? The notion of quantum
Darwinism, put forward and popularized by Wojciech Zurek, is a path in
that direction. In this talk I will discuss the main ideas behind
quantum Darwinism, explore it in some simple models, and briefly discuss
its relationship (or the lack of it) with the notion of non-Markovianity
of quantum dynamical systems.

### Thursday, Nov 21th, 2019, 12:30 - 1:30 pm in Keck 370, refreshments from 12:15 (same room)

#### *Speaker:* **Sandu Popescu, University of Bristol, UK**

*Title:* **Exploring the limits of no-backward-in-time signaling**

*Abstract:* One of the most routine observations that we make about our world is that we cannot signal backwards in time. So ubiquitous is this understanding that it is often taken as one of the basic laws of Nature. At first glance, this remark seems straightforward. However, as I will show, in probabilistic theories such as quantum mechanics, the consequences of such an assertion are far more involved. In fact, we will see that there is a surprising amount of liberty: some theories even allow the future to affect the past, nevertheless without signaling backwards in time.

### Friday, Nov 8th, 2019, 3 - 4 pm in Keck 171, refreshments from 2:45 (same room)

#### *Speaker:* **Fredrik Dahlqvist, University College London, UK**

*Title:* **A probabilistic approach to floating point arithmetic**

*Abstract:* Finite-precision floating point arithmetic introduces rounding errors which are traditionally bounded using a worst-case analysis. However, worst-case analysis might be overly conservative because worst-case errors can be extremely rare events in practice. Here we develop a probabilistic model of rounding errors with which it becomes possible to quantify the likelihood that the rounding error of an algorithm lies within a given interval.

Given an input distribution, the model requires the distribution of rounding errors. We show how to exactly compute this distribution for low precision arithmetic. For high precision arithmetic we derive a simple but surprisingly useful approximation. The model is then entirely compositional: given a numerical program written in a simple imperative programing language we can recursively compute the distribution of rounding errors at each step and propagate it through each program instruction. This is done by applying a formalism originaly developed by Kozen to understand the semantics of probabilistic programs, for example how probability distributions gets transformed by assignments or "if then else" statements.

### Wednesday, Nov 6th, 2019, 4 - 5 pm in Keck 370, refreshments from 3:45 (same room)

#### *Speaker:* **Jacques Pienaar, International Institute of Physics in Natal, Brazil**

*Title:* **The new question in quantum foundations: what is causality?**

*Abstract:* Work in quantum foundations has tended to concentrate on Bell's Theorem,
but recently a particular aspect of that theorem has taken on a life of
its own: the question of what "causality" means in quantum mechanics,
and how to model it. This movement has been driven in large part by
recent advances in causal modeling in the statistical and Artificial
Intelligence communities, which have had a heavy influence on quantum
causal modeling. These "classical" approaches tend to define causality
as a probabilistic and action-centered concept, which fits well with
modern information-theoretic treatments of quantum mechanics. However,
the classical approaches also tend to emphasize notions of underlying
determinism and objective mechanisms that do not sit so well in the
quantum context. In this talk I will step back from most of the
technical jargon and try to get to the heart of some of the conceptual
issues involved in "quantizing" the concept of causality, critically
reviewing some key findings, pointing out potential inconsistencies and
outlining possible directions for further inquiry into this fascinating
problem.

### Friday, October 18th, 2019, 12 - 1 pm in Keck 370, refreshments from 11:45 (same room)

#### *Speaker:* **Prof. Lev Vaidman, Alex Maguy-Glass Chair in Physics of Complex Systems, Tel Aviv University, Israel**

*Title:* **The past of a quantum particle**

*Abstract:* Textbooks of quantum mechanics lack the concept of the past of quantum systems. Few years ago I proposed to define the past of a quantum particle according to the trace it leaves. While in many cases this definition provides a reasonable description, for a nested Mach-Zehnder interferometer it leads to a picture seemingly contradicting common sense: the particle leaves a trace in a place through which it could not pass. I will discuss recent theoretical and experimental studies of this controversial issue.

### Friday, October 11th, 2019, 1 - 2 pm in Keck 370, refreshments from 12:45 (same room)

#### *Speaker:* **Prof. Thomas Curtright, University of Miami**

*Title:* **Massive Dual Gravity Revisited**

*Abstract:* I will describe a highly speculative model of gravity as a massive,
pure spin 2 field, which is "dual" to the usual description in terms of
a spacetime metric tensor.

In the dual description, for weak fields, the metric emerges as the field strength of an underlying fundamental field. More generally, if the gravitational field is not weak, the metric emerges as a nonlinear mixture involving the energy momentum tensor.

### Wednesday, October 9th, 2019, 4 - 5 pm in Keck 370, refreshments from 3:45 (same room)

#### *Speaker:* **Alain Hénaut, Institut de Mathématiques de Bordeaux, Université de Bordeaux, France**

*Title:* **On planar web geometry**

*Abstract:* Web geometry deals with foliations in general position. In the planar case and the complex setting, a $d$-web is given by the generic family of integral curves of an analytic or an algebraic differential equation $F(x,y,y')=0$ with $y'$-degree $d$. Invariants of these configurations as abelian relations (related to Abel's addition theorem), Lie symmetries or Godbillon-Vey sequences are investigated. This viewpoint enlarges the qualitative study of differential equations and their moduli. In the nonsingular case and through the singularities, Cartan-Spencer and meromorphic connections methods will be used. Basic examples will be given from different domains including classic algebraic geometry and WDVV-equations. Standard results and open problems will be mentioned. Illustration of the interplay between differential and algebraic geometry, new results will be presented.

### Thursday, September 5th, 2019, 4 - 5 pm in Keck 171, refreshments from 3:45 (same room)

#### *Speaker:* **Nicole Yunger Halpern, Harvard-Smithsonian ITAMP (Institute for Theoretical Atomic, Molecular, and Optical Physics) Harvard University Department of Physics**

*Title:* **Entropic uncertainty relations for quantum-information scrambling**

*Abstract:* How violently do two quantum operators disagree? Different subfields of physics feature different notions of incompatibility: (i) In quantum information theory, uncertainty relations are cast in terms of entropies. These entropic uncertainty relations constrain measurement outcomes. (ii) Condensed matter and high-energy physics feature interacting quantum many-body systems, such as spin chains. A local perturbation, such as a Pauli operator on one side of a chain, spreads through many-body entanglement. The perturbation comes to overlap, and to disagree, with probes localized on the opposite side of the system. This disagreement signals that quantum information about the perturbation has scrambled, or become hidden in highly nonlocal correlations. I will unite these two notions of quantum operator disagreement, presenting an entropic uncertainty relation for quantum-information scrambling. The entropies are of distributions over weak and strong measurements’ possible outcomes. The uncertainty bound strengthens when a spin chain scrambles in numerical simulations. Hence the subfields—quantum information, condensed matter, and high-energy physics—can agree about when quantum operations disagree. Our relation can be tested experimentally with superconducting qubits, trapped ions, and quantum dots.

NYH, Bartolotta, and Pollack, Comms. Phys. 2, 92 (2019). https://www.nature.com/articles/s42005-019-0179-8

### Monday, August 26, 2019, 7 - 8:30 pm, in Argyros Forum, Room 209 A&B, networking from 6:15 to 7pm (same room)

#### *Speaker:* **Frederick Eberhardt, Professor of Philosophy in the Division of the Humanities and Social Sciences at the California Institute of Technology**

*Title:* **Computing Causal Relations at Scale or Causality: From Aristotle through Computing to Zebrafish**

*Abstract:* What causes what? How do we untangle the “why” behind processes that regulate the brain, the climate or the economy? If “Correlation does not imply causation" is the standard mantra in science, how can we ever discover causal relationships behind the data? Will it ever be possible for intelligent AI to make its own deductions and predictions? In recent years researchers have developed mathematical techniques that give us the power to infer the underlying “why” behind scientific data. What’s more, we’ve learned that we can discover these causes without performing experiments. Starting with a little practical example with lightbulbs that can be worked out by hand we will see how the problem scales as the number of variables increases. To compute the neural connections in a zebrafish brain, high performance computing is essential.

*Bio:* Frederick Eberhardt is Professor of Philosophy in the Division of the Humanities and Social Sciences at the California Institute of Technology. Before coming to Caltech he was Assistant Professor in the Philosophy-Neuroscience-Psychology (PNP) program and the Department of Philosophy at Washington University in St. Louis and a postdoc at the Institute of Cognitive and Brain Sciences at the University of California, Berkeley. As an undergraduate he attended the London School of Economics for a Bachelor in Philosophy & Mathematics. He received his PhD in philosophy from Carnegie Mellon University, where he also completed a Masters in Machine Learning.

His research interests lie at the intersection of philosophy of science, machine learning and statistics. He is particularly interested in the development of methods for causal discovery from statistical data.

## Spring 2019

The seminar talks are held in **Keck Center for Science and Engineering, KC 171** (Center St. Orange, CA 92866, intersection of Center St. And Sycamore St.), **usually on Wednesday at 4 pm**.
Sometimes there will be a change of time or venue and the announcement will reflect this change.

See [http://www.chapman.edu/about/maps-directions/index.aspx Maps and directions], Keck Center is Building 28 on the Campus Map [https://www.chapman.edu/about/_files/maps-and-directions/current-maps/campus-map.pdf Campus map]

### Thursday, May 16, 2019 at 3 pm, in Keck 370, tea and cookies at 2:40pm (same room)

#### *Speaker:* **Dr. Purbita Jana, The Institute of Mathematical Sciences, Chennai, India**

*Title:* **Intuitionistic topological systems and their connections with Heyting algebra and Gödel algebra**

*Abstract:* (joint work with Antonio Di Nola and Revaz Grigolia) This talk will suggest a new approach of representation of a Heyting algebra as an I-topological system. I-topological systems will be introduced following the notion of topological system introduced by S. Vickers which is a triple (X, |=, A) consisting of a non-empty set X, a frame A and a relation between X and A satisfying logic of finite observations or geometric logic. It is well known that frame is the Lindenbaum algebra of geometric logic where as the Lindenbaum algebra of intuitionistic logic is a Heyting algebra. Hence to define I-topological systems, intuitionistic logic plays a crucial role. Moreover, we will focus on the categorical relationships between the I-topological system, Esakia space and Heyting algebra (and its particular case of Gödel algebra).

### Tuesday, April 16, 2019 at 5 pm, in Keck 370, tea and cookies at 3:40pm (same room)

#### *Speaker:* **Dr. Michele Piazzai, University of Amsterdam, Netherlands**

*Title:* **The Ecology of Ideas: Computational Evidence of Self-Organization**

*Abstract:* Concepts are cognitive structures that people use to categorize objects and organize their knowledge of the world. Common examples include food and furniture, but also rock'n'roll, quantum physics, and blockchain. For individuals, concepts are the main tools of learning and inference; for groups of individuals (i.e., social systems), they are indispensable instruments of coordination and exchange. But how can the idiosyncratic knowledge of individuals in a social system converge on shared concepts, so as to allow coordination? Conventional wisdom in social science suggests that designated mediators, such as music critics, facilitate this convergence by endorsing the learning and usage of particular concepts. By contrast, I will argue that rational decisions embedded in everyday social interaction provide the necessary and sufficient conditions for concepts' convergence. No mediators are needed for the agents to develop a shared conceptual structure: as commonly occurs in complex adaptive systems, design emerges in the absence of centralized designers. Simulation results are presented that illustrate this point.

### Thursday, April 11, 2019 at 4 pm, in Keck 370, tea and cookies at 3:40pm (same room)

#### *Speaker:* **Dr. Michele Piazzai, University of Amsterdam, Netherlands**

*Title:* **The Ecology of Ideas: Empirical Evidence of Evolutionary Pressure**

*Abstract:* Building on the notion that the collective cognition of agents engaged in social interaction constitutes a complex adaptive system, I present a theoretical framework to describe agents' shared conceptual structure as a self-replicating network of objects, ideas, and decisions. Finite cognitive resources at the agents' disposal create a competitive environment where concepts vie to be assigned to objects in categorization decisions, so as to survive in the agents' memory. This struggle for survival gives the socio-cognitive system traits characteristic of an ecology. I identify two sources of ecological interdependence among concepts, including horizontal relations between concepts located at the same level of abstraction and vertical relations between concepts located at different levels. An empirical model is developed to test whether these relations affect the selection of concepts in agents' decisions and the evolution of their shared conceptual structure. To test this model, I analyze one decade's worth of categorization decisions made by users of a crowd-sourced music encyclopedia whose objective is sorting records into genres and sub-genres.

### Thursday, March 28, 2019 at 5 pm, in Keck 171, tea and cookies at 4:40pm (same room)

#### *Speaker:* **Prof. Giovanni Sambin, University of Padova, Italy**

*Title:* **Half way to Grothendieck's aim (embedding of pointwise topology into pointfree topology)**

*Abstract:* For a mathematician, the most interesting motivation for a different foundation is when it offers a new perspective or even a solution to an old problem. Developing topology over a very "weak" foundation offers a precise mathematical expression and proof to the expectation, which is well present since the time of Grothendieck, of a notion of space generalising that of topological space.
The well-known link between pointfree and standard topology (with points) is a categorical adjunction between topological spaces Top and locales Loc. Using a "weak" foundation to develop topology, one is compelled to preserve more information than usual, and thus replace topological spaces with "concrete spaces", and locales with "positive topologies". Contrary to a common expectation, it is precisely this "useless" information what offers a solution to Grothendieck's aim. That is, the above adjunction becomes a categorical embedding of concrete spaces into positive topologies.
This embedding does not look possible without exploiting the presence of the new (intrinsically existential) notion of positivity relation.
Half way because this leaves the problem open of finding a suitable notion of positivity (formal closed subsets) in the framework of arbitrary Grothendieck topologies.

### Wednesday, March 13, 2019 at 4 pm, in Keck 171, tea and cookies at 3:45pm in Keck 370

#### *Speaker:* **Dr. Matt Pusey, University of Oxford**

*Title:* **Anomalous weak values and contextuality: robustness and imaginary**
parts

*Abstract:* I will discuss extensions to my previous work linking anomalous weak
values with contextuality. In particular, I will show that using
transformation noncontextuality obviates the experimentally
problematic requirement that the post-selection is projective. I will
also discuss the status of the imaginary part of weak values. My talk
will be based on joint work with Ravi Kunjwal and Matteo Lostaglio:
https://arxiv.org/abs/1812.06940

### Wednesday, February 27, 2019 at 4 pm, in Keck 171, tea and cookies at 3:45pm in Keck 370

#### *Speaker:* **Prof. Christos Tzounis, California State Polytechnic University, Pomona**

*Title:* **Metric of an Evaporating Black Hole**

*Abstract:* We present an approximate time-dependent metric in ingoing Eddington-Finkelstein coordinates for an evaporating nonrotating black hole as a first-order perturbation of the Schwarzschild metric, using the linearized back reaction from a realistic approximation to the stress-energy tensor for the Hawking radiation in the Unruh quantum state.

### Wednesday, February 13, 2019 at 4 pm, in Keck 171, tea and cookies at 3:45pm in Keck 370

#### *Speaker:* **Prof. Peter Rakitzis, Institute of Electronic Structure and Laser, Heraklion, Greece**

*Title:* **Ultrahigh-Density Spin-Polarized H and D Atoms Observed via Magnetization Quantum Beats**

*Abstract:* We measure nuclear and electron spin-polarized H and D densities of at least $10^{20}$ cm$^{−3}$ and $10^{19}$ cm$^{−3}$, i.e., at pressures of 5 bar and 0.5 bar, respectively, with ∼10-50 ns lifetimes, from the photodissociation of HBr and DI with circularly polarized UV light pulses. We observe the hyperfine quantum beating of the H and D magnetization with a pickup coil, i.e., the respective 0.7 and 3 ns periodic transfer of polarization from the electrons to the nuclei and back. These pulsed densities are ∼7 orders of magnitude higher than that produced by conventional continuous-production methods, and are sufficient for three novel applications: (a) laser-driven ion acceleration of spin-polarized electrons, protons, or deuterons, (b) the preparation of nuclear-spin-polarized molecules, and (c) the demonstration of spin-polarized D-T or D$^{-3}$He laser fusion at large laser facilities such as NIF, for which a reactivity enhancement of ∼50% is expected.

### Tuesday, February 5 -- Saturday, February 9, 2019 in Keck 370

#### **11th Annual Chapman University CECAT Workshop on Pointfree Mathematics**

TUESDAY, February 5th:

1:00 - 2:00pm **M. Andrew Moshier**, *Chapman University*, Weakening relations

THURSDAY, February 7th:

10:30 - 11:30am: **Peter Jipsen**, *Chapman University*, Involutive residuated lattices and relation algebras

11:30 - 12:30pm: **Sara Vannucci**, *University of Salerno*, Semiring and Semimodule Issues in Residuated Lattices

1:00 - 2:15pm: **Rick Ball**, *University of Denver*, Pointfree Integration

2:15 - 3:15pm: **Anna Laura Suarez**, *University of Birmingham*, D-frames and frame coproducts

3:30 - 4.30pm: **Alex Kurz**, *Chapman University*, On the duality theory of weakening relations

FRIDAY, February 8th

10:30 - 11:30am: **Ales Pultr**, *Charles University*, Strong Hausdorff properties of frames

11:30 - 12:30pm: **Diego Valota**, *University of Milan*, Computing Spectra via Dualities in the MTL hierarchy

2:00 - 3:00pm: **Olim Tuyt**, *University of Bern*, Algebraic finite model property of a modal Gödel logic

3:00 - 4:15pm: **Rick Ball**, *University of Denver*, Pointfree Integration, continued

SATURDAY, February 9th

1:00 - 5:00pm: Discussion session