Mathematics Department,
Instituto
Superior
Técnico
Center for Mathematical Analysis, Geometry, and Dynamical Systems |
Postal address: Departamento de Matemática Instituto Superior Técnico Av. Rovisco Pais 1049-001 Lisboa, Portugal |
Tel: +351-218417131 Fax: +351-218417035 / 598 E-mail: pmr (at) math.tecnico.ulisboa.ptWeb: http://www.math.tecnico.ulisboa.pt/~pmr/ Office: 4.28 |

Selected publications and preprints

- Effective equivalence relations and principal quantales, with Juan Pablo Quijano, Semigroup Forum (to appear), DOI 10.1007/s00233-019-09994-z; arXiv:1807.08979.
- Functoriality of groupoid quantales II, with Juan Pablo Quijano (2018); arXiv:1803.01075.
- The many groupoids of a stably Gelfand quantale, J. Algebra 498 (2018) 197-210, DOI 10.1016/j.jalgebra.2017.11.042; arXiv:1706.06545.
- Open maps of involutive quantales, Appl. Categ. Structures 26(4) (2018) 631-644, DOI 10.1007/s10485-017-9506-y; arXiv:1706.04909.
- Quantales and Fell bundles, Adv. Math. 325 (2018) 312-374, DOI 10.1016/j.aim.2017.12.001; arXiv:1701.08653.
- Linear structures on locales, with João Paulo Santos, Theory Appl. Categ. 31 (2016) 502-541.
- Open quotients of trivial vector bundles, with João Paulo Santos, Topol. Appl. 224C (2017) 19-47, DOI 10.1016/j.topol.2017.04.001; arXiv:1510.06329.
- Invariant means on Boolean inverse monoids, with Ganna Kudryavtseva, Mark V. Lawson, Daniel H. Lenz, Semigroup Forum 92 (2016) 77-101, DOI 10.1007/s00233-015-9768-3; arXiv:1503.03733.
- Functoriality of groupoid quantales I, J. Pure Appl. Algebra 219 (2015) 3089-3109, DOI 10.1016/j.jpaa.2014.10.004; arXiv:1401.8116.
- Quantales of open groupoids, with M.C. Protin, J. Noncommutative Geom. 6 (2012) 199-247, DOI 10.4171/JNCG/90; arXiv:0811.4539v2.
- Groupoid sheaves as quantale sheaves, J. Pure Appl. Algebra 216 (2012) 41-70, DOI 10.1016/j.jpaa.2011.05.002; arXiv: 0807.4848v3.
- Étale groupoids as germ groupoids and their base extensions, with D. Matsnev, Proc. Edinb. Math. Soc. 53 (2010) 765-785; arXiv: 0807.3916.
- Sheaves as modules, with E. Rodrigues, Appl. Categ. Structures 18 (2010) 199-217; arXiv:0711.4401.
- An algebraic generalization of Kripke structures, with S.
Marcelino,
*Math. Proc. Cambridge Philos. Soc.*145 (2008) 549-577; arXiv:0704.1886. - Étale groupoids and their quantales, Adv. Math. 208 (2007) 147-209, DOI 10.1016/j.aim.2006.02.004; arXiv:math/0412478.
- A note on infinitely distributive inverse semigroups, Semigroup Forum 73 (2006) 156-158, DOI 10.1007/s00233-005-0547-4; arXiv:math/0506454.

- Lectures on étale groupoids, inverse semigroups and quantales, Lecture Notes for the GAMAP IP Meeting, Antwerp, 4-18 September, 2006, 115 pp.
- A noncommutative theory of Penrose tilings, with C.J. Mulvey, Internat. J. Theoret. Phys. 44 (2005) 655-689.
- On quantales that classify C*-algebras, with D. Kruml, Cah. Topol. Géom. Différ. Catég. 45 (2004) 287-296.
- Sup-lattice 2-forms and quantales, J. Algebra 276 (2004) 143-167.
- On quantales and spectra of C*-algebras, with D. Kruml, J.W. Pelletier, J. Rosický, Appl. Categ. Structures 11 (2003) 543-560.
- Localic sup-lattices and tropological systems, with S.J. Vickers, Theoret. Comput. Sci. 305 (2003) 311-346.
- Tropological systems are points of quantales, J. Pure Appl. Algebra 173 (2002) 87-120.
- Quantales, finite observations and strong bisimulation, Theoret. Comput. Sci. 254 (2001) 95-149.
- Quantales and observational semantics, In B. Coecke, D. Moore, A. Wilce, editors, Current Research in Operational Quantum Logic: Algebras, Categories and Languages, 263-288, Fund. Theories Phys. 111, Kluwer Acad. Publ., 2000.
- Modular specification of concurrent systems with observational logic, In J.L. Fiadeiro, editor, Recent Trends in Algebraic Development Techniques (Lisbon 1998), 310-325, Lecture Notes in Comput. Sci. 1589, Springer, Berlin, 1999.

Selected talks and short courses

- Sketches of a mathematical theory of qualia, Workshop on Models of Consciousness, Univ. Oxford, September 9-12, 2019.
- Generalized C*-diagonals via maps of quantales, Workshop on Analysis and Synthesis for Operator Algebras, IWOTA 2019, Instituto Superior Técnico, Lisboa, Portugal, July 22-26, 2019.
- A mathematical model of qualia, poster at 26th International Interdisciplinary Conference on The Science of Consciousness, Interlaken, Switzerland, June 25-28, 2019.
- A quantale model of cognition, Workshop on Algebra, Logic and Topology, Univ. Coimbra, September 27-29, 2018.
- Quanta and qualia, Workshop on Combining Viewpoints in Quantum Theory, ICMS, Edinburgh, March 19-22, 2018.
- Quantales and Fell bundles, Workshop on Applications of Operator Algebras: order, disorder and symmetry, ICMS, Edinburgh, June 26-30, 2017.
- Groupoid quantales, 37th Linz Seminar on Fuzzy Set Theory - Enriched Category Theory and Related Topics, Bildungszentrum St. Magdalena, Linz, Austria, February 7-10, 2017.
- Linear structures on locales, Workshop on Dualities 2016, Univ. Coimbra, September 19-21, 2016.
- Stably Gelfand quantales and C*-algebras, Fields Institute Workshop on New Directions in Inverse Semigroups, Univ. Ottawa, Canada, 1-4 June, 2016.
- Étale groupoids and their quantales: functoriality, Workshop on Categorical Methods in Algebra and Topology, Univ. Coimbra, January 24-26, 2014.
- Inverse semigroups and groupoids via quantales, Edinburgh Workshop on Semigroup Representations 2013, Univ. Herriot-Watt, 10-12 April 2013.
- Quantales of open groupoids, Fields Institute Workshop on Semigroups and Categories, Univ. Ottawa, Canada, 2-4 Mat 2010.
- Stably Gelfand quantales, groupoids and Cartan sub-C*-algebras, Fifth Workshop on Categories, Logic and Foundations of Physics, Imperial College, London, August 6, 2009.
- Sheaves on groupoids and quantales (2 x 45 mins.), Workshop on Sheaves in Geometry and Quantum Theory, Nijmegen, Netherlands, 3-5 September 2008.
- Noncommutative geometry and Bohr's doctrine of classical concepts, Applications of Noncommutative Geometry, Minisymposium of 5th European Congress of Mathematics (5ECM), Amsterdam, Netherlands, 17 July 2008.
- Quantales and toposes, International category Theory Conference, Calais, France, 22-28 June 2008.
- Quantale-valued sets, quantale modules, and groupoid actions, International Category Thery Conference, Carvoeiro, Portugal, 17-23 June 2007.<\li>
- Quantales, groupoids, and C*-algebras (poster), Workshop on Trends in Noncommutative Geometry, Newton Institute, Cambridge, UK, 18-22 December 2006.
- Étale groupoids, inverse semigroups and quantales, nine hour course at GAMAP: Geometric and Algebraic Methods oh Physics and Applications, Antwerp, 4-198 September 2006.
- Applications of quantale theory to groupoids and inverse semigroups, Canadian Math. Soc. Summer 2006 Meeting (Category Theory session), Calgary, Canada, 3-5 June 2006.
- From groupoids and inverse semigroups to supported quantales, Categories and Semigroups Workshop, Univ. Calgary, Canada, June 2, 2006.
- Quantales, groupoids, and C*-algebras, Operator Algebra Workshop, Queen's Univ., Belfast, UK, 5-6 May 2006.
- Quantale theory, six hour course at GAMAP: Geometric and Algebraic Methods oh Physics and Applications, Antwerp, 5-15 September 2005.
- Localic germ groupoids of inverse semigroups, Workshop on Categorical Methods in Algebra, Geometry and Mathematical Physics, Australian National University, Canberra, 18-21 July 2005.
- Localic germ groupoids of inverse semigroups, 81st Peripatetic Seminar on Sheaves and Logic (PSSL81), Univ. Coimbra, 9-10 April 2005.
- Supported quantales: an interface between modal logic and geometry, Workshop on Logic from Quantales/OASIS seminar, Univ. Oxford, 21 January 2005.
- Groupoid quantales, International Category Theory Conference, Univ. British Columbia, Vancouver, 18-24 July 2004.
- A quantale that classifies Penrose tilings, European Category Theory Meeting, Haute-Bodeux, Belgium, 7-13 September 2003.
- Quantales=1/2-quantization?, Sheaves and Topoi in Theoretical physics, Imperial College, London, 17-18 July 2003.
- From algebras to quantales and back, Workshop on Categorical Structures for Descent and Galois Theory, Hopf Algebras and Semiabelian Categories, Fields Institute in Toronto, Canada, 23-28 September 2002.
- Quantales, C*-algebras, and Penrose tilings, Workshop on Sheaves and Topoi in Physics, Imperial College, London, 11-12 July 2002.
- How quantum are quantales? International Conference on Quantum Structures, Technical Univ. Vienna, Austria, 1-7 July 2002.
- On quantales and spectra of C*-algebras, IV Iberoamerican Conference on Topology and Its Applications, Coimbra, 18-21 April 2001.

Grants

- FCT EXCL/MAT-GEO/0222/2012,
*Geometry and Mathematical Physics*, 1 May 2013 - 30 April 2016 (non-PI). - FCT POCI/MAT/55958/2004:
*Groupoids and quantales in geometry and analysis*, 14 Jul. 2005 - 13 Jul. 2008 - Participation of UTL/IST at the
SOCRATES
IP
103466-IC-1-2003-1-BE-ERASMUS-IPUC-3: GAMAP:
Geometric
and
Algebraic Methods of Physics and Applications
(coordinated by Univ. Antwerp).

- CRUP / British Council (Treaty of Windsor) Nos. B-22/04, B-68/05:
*Quantalic aspects of quasiperiodic tilings*, 1 Apr. 2004 - 30 Dec. 2006

- FCT POCTI/1999/MAT/33018:
*Quantales,*1 Oct. 2000 - 30 Jun. 2003 - CRUP
/
British Council (Treaty of Windsor) No. B-29/99:
*Dynamic observational logic*, 1 Apr. 1999 - 30 Apr. 2000

Events

- GAMAP IP meeting, Antwerp, September 4-18, 2006.
- GAMAP IP meeting,
Antwerp, September 5-15, 2005.

- Short course: Directed combinatorial homology and noncommutative geometry, Lisbon, February 9-13, 2004.
- Meeting on the Theory and Applications of Quantales, Lisbon, July 16-18, 2002.

Seminars

- Analysis, Geometry and Dynamical Systems
- Groupoids and Noncommutative Geometry (Oct. 2005 - )
- Algebra Seminar (Feb. 2001 - )
- "Geometria em Lisboa"
- Colloquium
(Oct. 2001 - Dec. 2004)

- "Seminário Diagonal" (Mar. 2001 - Jun. 2003)

Editorship

Current teaching

- On leave.

Past teaching

- Differential and Integral Calculus (CDI-II) (MEMec+MEM+LENO: Spring 2019)
- Differential and Integral Calculus (CDI-II) (LEIC+MEC+LEGM: Spring 2017)
- Differential and Integral Calculus (CDI-II) (MEC+LEGM: Spring 2016)
- Lie Groups and Lie Algebras (Math PhD: Fall 2015)
- Differential and Integral Calculus (CDI-II) (MEC+LEGM: Spring 2015)
- Differential and Integral Calculus (CDI-II) (MEC+MEFT+LEGM: Fall 2014)
- Differential and Integral Calculus (CDI-II) (MEMec: Spring 2014)
- Linear Algebra (LMAC+MEFT+MEBiom: Fall 2010)
- Topology (LMAC: Spring 2010)
- Linear Algebra (AL-3) (MEEC: Fall 2009)
- Topology (LMAC: Spring 2009)
- Linear Algebra (AL-10) (MEEC: Fall 2008)
- Topology (LMAC: Spring 2008)
- Linear Algebra (LEIC: Fall 2007)
- Linear
Algebra (
AL-14,
AL-23:
Spring
2007)

- Linear Algebra (LEIC: Fall 2006)
- Category Theory (LMAC, MMA: Fall 2001, Fall 2003, Spring 2005, Spring 2006)
- Algebra for Logic and Computer Science (LMAC, LCI, LEIC: Spring 2005, Spring 2006)
- Theory of Computation (LEIC: Fall 2005)
- Noncommutative Algebra (LMAC, MMA: Spring 2003)
- Ordered Algebraic Structures (LMAC, MMA: Fall 2000, Spring 2002)
- Algorithms and Data Structures (LEIC: Spring 1997-2001)
- Concurrent Processes (LMAC, MEIC: Fall 1996-1998, Fall 2003)

Postdocs

- Rachel Dawe Martins, PhD in Mathematical Physics, Nottingham University, 2006. From 2006 to 2014 (with interruptions).
- Dmitry Matsnev, PhD in Mathematics, Penn State University, 2005. From 2006 to 2008.
- Radu Popescu, Doctorat de Mathematiques, Université Claude Bernard, Lyon 1, France, 2000. From 2005 to 2008.
- David Kruml, PhD Math., Univ. Masaryk, Brno, Czech Rep., 2002. From 2003 to 2004.

PhD students

- Juan Pablo Quijano. Sheaves and functoriality of groupoid quantales, 2018.
- Elias Rodrigues. Sheaves, local homeomorphisms and local sets as Hilbert locale modules, 2009.
- Misha Protin. Quantales of open groupoids, 2008.

MSc students

- João Carvalho, A not so short introduction to Grothendieck topoi, 2013. Co-supervisor Wojciech Chachólski (KTH).
- Cátia Vaz, Supports of quantales and applications to modal logic, 2004.
- Vanda Lourenço, Involutive quantales and modules in a non-commutative generalization of the Gelfand representation, 2004.