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enPropositional equality, identity types, and computational paths [Math. Logic]
http://cmafcio.ciencias.ulisboa.pt/node/80
<span>Propositional equality, identity types, and computational paths [Math. Logic]</span>
<div><time datetime="2017-07-25T15:00:00Z">Tue, 07/25/2017 - 15:00</time></div>
<span><span lang="" about="http://cmafcio.ciencias.ulisboa.pt/user/13" typeof="schema:Person" property="schema:name" datatype="" xml:lang="">pjmendes@fc.ul.pt</span></span>
<span>Wed, 07/19/2017 - 13:01</span>
<div><p><strong>Ruy de Queiroz</strong> (Universidade Federal de Pernambuco)</p>
<p>In structural proof theory the notion of canonical proof is rather basic, and it is usually taken for granted that a canonical proof of a sentence must be unique up to certain minor syntactical details. When setting up a proof theory for equality one is faced with a somewhat unexpected situation where there may not be a unique canonical proof of an equality statement. Indeed, in a (1994) proposal for the formalisation of proofs of propositional equality in the Curry-Howard style, we have already uncovered such a peculiarity. Totally independently, and in a different setting, Hofmann & Streicher (1994) have shown how to build a model of Martin-Löf's Type Theory (with the so-called Identity Type) in which uniqueness of canonical proofs of identity statements does not hold. The intention here is to show that, by considering proofs of equality as sequences of rewrites and substitution, it comes a rather natural fact that two distinct proofs may be canonical and yet none is to be preferred over the other. By looking at proofs of equality as rewriting (or computational) paths this approach is in line with the recently proposed connections between type theory and homotopy theory via identity types, since elements of identity types will be, concretely, paths (or homotopies), with unbounded iteration of this (paths between paths, etc.). By introducing terms representing paths, the notion of 'paths between paths' becomes rather natural, and thus a syntactical counterpart to the notion of homotopy emerges quite straightforward. Recent results giving a categorical interpretation of identity types as types of computational paths will also be touched upon, time permitting.</p>
<p><a href="https://www.google.pt/maps/place/Edificio+C6+-+Faculdade+de+Ci%C3%AAncias+da+Universidade+de+Lisboa/@38.7557128,-9.1593984,17z">Room 6.2.33</a> [⤴]<img alt="Fundação para a Ciência e Tecnologia" data-entity-type="file" data-entity-uuid="2e09c74e-2138-4d43-8d43-3b0d8e16af3d" src="http://cmafcio.ciencias.ulisboa.pt/sites/cmafcio/files/inline-images/2015_FCT_H_cinza_1.jpg" class="align-right" /></p></div>
Wed, 19 Jul 2017 13:01:31 +0000pjmendes@fc.ul.pt80 at http://cmafcio.ciencias.ulisboa.ptG2-instantons with symmetry [Geometry]
http://cmafcio.ciencias.ulisboa.pt/node/79
<span>G2-instantons with symmetry [Geometry]</span>
<div><time datetime="2017-07-21T11:00:00Z">Fri, 07/21/2017 - 11:00</time></div>
<span><span lang="" about="http://cmafcio.ciencias.ulisboa.pt/user/13" typeof="schema:Person" property="schema:name" datatype="" xml:lang="">pjmendes@fc.ul.pt</span></span>
<span>Wed, 07/12/2017 - 14:29</span>
<div><p><strong>Gonçalo Oliveira</strong> (Duke University - E.U.A.)</p>
<p>G2-instantons, are 7-dimensional analogs of anti-self-dual connections in 4 dimensions, which however require the underlying manifold to have a so called G2-structure. The interest on these G2-instantons comes from the suggestion, of Donaldson and Segal, that it may be possible to use them to define enumerative invariants of G2-holonomy Riemannian metrics. In the talk, I will explain these ideas and report on joint work with Jason Lotay where we give existence, non-existence and classification results for these instantons. I will focus in the particular case of R<sup>4</sup> x S<sup>3</sup>, with its two explicitly known distinct G2-holonomy metrics, and exhibit the different behavior of their instantons. I will also explain an explicit example of sequences of G2-instantons where “bubbling” and “removable singularity” phenomena occur in the limit. All this is joint work with Jason Lotay.</p>
<p><a href="https://www.google.pt/maps/place/Edificio+C6+-+Faculdade+de+Ci%C3%AAncias+da+Universidade+de+Lisboa/@38.7557128,-9.1593984,17z">Room 6.2.33</a> [⤴]<img alt="Fundação para a Ciência e Tecnologia" data-entity-type="file" data-entity-uuid="2e09c74e-2138-4d43-8d43-3b0d8e16af3d" src="http://cmafcio.ciencias.ulisboa.pt/sites/cmafcio/files/inline-images/2015_FCT_H_cinza_1.jpg" class="align-right" /></p></div>
Wed, 12 Jul 2017 14:29:21 +0000pjmendes@fc.ul.pt79 at http://cmafcio.ciencias.ulisboa.ptA quantitative analysis of a theorem by F.E.Browder guided by the bounded functional interpretation [Math. Logic]
http://cmafcio.ciencias.ulisboa.pt/node/78
<span>A quantitative analysis of a theorem by F.E.Browder guided by the bounded functional interpretation [Math. Logic]</span>
<div><time datetime="2017-07-07T15:00:00Z">Fri, 07/07/2017 - 15:00</time></div>
<span><span lang="" about="http://cmafcio.ciencias.ulisboa.pt/user/13" typeof="schema:Person" property="schema:name" datatype="" xml:lang="">pjmendes@fc.ul.pt</span></span>
<span>Mon, 07/03/2017 - 09:21</span>
<div><p><strong>Pedro Pinto</strong> (Universidade de Lisboa)</p>
<p>In [2], Kohlenbach did an analysis of the proof of Browder's theorem (in [1]) via the monotone functional interpretation. I will be following the same outline but guided by the bounded functional interpretation ([3], [4]). Although the bounds obtained are the same, this example provides a first look at how the bounded functional interpretation works in practice.</p>
<p>[1] Browder, Felix E, Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach spaces, Archive for Rational Mechanics and Analysis, vol. 24 (1967), no. 1, pp. 82-90.<br />
[2] Kohlenbach, Ulrich, On quantitative versions of theorems due to F.E. Browder and R. Wittmann, Advances in Mathematics, vol. 226 (2011), no. 3, pp. 2764-2795.<br />
[3] Ferreira, Fernando and Oliva, Paulo, Bounded functional interpretation, Annals of Pure and Applied Logic, vol. 135 (2005), no. 1-3, pp. 73-112.<br />
[4] Ferreira, Fernando, Injecting uniformities into Peano arithmetic, Annals of Pure and Applied Logic, vol. 157 (2009), no. 2, pp. 122-129.</p>
<p><a href="https://www.google.pt/maps/place/Edificio+C6+-+Faculdade+de+Ci%C3%AAncias+da+Universidade+de+Lisboa/@38.7557128,-9.1593984,17z">Room 6.2.38</a> [⤴]<img alt="Fundação para a Ciência e Tecnologia" data-entity-type="file" data-entity-uuid="2e09c74e-2138-4d43-8d43-3b0d8e16af3d" src="http://cmafcio.ciencias.ulisboa.pt/sites/cmafcio/files/inline-images/2015_FCT_H_cinza_1.jpg" class="align-right" /></p></div>
Mon, 03 Jul 2017 09:21:06 +0000pjmendes@fc.ul.pt78 at http://cmafcio.ciencias.ulisboa.ptEquational Hybrid Propositional Type Theory [Math. Logic]
http://cmafcio.ciencias.ulisboa.pt/node/77
<span>Equational Hybrid Propositional Type Theory [Math. Logic]</span>
<div><time datetime="2017-06-30T15:00:00Z">Fri, 06/30/2017 - 15:00</time></div>
<span><span lang="" about="http://cmafcio.ciencias.ulisboa.pt/user/13" typeof="schema:Person" property="schema:name" datatype="" xml:lang="">pjmendes@fc.ul.pt</span></span>
<span>Mon, 06/26/2017 - 10:26</span>
<div><p><strong>Manuel Martins</strong> (Universidade de Aveiro)</p>
<p>Equational Hybrid Propositional Type Theory is a combination of propositional type theory, equational logic and hybrid modal logic. The structures used to interpret the language contain a hierarchy of propositional types, an algebra and a Kripke frame. In this talk we present a calculus for this logic that combines the axiomatics of its three components. The completeness proof is based on the three proofs Henkin published last century: (i) Completeness in type theory (ii) The completeness of the ﬁrst-order functional calculus and (iii) Completeness in propositional type theory. The challenge was to deal with all the heterogeneous components in an integrated system. (This is joint work with Maria Manzano and Antonia Huertas)</p>
<p><a href="https://www.google.pt/maps/place/Edificio+C6+-+Faculdade+de+Ci%C3%AAncias+da+Universidade+de+Lisboa/@38.7557128,-9.1593984,17z">Room 6.2.33</a> [⤴]<img alt="Fundação para a Ciência e Tecnologia" data-entity-type="file" data-entity-uuid="2e09c74e-2138-4d43-8d43-3b0d8e16af3d" src="http://cmafcio.ciencias.ulisboa.pt/sites/cmafcio/files/inline-images/2015_FCT_H_cinza_1.jpg" class="align-right" /></p></div>
Mon, 26 Jun 2017 10:26:20 +0000pjmendes@fc.ul.pt77 at http://cmafcio.ciencias.ulisboa.ptA unifying approach for (reflexive) autoepistemic logic, nonmonotonic S4F and ASP [Math. Logic]
http://cmafcio.ciencias.ulisboa.pt/node/76
<span>A unifying approach for (reflexive) autoepistemic logic, nonmonotonic S4F and ASP [Math. Logic]</span>
<div><time datetime="2017-06-23T15:00:00Z">Fri, 06/23/2017 - 15:00</time></div>
<span><span lang="" about="http://cmafcio.ciencias.ulisboa.pt/user/13" typeof="schema:Person" property="schema:name" datatype="" xml:lang="">pjmendes@fc.ul.pt</span></span>
<span>Mon, 06/19/2017 - 09:13</span>
<div><p><strong>Ezgi Su</strong> (Universidade de Lisboa) </p>
<p>Autoepistemic logic is an important form of nonmonotonic reasoning, introduced by Moore in order to allow an agent to reason about his own knowledge. It extends classical propositional logic by an epistemic modal operator L. Schwarz has shown that autoepistemic logic (AEL) and reflexive autoepistemic logic (RAEL) are the nonmonotonic variants of respectively the modal logics KD45 and SW5. In this talk, we implement the same strategy, embedding nonmonotonic S4F into a monotonic bimodal logic MLF*, to AEL and RAEL. This way, we propose a unifying mechanism for these major forms of nonmonotonic reasoning. Pearce’s equilibrium logic (EL) is a general-purpose nonmonotonic formalism, principally proposed as a semantical framework for answer set programming (ASP). Farinas et al. has recently embedded EL into a monotonic bimodal logic called MEM. Finally, in this talk, we discuss the relations between these two approaches, and try to find out the potential capability for capturing MEM under MLF*.</p>
<p><a href="https://www.google.pt/maps/place/Edificio+C6+-+Faculdade+de+Ci%C3%AAncias+da+Universidade+de+Lisboa/@38.7557128,-9.1593984,17z">Room 6.2.33</a> [⤴]<img alt="Fundação para a Ciência e Tecnologia" data-entity-type="file" data-entity-uuid="2e09c74e-2138-4d43-8d43-3b0d8e16af3d" src="http://cmafcio.ciencias.ulisboa.pt/sites/cmafcio/files/inline-images/2015_FCT_H_cinza_1.jpg" class="align-right" /></p></div>
Mon, 19 Jun 2017 09:13:29 +0000pjmendes@fc.ul.pt76 at http://cmafcio.ciencias.ulisboa.ptNeural Networks: recent advances - Deep learning - and some (mathematical ?) problems [Dyn. Systems]
http://cmafcio.ciencias.ulisboa.pt/node/75
<span>Neural Networks: recent advances - Deep learning - and some (mathematical ?) problems [Dyn. Systems]</span>
<div><time datetime="2017-06-30T10:30:00Z">Fri, 06/30/2017 - 10:30</time></div>
<span><span lang="" about="http://cmafcio.ciencias.ulisboa.pt/user/13" typeof="schema:Person" property="schema:name" datatype="" xml:lang="">pjmendes@fc.ul.pt</span></span>
<span>Fri, 06/16/2017 - 10:32</span>
<div><p><strong>Rui Alberto Pimenta Rodrigues</strong> (Faculdade de Ciências e Tecnologia - Universidade Nova)</p>
<p>Abstract: (Deep) Neural networks seem to be achieving the targets drawn for Artificial Intelligence in the early 60's. We will present the practical accomplishments. The underlying mathematical models will be discussed: • Feed forward neural networks • Recurrent neural networks • Generative neural networks (Boltzmann Machines, Variational AutoEncoders, Generative Adversary Networks). All these models have sometimes millions of parameters that are learned (estimated) from examples (training data). The learning procedure will be discussed. Training data is always a small fraction of real data where these models are applied. To avoid overfitting the model parameters to training data, and poor generalization to large real data, regularization techniques are used.</p>
<p><a href="https://www.google.pt/maps/place/Edificio+C6+-+Faculdade+de+Ci%C3%AAncias+da+Universidade+de+Lisboa/@38.7557128,-9.1593984,17z">Room 6.2.33</a> [⤴]<img alt="Fundação para a Ciência e Tecnologia" data-entity-type="file" data-entity-uuid="2e09c74e-2138-4d43-8d43-3b0d8e16af3d" src="http://cmafcio.ciencias.ulisboa.pt/sites/cmafcio/files/inline-images/2015_FCT_H_cinza_1.jpg" class="align-right" /></p></div>
Fri, 16 Jun 2017 10:32:17 +0000pjmendes@fc.ul.pt75 at http://cmafcio.ciencias.ulisboa.ptTopological Lattice Models in Geometry and Physics [Geometry]
http://cmafcio.ciencias.ulisboa.pt/node/74
<span>Topological Lattice Models in Geometry and Physics [Geometry]</span>
<div><time datetime="2017-06-23T13:30:00Z">Fri, 06/23/2017 - 13:30</time></div>
<span><span lang="" about="http://cmafcio.ciencias.ulisboa.pt/user/13" typeof="schema:Person" property="schema:name" datatype="" xml:lang="">pjmendes@fc.ul.pt</span></span>
<span>Wed, 06/14/2017 - 14:34</span>
<div><p><strong>Björn Gohla</strong> (GFM - Universidade de Lisboa)</p>
<p>Lattice models arise in physics as discrete approximations of quantum field theories (QFT). Topological quantum field theories (TQFT) on the other hand by definition are QFTs, that can be defined on smooth or topological space-times, as opposed to the usual pseudo-Riemannian space-times required by QFTs. Interesting TQFTs can be defined as lattice models, giving exactly solvable models in 2 dimensions for example. As an example in mathematics the Turaev-Viro TQFT gives a well-defined topological invariant of 3-manifolds. We will give an overview of these phenomena and present our work on a 4-dimensional gravitational lattice model.</p>
<p><a href="https://www.google.pt/maps/place/Edificio+C6+-+Faculdade+de+Ci%C3%AAncias+da+Universidade+de+Lisboa/@38.7557128,-9.1593984,17z">Room 6.2.33</a> [⤴]<img alt="Fundação para a Ciência e Tecnologia" data-entity-type="file" data-entity-uuid="2e09c74e-2138-4d43-8d43-3b0d8e16af3d" src="http://cmafcio.ciencias.ulisboa.pt/sites/cmafcio/files/inline-images/2015_FCT_H_cinza_1.jpg" class="align-right" /></p></div>
Wed, 14 Jun 2017 14:34:44 +0000pjmendes@fc.ul.pt74 at http://cmafcio.ciencias.ulisboa.ptClassification of vector bundles over the Riemann Sphere [Geometry]
http://cmafcio.ciencias.ulisboa.pt/node/73
<span>Classification of vector bundles over the Riemann Sphere [Geometry]</span>
<div><time datetime="2017-06-16T14:00:00Z">Fri, 06/16/2017 - 14:00</time></div>
<span><span lang="" about="http://cmafcio.ciencias.ulisboa.pt/user/13" typeof="schema:Person" property="schema:name" datatype="" xml:lang="">pjmendes@fc.ul.pt</span></span>
<span>Wed, 06/14/2017 - 14:32</span>
<div><p><strong>Javier Alcaide</strong> (Mestrado em Matemática - FCUL)</p>
<p>We will give a brief introduction to the theory of compact Riemann surfaces, as an approach to algebraic geometry. Objects like sheaves and vector bundles play an important role in this subject, as they allow us to understand global properties of a surface looking only at local information. After giving some important results of sheaf cohomology, like Serre's duality or the Riemann-Roch theorem, we will use these tools to classify holomorphic vector bundles over the Riemann sphere.</p>
<p><a href="https://www.google.pt/maps/place/Edificio+C6+-+Faculdade+de+Ci%C3%AAncias+da+Universidade+de+Lisboa/@38.7557128,-9.1593984,17z">Room 6.2.33</a> [⤴]<img alt="Fundação para a Ciência e Tecnologia" data-entity-type="file" data-entity-uuid="2e09c74e-2138-4d43-8d43-3b0d8e16af3d" src="http://cmafcio.ciencias.ulisboa.pt/sites/cmafcio/files/inline-images/2015_FCT_H_cinza_1.jpg" class="align-right" /></p></div>
Wed, 14 Jun 2017 14:32:56 +0000pjmendes@fc.ul.pt73 at http://cmafcio.ciencias.ulisboa.ptSIR-Network model: epidemics dynamics in a city & climate variations [Analysis & Diff. Eqs.]
http://cmafcio.ciencias.ulisboa.pt/node/72
<span>SIR-Network model: epidemics dynamics in a city & climate variations [Analysis & Diff. Eqs.]</span>
<div><time datetime="2017-06-08T13:30:00Z">Thu, 06/08/2017 - 13:30</time></div>
<span><span lang="" about="http://cmafcio.ciencias.ulisboa.pt/user/13" typeof="schema:Person" property="schema:name" datatype="" xml:lang="">pjmendes@fc.ul.pt</span></span>
<span>Tue, 06/06/2017 - 15:44</span>
<div><p><strong>Stefanella Boatto</strong> (INRIA-CentraleSupelec & Universidade Federal de Rio de Janeiro)</p>
<p><br />
The SIR-network mode deals with the propagation of disease epidemics in highly populated cities. The nodes, or vertices, are the city’s neighborhoods, in which the local populations are assumed to be well-mixed. The directed edges represent the fractions of people moving from their neighborhoods of residence to those of daily activities. First, we present some fundamental properties of the basic reproduction number (Ro) for this model. In particular, we focus on how Ro depends upon the geometry and the heterogeneity (different infection rates in each vertex) of the network. This allows us to conclude whether an epidemic outbreak can be expected or not. Second, we submit the SIR-network model to data fitting, using data collected during the 2008 Rio de Janeiro dengue fever epidemic. Important conclusions are drawn from the fitted parameters, and we show that improved results are found when a time-dependent infection parameter is introduced. This work is in collaboration with Lucas Stolerman (UFRJ/IMPA, Brazil) and Daniel Coombs (UBC, Canada). Finally we can show some recent results, in collaboration with Catherine Bonnet (INRIA-CentraleSupelec), Bernard Cazelles (UPMC) and Frederic Mazenc (INRIA-CentraleSupelec), about existence and approximation of attractors in the case of a periodic or bounded time dependent infectious parameter.</p>
<p><a href="https://www.google.pt/maps/place/Edificio+C6+-+Faculdade+de+Ci%C3%AAncias+da+Universidade+de+Lisboa/@38.7557128,-9.1593984,17z">Room 6.2.33</a> [⤴]<img alt="Fundação para a Ciência e Tecnologia" data-entity-type="file" data-entity-uuid="2e09c74e-2138-4d43-8d43-3b0d8e16af3d" src="http://cmafcio.ciencias.ulisboa.pt/sites/cmafcio/files/inline-images/2015_FCT_H_cinza_1.jpg" class="align-right" /></p></div>
Tue, 06 Jun 2017 15:44:46 +0000pjmendes@fc.ul.pt72 at http://cmafcio.ciencias.ulisboa.ptNon-local quantities and their application [Analysis & Diff. Eqs.]
http://cmafcio.ciencias.ulisboa.pt/node/71
<span>Non-local quantities and their application [Analysis & Diff. Eqs.]</span>
<div><time datetime="2017-06-08T13:30:00Z">Thu, 06/08/2017 - 13:30</time></div>
<span><span lang="" about="http://cmafcio.ciencias.ulisboa.pt/user/13" typeof="schema:Person" property="schema:name" datatype="" xml:lang="">pjmendes@fc.ul.pt</span></span>
<span>Tue, 06/06/2017 - 14:40</span>
<div><p><strong>Marco Caroccia</strong> (CNA - Carnegie Melon University - Pittsburgh - USA)</p>
<p>Non-local quantities and fractional Sobolev norms has received an incresing amount of attention in the last decade due to the interesting property they present and their various application in different fields, from graph theory to PDEs. In the first part of the talk I will introduce these topics by starting with a general overview on such quantities and by talking about the state of arts. Then in the second part of my seminar I will talk about two results obtained in collaboration with Dejan Slepcev and Matteo Rinaldi, involving a non-local version of Mumford-Shah functional for graph and the non-local perimeter.</p>
<p><a href="https://www.google.pt/maps/place/Edificio+C6+-+Faculdade+de+Ci%C3%AAncias+da+Universidade+de+Lisboa/@38.7557128,-9.1593984,17z">Room 6.2.33</a> [⤴]<img alt="Fundação para a Ciência e Tecnologia" data-entity-type="file" data-entity-uuid="2e09c74e-2138-4d43-8d43-3b0d8e16af3d" src="http://cmafcio.ciencias.ulisboa.pt/sites/cmafcio/files/inline-images/2015_FCT_H_cinza_1.jpg" class="align-right" /></p></div>
Tue, 06 Jun 2017 14:40:19 +0000pjmendes@fc.ul.pt71 at http://cmafcio.ciencias.ulisboa.pt