This is joint work with V. Turaev.\n LOCATION:https://researchseminars.org/talk/TQFT/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Ricardo Campos (CNRS - University of Montpellier) DTSTART;VALUE=DATE-TIME:20200710T160000Z DTEND;VALUE=DATE-TIME:20200710T170000Z DTSTAMP;VALUE=DATE-TIME:20240423T102923Z UID:TQFT/9 DESCRIPTION:Title: The homotopy type of associative and commutative algebras\nby Ricardo Cam pos (CNRS - University of Montpellier) as part of Topological Quantum Fiel d Theory Club (IST\, Lisbon)\n\n\nAbstract\nGiven a topological space\, ho w much of its homotopy type is captured by its algebra of singular cochain s? The experienced rational homotopy theorist will argue that one should c onsider instead a commutative algebra of forms. This raises the more algeb raic question "Given a dg commutative algebra\, how much of its homotopy t ype (quasi-isomorphism type) is contained in its associative part?" Despit e its elementary formulation\, this question turns out to be surprisingly subtle and has important consequences.\nIn this talk\, I will show how one can use operadic deformation theory to give an affirmative answer in char acteristic zero.\nWe will also see how the Koszul duality between Lie alge bras and commutative algebras allows us to use similar arguments to deduce that under good conditions Lie algebras are determined by the (associativ e algebra structure of) their universal enveloping algebras.\n\n\n(Joint w ith Dan Petersen\, Daniel Robert-Nicoud and Felix Wierstra and based on ar Xiv:1904.03585)\n LOCATION:https://researchseminars.org/talk/TQFT/9/ END:VEVENT BEGIN:VEVENT SUMMARY:Pedro Boavida de Brito (Instituto Superior Técnico and CAMGSD) DTSTART;VALUE=DATE-TIME:20200717T160000Z DTEND;VALUE=DATE-TIME:20200717T170000Z DTSTAMP;VALUE=DATE-TIME:20240423T102923Z UID:TQFT/10 DESCRIPTION:Title: Ga lois symmetries of knot spaces\nby Pedro Boavida de Brito (Instituto S uperior Técnico and CAMGSD) as part of Topological Quantum Field Theory C lub (IST\, Lisbon)\n\n\nAbstract\nI’ll describe how the absolute Galois group of the rationals acts on a space which is closely related to the spa ce of all knots. The path components of this space form a finitely generat ed abelian group which is\, conjecturally\, a universal receptacle for int egral finite-type knot invariants. The added Galois symmetry allows us to extract new information about its homotopy and homology beyond characteris tic zero. I will then discuss some work in progress concerning higher-dime nsional variants.\n\nThis is joint work with Geoffroy Horel.\n LOCATION:https://researchseminars.org/talk/TQFT/10/ END:VEVENT BEGIN:VEVENT SUMMARY:André Henriques (University of Oxford) DTSTART;VALUE=DATE-TIME:20200925T160000Z DTEND;VALUE=DATE-TIME:20200925T170000Z DTSTAMP;VALUE=DATE-TIME:20240423T102923Z UID:TQFT/11 DESCRIPTION:Title: Re ps of relative mapping class groups via conformal nets\nby André Henr iques (University of Oxford) as part of Topological Quantum Field Theory C lub (IST\, Lisbon)\n\n\nAbstract\nGiven a surface with boundary Σ\, its r elative mapping class group is the quotient of Diff(Σ) by the subgroup of maps which are isotopic to the identity via an isotopy that fixes the bou ndary pointwise. (If Σ has no boundary\, then that's the usual mapping cl ass group\; if Σ is a disc\, then that's the group Diff(S¹) of diffeomor phisms of S¹.)\n\nConformal nets are one of the existing axiomatizations of chiral conformal field theory (vertex operator algebras being another o ne). We will show that\, given an arbitrary conformal net and a surface wi th boundary Σ\, we get a continuous projective unitary representation of the relative mapping class group (orientation reversing elements act by an ti-unitaries). When the conformal net is rational and Σ is a closed surfa ce (i.e. ∂Σ = ∅)\, then these representations are finite dimensional and well known. When the conformal net is not rational\, then we must requ ire ∂Σ ≠ ∅ for these representations to be defined. We will try to explain what goes wrong when Σ is a closed surface and the conformal net is not rational.

\n\nThe material presented in this talk is partially based on my paper arXiv:1409.8672 with Arthur Bartels and Chris Douglas.\n LOCATION:https://researchseminars.org/talk/TQFT/11/ END:VEVENT BEGIN:VEVENT SUMMARY:Davide Masoero (Group of Mathematical Physics\, University of Lisb on) DTSTART;VALUE=DATE-TIME:20201002T160000Z DTEND;VALUE=DATE-TIME:20201002T170000Z DTSTAMP;VALUE=DATE-TIME:20240423T102923Z UID:TQFT/13 DESCRIPTION:Title: Co unting Monster Potentials\nby Davide Masoero (Group of Mathematical Ph ysics\, University of Lisbon) as part of Topological Quantum Field Theory Club (IST\, Lisbon)\n\n\nAbstract\nThe monster potentials were introduced by Bazhanov-Lukyanov-Zamolodchikov in the framework of the ODE/IM corres pondence. They should in fact be in 1:1 correspondence with excited states of the Quantum KdV model (an Integrable Conformal Field Theory) since the y are the most general potentials whose spectral determinant solves the Be the Ansatz equations of such a theory. By studying the large momentum limi t of the monster potentials\, I retrieve that:\n\n1) The poles of the mons ter potentials asymptotically condensate about the complex equilibria of t he ground state potential.\n\n2) The leading correction to such asymptotic s is described by the roots of Wronskians of Hermite polynomials.\n\nThis allows me to associate to each partition of N a unique monster potential with N roots\, of which I compute the spectrum. As a consequence\, I prov e up to a few mathematical technicalities that\, fixed an integer N\, the number of monster potentials with N roots coincide with the number of inte ger partitions of N\, which is the dimension of the level N subspace of th e quantum KdV model. In striking accordance with the ODE/IM correspondence .\n\nThe talk is based on the preprint https://arxiv.org/abs/2009.14638 \ , written in collaboration with Riccardo Conti (Group of Mathematical Phys ics of Lisbon University).\n LOCATION:https://researchseminars.org/talk/TQFT/13/ END:VEVENT BEGIN:VEVENT SUMMARY:Tom Sutherland (Group of Mathematical Physics\, University of Lisb on) DTSTART;VALUE=DATE-TIME:20200703T160000Z DTEND;VALUE=DATE-TIME:20200703T170000Z DTSTAMP;VALUE=DATE-TIME:20240423T102923Z UID:TQFT/14 DESCRIPTION:Title: Mi rror symmetry for Painlevé surfaces\nby Tom Sutherland (Group of Math ematical Physics\, University of Lisbon) as part of Topological Quantum Fi eld Theory Club (IST\, Lisbon)\n\n\nAbstract\nThis talk will survey aspect s of mirror symmetry for ten families of non-compact hyperkähler manifold s on which the dynamics of one of the Painlevé equations is naturally def ined. They each have a pair of natural realisations: one as the complement of a singular fibre of a rational elliptic surface and another as the com plement of a triangle of lines in a (singular) cubic surface. The two real isations relate closely to a space of stability conditions and a cluster v ariety of a quiver respectively\, providing a perspective on SYZ mirror sy mmetry for these manifolds. I will discuss joint work in progress with Hel ge Ruddat studying the canonical basis of theta functions on these cubic s urfaces.\n LOCATION:https://researchseminars.org/talk/TQFT/14/ END:VEVENT BEGIN:VEVENT SUMMARY:Alexander Shapiro (University of Notre Dame) DTSTART;VALUE=DATE-TIME:20201009T160000Z DTEND;VALUE=DATE-TIME:20201009T170000Z DTSTAMP;VALUE=DATE-TIME:20240423T102923Z UID:TQFT/15 DESCRIPTION:Title: Cl uster realization of quantum groups and higher Teichmüller theory\nby Alexander Shapiro (University of Notre Dame) as part of Topological Quant um Field Theory Club (IST\, Lisbon)\n\n\nAbstract\nQuantum higher Teichmü ller theory\, as described by Fock and Goncharov\, endows a quantum charac ter variety on a surface $S$ with a cluster structure. The latter allows o ne to construct a canonical representation of the character variety\, whic h happens to be equivariant with respect to an action of the mapping class group of $S$. It was conjectured by the authors that these representation s behave well with respect to cutting and gluing of surfaces\, which in tu rn yields an analogue of a modular functor. In this talk\, I will show how the quantum group and its positive representations arise in this context. I will also explain how the modular functor conjecture is related to the closedness of positive representations under tensor products as well as to the non-compact analogue of the Peter-Weyl theorem. If time permits\, I w ill say a few words about the proof of the conjecture.\n\nThis talk is bas ed on joint works with Gus Schrader.\n LOCATION:https://researchseminars.org/talk/TQFT/15/ END:VEVENT BEGIN:VEVENT SUMMARY:Miranda Cheng (University of Amsterdam) DTSTART;VALUE=DATE-TIME:20201016T160000Z DTEND;VALUE=DATE-TIME:20201016T170000Z DTSTAMP;VALUE=DATE-TIME:20240423T102923Z UID:TQFT/16 DESCRIPTION:Title: Qu antum modular forms and $3$-manifolds\nby Miranda Cheng (University of Amsterdam) as part of Topological Quantum Field Theory Club (IST\, Lisbon )\n\n\nAbstract\nQuantum modular forms are functions on rational numbers t hat have rather mysterious weak modular properties. Mock modular forms and false theta functions are examples of holomorphic functions on the upper- half plane which lead to quantum modular forms. Inspired by the $3d-3d$ co rrespondence in string theory\, new topological invariants named homologic al blocks for (in particular plumbed) three-manifolds have been proposed a few years ago. My talk aims to explain the recent observations on the qua ntum modular properties of the homological blocks\, as well as the relatio n to logarithmic vertex algebras.\n\nThe talk will be based on a series of work in collaboration with Sungbong Chun\, Boris Feigin\, Francesca Ferra ri\, Sergei Gukov\, Sarah Harrison\, and Gabriele Sgroi.\n LOCATION:https://researchseminars.org/talk/TQFT/16/ END:VEVENT BEGIN:VEVENT SUMMARY:Marco Mackaay (University of Algarve) DTSTART;VALUE=DATE-TIME:20201106T170000Z DTEND;VALUE=DATE-TIME:20201106T180000Z DTSTAMP;VALUE=DATE-TIME:20240423T102923Z UID:TQFT/17 DESCRIPTION:Title: Th e double-centralizer theorem in 2-representation theory and its applicatio ns\nby Marco Mackaay (University of Algarve) as part of Topological Qu antum Field Theory Club (IST\, Lisbon)\n\n\nAbstract\nFinitary birepresent ation theory of finitary bicategories is a categorical analog of finite-di mensional representation theory of finite-dimensional algebras. The role o f the simples is played by the so-called simple transitive birepresentatio ns and the classification of the latter\, for any given finitary bicategor y\, is a fundamental problem in finitary birepresentation theory (the clas sification problem). \nAfter briefly reviewing the basics of birepresentat ion theory\, I will explain an analog of the double centralizer theorem fo r finitary bicategories\, which was inspired by Etingof and Ostrik's doubl e centralizer theorem for tensor categories. As an application\, I will sh ow how it can be used to (almost completely) solve the classification prob lem for Soergel bimodules in any finite Coxeter type.\n LOCATION:https://researchseminars.org/talk/TQFT/17/ END:VEVENT BEGIN:VEVENT SUMMARY:João Faria Martins (University of Leeds) DTSTART;VALUE=DATE-TIME:20201030T170000Z DTEND;VALUE=DATE-TIME:20201030T180000Z DTSTAMP;VALUE=DATE-TIME:20240423T102923Z UID:TQFT/18 DESCRIPTION:Title: Cr ossed modules\, homotopy 2-types\, knotted surfaces and welded knots\n by João Faria Martins (University of Leeds) as part of Topological Quantu m Field Theory Club (IST\, Lisbon)\n\n\nAbstract\nI will review the constr uction of invariants of knots\, loop braids and knotted surfaces derived f rom finite crossed modules. I will also show a method to calculate the alg ebraic homotopy 2-type of the complement of a knotted surface $\\Sigma$ em bedded in the 4-sphere from a movie presentation of $\\Sigma$. This will e ntail a categorified form of the Wirtinger relations for a knot group. Alo ng the way I will also show applications to welded knots in terms of a biq uandle related to the homotopy 2-type of the complement of the tube of a w elded knots.\n\nThe last stages of this talk are part of the framework of the Leverhulme Trust research project grant: RPG-2018-029: “Emergent Ph ysics From Lattice Models of Higher Gauge Theory.\n LOCATION:https://researchseminars.org/talk/TQFT/18/ END:VEVENT BEGIN:VEVENT SUMMARY:Tudor Dimofte (University of California\, Davis) DTSTART;VALUE=DATE-TIME:20201120T170000Z DTEND;VALUE=DATE-TIME:20201120T180000Z DTSTAMP;VALUE=DATE-TIME:20240423T102923Z UID:TQFT/19 DESCRIPTION:Title: $3 d$ A and B models and link homology\nby Tudor Dimofte (University of C alifornia\, Davis) as part of Topological Quantum Field Theory Club (IST\, Lisbon)\n\n\nAbstract\nI will discuss some current work (with Garner\, Hi lburn\, Oblomkov\, and Rozansky) on new and old constructions of HOMFLY-PT link homology in physics and mathematics\, and new connections among them . In particular\, we relate the classic proposal of Gukov-Schwarz-Vafa\, i nvolving M-theory on a resolved conifold\, to constructions in $3d$ TQFT's . In the talk\, I will focus mainly on the $3d$ part of the story. I'll re view general aspects of $3d$ TQFT's\, in particular the "$3d$ A and B mode ls" that play a role here\, and how link homology appears in them.\n LOCATION:https://researchseminars.org/talk/TQFT/19/ END:VEVENT BEGIN:VEVENT SUMMARY:Victor Ostrik (University of Oregon) DTSTART;VALUE=DATE-TIME:20201204T170000Z DTEND;VALUE=DATE-TIME:20201204T180000Z DTSTAMP;VALUE=DATE-TIME:20240423T102923Z UID:TQFT/20 DESCRIPTION:Title: Tw o dimensional topological field theories and partial fractions\nby Vic tor Ostrik (University of Oregon) as part of Topological Quantum Field The ory Club (IST\, Lisbon)\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/TQFT/20/ END:VEVENT BEGIN:VEVENT SUMMARY:Penka Georgieva (Sorbonne Université) DTSTART;VALUE=DATE-TIME:20201218T170000Z DTEND;VALUE=DATE-TIME:20201218T180000Z DTSTAMP;VALUE=DATE-TIME:20240423T102923Z UID:TQFT/21 DESCRIPTION:Title: Kl ein TQFT and real Gromov-Witten invariants\nby Penka Georgieva (Sorbon ne Université) as part of Topological Quantum Field Theory Club (IST\, Li sbon)\n\n\nAbstract\nIn this talk I will explain how the Real Gromov-Witte n theory of local 3-folds with base a Real curve gives rise to an extensio n of a 2d Klein TQFT. The latter theory is furthermore semisimple which al lows for complete computation from the knowledge of a few basic elements w hich can be computed explicitly. As a consequence of the explicit expressi ons we find in the Calabi-Yau case\, we obtain the expected Gopukumar-Vafa formula and relation to SO/Sp Chern-Simons theory.\n LOCATION:https://researchseminars.org/talk/TQFT/21/ END:VEVENT BEGIN:VEVENT SUMMARY:Vladimir Dragović (Univ. Texas at Dallas) DTSTART;VALUE=DATE-TIME:20201113T170000Z DTEND;VALUE=DATE-TIME:20201113T180000Z DTSTAMP;VALUE=DATE-TIME:20240423T102923Z UID:TQFT/22 DESCRIPTION:Title: El lipsoidal billiards\, extremal polynomials\, and partitions\nby Vladim ir Dragović (Univ. Texas at Dallas) as part of Topological Quantum Field Theory Club (IST\, Lisbon)\n\n\nAbstract\n

A comprehensive study of peri odic trajectories of the billiards within ellipsoids in the d-dimensional Euclidean space is presented. The novelty of the approach is based on a re lationship established between the periodic billiard trajectories and the extremal polynomials of the Chebyshev type on the systems of d intervals o n the real line. Classification of periodic trajectories is based on a ne w combinatorial object: billiard partitions.

\nThe case study of tra jectories of small periods T\, d ≤ T ≤ 2d is given. In particular\, it is proven that all d-periodic trajectories are contained in a coordinate- hyperplane and that for a given ellipsoid\, there is a unique set of caust ics which generates d + 1-periodic trajectories. A complete catalog of bil liard trajectories with small periods is provided for d = 3.

\nThe talk is based on the following papers:

\nV. Dragović\, M. Radnović \, Periodic ellipsoidal billiard trajectories and extremal polynomials\, Communications Mathematical Physics\, 2019\, Vol. 372\, p. 183-211.

\n< p>G. Andrews\, V. Dragović\, M. Radnović\, Combinatorics of the periodic billiards within quadrics\, \narXiv: 1908.01026\, The Ramanujan Journal\, DOI: 10.1007/s11139-020-00346-y.\n LOCATION:https://researchseminars.org/talk/TQFT/22/ END:VEVENT BEGIN:VEVENT SUMMARY:Brent Pym (McGill University) DTSTART;VALUE=DATE-TIME:20210115T170000Z DTEND;VALUE=DATE-TIME:20210115T180000Z DTSTAMP;VALUE=DATE-TIME:20240423T102923Z UID:TQFT/23 DESCRIPTION:Title: Mu ltiple zeta values in deformation quantization\nby Brent Pym (McGill U niversity) as part of Topological Quantum Field Theory Club (IST\, Lisbon) \n\n\nAbstract\nIn 1997\, Kontsevich gave a universal solution to the defo rmation quantization problem in mathematical physics: starting from any Po isson manifold (the classical phase space)\, it produces a noncommutative algebra of quantum observables by deforming the ordinary\nmultiplication o f functions. His formula is a Feynman expansion whose Feynman integrals gi ve periods of the moduli space of marked holomorphic disks. I will describ e joint work with Peter Banks and Erik Panzer\, in which we prove that Kon tsevich's integrals evaluate to integer-linear\ncombinations of multiple z eta values\, building on Francis Brown's theory of polylogarithms on the m oduli space of genus zero curves.\n LOCATION:https://researchseminars.org/talk/TQFT/23/ END:VEVENT BEGIN:VEVENT SUMMARY:Anna Beliakova (University of Zürich) DTSTART;VALUE=DATE-TIME:20201211T170000Z DTEND;VALUE=DATE-TIME:20201211T180000Z DTSTAMP;VALUE=DATE-TIME:20240423T102923Z UID:TQFT/24 DESCRIPTION:Title: Cy clotomic expansions of the $gl_N$ knot invariants\nby Anna Beliakova ( University of Zürich) as part of Topological Quantum Field Theory Club (I ST\, Lisbon)\n\n\nAbstract\nNewton’s interpolation is a method to recons truct a function from its values at different points. In the talk I will e xplain how one can use this method to construct an explicit basis for the center of quantum $gl_N$ and to show that the universal $gl_N$ knot invari ant expands in this basis. This will lead us to an explicit construction o f the so-called unified invariants for integral homology 3-spheres\, that dominate all Witten-Reshetikhin-Turaev invariants. This is a joint work wi th Eugene Gorsky\, that generalizes famous results of Habiro for $sl_2$.\n LOCATION:https://researchseminars.org/talk/TQFT/24/ END:VEVENT BEGIN:VEVENT SUMMARY:Severin Bunk (Univ. Hamburg) DTSTART;VALUE=DATE-TIME:20210129T170000Z DTEND;VALUE=DATE-TIME:20210129T180000Z DTSTAMP;VALUE=DATE-TIME:20240423T102923Z UID:TQFT/25 DESCRIPTION:Title: Un iversal Symmetries of Gerbes and Smooth Higher Group Extensions\nby Se verin Bunk (Univ. Hamburg) as part of Topological Quantum Field Theory Clu b (IST\, Lisbon)\n\n\nAbstract\nGerbes are geometric objects describing th e third integer cohomology group of a manifold and the B-field in string t heory\; they can essentially be understood as bundles of categories whose fibre is equivalent to the category of vector spaces. Starting from a hand s-on example\, I will explain gerbes and their categorical features. The m ain topic of this talk will then be the study of symmetries of gerbes in a universal manner. We will see that these symmetries are completely encode d in an extension of smooth 2-groups. In the last part\, I will survey how this construction can be used to provide a new smooth model for the strin g group\, via a theory of group extensions in $\\infty$-topoi.\n LOCATION:https://researchseminars.org/talk/TQFT/25/ END:VEVENT BEGIN:VEVENT SUMMARY:Renee Hoekzema (Univ. Oxford) DTSTART;VALUE=DATE-TIME:20210122T170000Z DTEND;VALUE=DATE-TIME:20210122T180000Z DTSTAMP;VALUE=DATE-TIME:20240423T102923Z UID:TQFT/26 DESCRIPTION:Title: Ma nifolds with odd Euler characteristic and higher orientability\nby Ren ee Hoekzema (Univ. Oxford) as part of Topological Quantum Field Theory Clu b (IST\, Lisbon)\n\n\nAbstract\nOrientable manifolds have even Euler chara cteristic unless the dimension is a multiple of 4. I give a generalisation of this theorem: $k$-orientable manifolds have even Euler characteristic (and in fact vanishing top Wu class)\, unless their dimension is $2^{k+1}m $ for some integer $m$. Here we call a manifold $k$-orientable if the $i^{ \\rm th}$ Stiefel-Whitney class vanishes for all $0 < i < 2^k$. This theor em is strict for $k=0\,1\,2\,3$\, but whether there exist 4-orientable man ifolds with an odd Euler characteristic is a new open question. Such manif olds would have dimensions that are a multiple of 32. I discuss manifolds of dimension high powers of 2 and present the results of calculations on t he cohomology of the second Rosenfeld plane\, a special 64-dimensional man ifold with odd Euler characteristic.\n LOCATION:https://researchseminars.org/talk/TQFT/26/ END:VEVENT BEGIN:VEVENT SUMMARY:Ben Elias (Univ. Oregon) DTSTART;VALUE=DATE-TIME:20210226T170000Z DTEND;VALUE=DATE-TIME:20210226T180000Z DTSTAMP;VALUE=DATE-TIME:20240423T102923Z UID:TQFT/27 DESCRIPTION:Title: In troduction to the Hecke category and the diagonalization of the full twist \nby Ben Elias (Univ. Oregon) as part of Topological Quantum Field The ory Club (IST\, Lisbon)\n\n\nAbstract\nThe group algebra of the symmetric group has a large commutative subalgebra generated by Young-Jucys-Murphy e lements\, which acts diagonalizably on any irreducible representation. The goal of this talk is to give an accessible introduction to the categorifi cation of this story. The main players are: Soergel bimodules\, which cate gorify the Hecke algebra of the symmetric group\; Rouquier complexes\, whi ch categorify the braid group where Young-Jucys-Murphy elements live\; and the Elias-Hogancamp theory of categorical diagonalization\, which allows one to construct projections to "eigencategories."\n LOCATION:https://researchseminars.org/talk/TQFT/27/ END:VEVENT BEGIN:VEVENT SUMMARY:Pedro Vaz (Université Catholique de Louvain\, Belgium) DTSTART;VALUE=DATE-TIME:20210108T170000Z DTEND;VALUE=DATE-TIME:20210108T180000Z DTSTAMP;VALUE=DATE-TIME:20240423T102923Z UID:TQFT/28 DESCRIPTION:Title: Ca tegorification of Verma Modules in low-dimensional topology\nby Pedro Vaz (Université Catholique de Louvain\, Belgium) as part of Topological Q uantum Field Theory Club (IST\, Lisbon)\n\n\nAbstract\nIn this talk I will review the program of categorification of Verma modules and explain their applications to low-dimensional topology\, namely to the construction of Khovanov invariants for links in the solid torus via a categorification of the blob algebra.\n\nThe material presented spreads along several collabo rations with Abel Lacabanne\, and Grégoire Naisse.\n LOCATION:https://researchseminars.org/talk/TQFT/28/ END:VEVENT BEGIN:VEVENT SUMMARY:Florio Ciaglia (MPI Leipzig) DTSTART;VALUE=DATE-TIME:20210212T170000Z DTEND;VALUE=DATE-TIME:20210212T180000Z DTSTAMP;VALUE=DATE-TIME:20240423T102923Z UID:TQFT/29 DESCRIPTION:Title: A groupoid-based perspective on quantum mechanics\nby Florio Ciaglia (MP I Leipzig) as part of Topological Quantum Field Theory Club (IST\, Lisbon) \n\n\nAbstract\nIn this talk\, I will expound a point of view on the th eoretical investigation of the foundations and mathematical formalism of q uantum mechanics which is based on Schwinger’s “Symbolism of atomic me asurement” [8] on the physical side\, and on the notion of groupoid on t he mathematical side. I will start by reviewing the “development” of q uantum mechanics and its formalism starting from Schrödinger’s wave mec hanics\, passing through the Hilbert space quantum mechanics\, and arrivin g at the $C^∗$-algebraic formulation of quantum mechanics in order to gi ve an intuitive idea of what is the “place” of the groupoid-based appr oach to quantum theories presented here. Then\, after (what I hope will be ) a highly digestible introduction to the notion of groupoid\, I will revi ew two historic experimental instances in which the shadow of the structur e of groupoid may be glimpsed\, namely\, the Ritz-Rydberg combination prin ciple\, and the Stern-Gerlach experiment. The last part of the talk will b e devoted to building a bridge between the groupoid-based approach to quan tum mechanics and the more familiar $C^∗$-algebraic one by analysing how to obtain a (possibly) non-commutative algebra out of a given groupoid. T wo relevant examples will be discussed\, and some comment on future direct ions (e.g.\, the composition of systems) will close the talk. The material presented is part of an ongoing project developed together with Dr. F. Di Cosmo\, Prof. A. Ibort\, and Prof. G. Marmo. In particular\, the discrete -countable theory has already appeared in [1\, 2\, 3\, 4\, 5\, 6\, 7].

\nReferences

\n[1] F. M. Ciaglia\, F. Di Cosmo\, A. Ibort\, and G . Marmo. Evolution of Classical and Quantum States in the Groupoid Pictur e of Quantum Mechanics. Entropy\, 11(22):1292 – 18\, 2020.

\n[2] F. M. Ciaglia\, F. Di Cosmo\, A. Ibort\, and G. Marmo. Schwinger’s Pict ure of Quantum Mechanics. International Journal of Geometric Methods in Mo dern Physics\, 17(04):2050054 (14)\, 2020.

\n[3] F. M. Ciaglia\, F . Di Cosmo\, A. Ibort\, and G. Marmo. Schwinger’s Picture of Quantum Mec hanics IV: Composition and independence. International Journal of Geometri c Methods in Modern Physics\, 17(04):2050058 (34)\, 2020.

\n[4] F. M. Ciaglia\, A. Ibort\, and G. Marmo. A gentle introduction to Schwinger ’s formulation of quantum mechanics: the groupoid picture. Modern Physi cs Letters A\, 33(20):1850122–8\, 2018.

\n[5] F. M. Ciaglia\, A. Ibort\, and G. Marmo. Schwinger’s Picture of Quantum Mechanics I: Group oids. International Journal of Geometric Methods in Modern Physics\, 16(08 ):1950119 (31)\, 2019.

\n[6] F. M. Ciaglia\, A. Ibort\, and G. Mar mo. Schwinger’s Picture of Quantum Mechanics II: Algebras and Observable s. International Journal of Geometric Methods in Modern Physics\, 16(09):1 950136 (32)\, 2019.

\n[7] F. M. Ciaglia\, A. Ibort\, and G. Marmo. Schwinger’s Picture of Quantum Mechanics III: The Statistical Interpret ation. International Journal of Geometric Methods in Modern Physics\, 16(1 1):1950165 (37)\, 2019.

\n[8] J. Schwinger. Quantum Mechanics\, Sym bolism of Atomic Measurements. Springer-Verlag\, Berlin\, 2001.

\n LOCATION:https://researchseminars.org/talk/TQFT/29/ END:VEVENT BEGIN:VEVENT SUMMARY:David Reutter (MPI Bonn) DTSTART;VALUE=DATE-TIME:20210219T170000Z DTEND;VALUE=DATE-TIME:20210219T180000Z DTSTAMP;VALUE=DATE-TIME:20240423T102923Z UID:TQFT/30 DESCRIPTION:Title: Se misimple topological field theories in even dimensions\nby David Reutt er (MPI Bonn) as part of Topological Quantum Field Theory Club (IST\, Lisb on)\n\n\nAbstract\nA major open problem in quantum topology is the constru ction of an oriented 4-dimensional topological quantum field theory (TQFT) in the sense of Atiyah-Segal which is sensitive to exotic smooth structur e. More generally\, how much manifold topology can a TQFT see? \n\nIn this talk\, I will answer this question for semisimple field theories in even dimensions — I will sketch a proof that such field theories can at most see the stable diffeomorphism type of a manifold and conversely\, that if two sufficiently finite manifolds are not stably diffeomorphic then they c an be distinguished by semisimple field theories. In this context\, `semis implicity' is a certain algebraic condition applying to all currently know n examples of vector-space-valued TQFTs\, including `unitary field theorie s’\, and `once-extended field theories' which assign algebras or linear categories to codimension 2 manifolds. I will discuss implications in dime nsion 4\, such as the fact that oriented semisimple field theories cannot see smooth structure\, while unoriented ones can. \n\nThroughout\, I will use the Crane-Yetter field theory associated to a ribbon fusion category\, as a guiding example.\n\nThis is based on arXiv:2001.02288 and joint work in progress with Chris Schommer-Pries.\n LOCATION:https://researchseminars.org/talk/TQFT/30/ END:VEVENT BEGIN:VEVENT SUMMARY:Ikshu Neithalath (UCLA\, California) DTSTART;VALUE=DATE-TIME:20210312T170000Z DTEND;VALUE=DATE-TIME:20210312T180000Z DTSTAMP;VALUE=DATE-TIME:20240423T102923Z UID:TQFT/31 DESCRIPTION:Title: Sk ein Lasagna modules of 2-handlebodies\nby Ikshu Neithalath (UCLA\, Cal ifornia) as part of Topological Quantum Field Theory Club (IST\, Lisbon)\n \n\nAbstract\nMorrison\, Walker and Wedrich recently defined a generalizat ion of Khovanov-Rozansky homology to links in the boundary of a 4-manifold . \nWe will discuss recent joint work with Ciprian Manolescu on computing the "skein lasagna module\," a basic part of MWW's invariant\, for a certa in class of 4-manifolds.\n LOCATION:https://researchseminars.org/talk/TQFT/31/ END:VEVENT BEGIN:VEVENT SUMMARY:Owen Gwilliam (Univ. Massachusetts\, Amherst) DTSTART;VALUE=DATE-TIME:20210305T140000Z DTEND;VALUE=DATE-TIME:20210305T150000Z DTSTAMP;VALUE=DATE-TIME:20240423T102923Z UID:TQFT/32 DESCRIPTION:Title: Bu lk-boundary correspondences with factorization algebras\nby Owen Gwill iam (Univ. Massachusetts\, Amherst) as part of Topological Quantum Field T heory Club (IST\, Lisbon)\n\n\nAbstract\nFactorization algebras provide a flexible language for describing the observables of a perturbative QFT\, a s shown in joint work with Kevin Costello. Those constructions extend to a manifold with boundary for a special class of theories. I will discuss wo rk with Eugene Rabinovich and Brian Williams that includes\, as an example \, a perturbative version of the correspondence between chiral ${\\rm U}(1 )$ currents on a Riemann surface and abelian Chern-Simons theory on a bulk 3-manifold\, but also includes a systematic higher dimensional version fo r higher abelian CS theory on an oriented smooth manifold of dimension $4n +3$ with boundary a complex manifold of complex dimension $2n+1$. Given ti me\, I will discuss how this framework leads to a concrete construction of the center of higher enveloping algebras of Lie algebras\, in work with G reg Ginot and Brian Williams.\n\nPLEASE NOTE THE UNUSUAL TIME!\n LOCATION:https://researchseminars.org/talk/TQFT/32/ END:VEVENT BEGIN:VEVENT SUMMARY:Fabio di Cosmo (Instituto de Ciencias Matemáticas\, Madrid) DTSTART;VALUE=DATE-TIME:20210319T170000Z DTEND;VALUE=DATE-TIME:20210319T174000Z DTSTAMP;VALUE=DATE-TIME:20240423T102923Z UID:TQFT/33 DESCRIPTION:Title: St atistical Interpretation in the Schwinger’s picture of Quantum Mechanics \nby Fabio di Cosmo (Instituto de Ciencias Matemáticas\, Madrid) as p art of Topological Quantum Field Theory Club (IST\, Lisbon)\n\n\nAbstract\ nIn this talk I will illustrate some ideas about the statistical interpret ation in the Schwinger’s picture of Quantum Mechanics. After a brief int roduction on the postulates assumed in this framework\, I will recall the basic ingredients of Connes’ non commutative integration theory. This la nguage allows me to define\, on one hand quantum measures on the groupoid associated with the quantum systems\, and on the other weights on the corr esponding groupoid von-Neumann algebra. In particular\, quantum measures a re a generalization of measures on sigma-algebras which is suited for the description of interference phenomena. Then\, the final part of the talk w ill be devoted to the statistical interpretation associated with both situ ations.\n\nFirst part of a double session\, followed by a 20 minute break for coffee and discussion\, before the second speaker\, Pedro Resende.\n LOCATION:https://researchseminars.org/talk/TQFT/33/ END:VEVENT BEGIN:VEVENT SUMMARY:Pedro Resende (Instituto Superior Técnico\, Lisbon) DTSTART;VALUE=DATE-TIME:20210319T180000Z DTEND;VALUE=DATE-TIME:20210319T184000Z DTSTAMP;VALUE=DATE-TIME:20240423T102923Z UID:TQFT/34 DESCRIPTION:Title: An abstract theory of physical measurements\nby Pedro Resende (Instituto Superior Técnico\, Lisbon) as part of Topological Quantum Field Theory C lub (IST\, Lisbon)\n\n\nAbstract\nSince its early days\, quantum mechanics has forced physicists to consider the interaction between quantum systems and classically described experimental devices — a fundamental tenet fo r Bohr was that the results of measurements need to be communicated using the language of classical physics.\n\nSeveral decades of progress have led to improved understanding\, but the tension between “quantum” and “ classical” persists. Ultimately\, how is classical information extracted from a measurement? Is classical information fundamental\, as in Wheeler ’s “it from bit”? In this talk\, which is based on ongoing work [1]\ , I approach the problem mathematically by considering spaces whose points are measurements\, abstractly conceived in terms of the classical informa tion they produce. Concretely\, measurement spaces are stably Gelfand quan tales [2] equipped with a compatible sober topology\, but essentially thei r definition hinges on just two binary operations\, called composition and disjunction\, whose intuitive meanings are fairly clear. Despite their si mplicity\, these spaces have interesting mathematical properties. C*-algeb ras yield measurement spaces of “quantum type\,” and Lie groupoids giv e us spaces of “classical type\,” such as those which are associated w ith a specific experimental apparatus. The latter also yield a connection to Schwinger’s selective measurements\, which have been recast in groupo id language by Ciaglia et al.\nAn interaction between the two types\, prov iding a mathematical approach to Bohr’s quantum/classical split\, can be described in terms of groupoid (or Fell bundle) C*-algebras as in [3]. I will illustrate the basic ideas with simple examples\, such as spin measur ements performed with a Stern–Gerlach apparatus.\n\nReferences\n\n[1] P. Resende\, An abstract theory of physical measurements (2021)\, available at \nhttps://arxiv.org/abs/2102.01712.\n\n[2] P. Resende\, The many groupo ids of a stably Gelfand quantale\, J. Algebra 498 (2018)\, 197–210\, \nD OI 10.1016/j.jalgebra.2017.11.042.\n\n[3] P. Resende\, Quantales and Fell bundles\, Adv. Math. 325 (2018)\, 312–374\, \nDOI 10.1016/j.aim.2017.12. 001. MR3742593\n\nSecond part of a double session\, followed by a 20 minut e discussion period.\n LOCATION:https://researchseminars.org/talk/TQFT/34/ END:VEVENT BEGIN:VEVENT SUMMARY:Roger Picken (Instituto Superior Técnico\, Lisbon) DTSTART;VALUE=DATE-TIME:20210409T160000Z DTEND;VALUE=DATE-TIME:20210409T170000Z DTSTAMP;VALUE=DATE-TIME:20240423T102923Z UID:TQFT/35 DESCRIPTION:Title: Li nk invariants from finite crossed modules and a lifting of the Eisermann i nvariant\nby Roger Picken (Instituto Superior Técnico\, Lisbon) as pa rt of Topological Quantum Field Theory Club (IST\, Lisbon)\n\n\nAbstract\n This talk is based on work with João Faria Martins (Univ. Leeds) [1] and several projects with students. I will describe the construction of an inv ariant of tangles and framed tangles which takes values in an arbitrary cr ossed module of finite groups. This involves the fundamental crossed modul e associated to a natural topological pair coming from a knot diagram\, an d a suitable class of morphisms from this fundamental crossed module to th e chosen finite crossed module. Our construction includes all rack and qua ndle cohomology (framed) link invariants\, as well as the Eisermann invari ant of knots [2-3]\, for which we also find a lifting. The Eisermann invar iant detects information about a suitable choice of meridian and longitude in the knot complement boundary.\n\n[1] João Faria Martins and Roger
Picken: Link invariants from finite categorical groups\, Homology\, Homoto
py and Applications\, 17(2) (2015)\, 205–233\; arXiv:1301.3803v2 [math.GT]\, arXiv:1612.03501v1 [math.GT]

\n[2] M. Eiserma
nn: Knot colouring polynomials\, Pacific J. Math. 231 (2007)\, no. 2\, 305
–336.

\n[3] M. Eisermann: Homological characterization of the unkno
t\, J. Pure Appl. Algebra 177 (2003)\, no. 2\, 131–157.

\n[1] J. Faria Martins and R. Picken\, Diff. Geom. Appl . 29\, 179 (2011)\, arXiv:0907.2566.\n

\n\n[2] T. Porter\, J. Lond. Math. Soc. (2)58\, No. 3\, 723 (1998)\, MR 1678163.\n

\n\n[3] F. Gir elli\, H. Pfeiffer and E. M. Popescu\, Jour. Math. Phys. 49\, 032503 (2008 )\, arXiv:0708.3051.\n

\n LOCATION:https://researchseminars.org/talk/TQFT/45/ END:VEVENT BEGIN:VEVENT SUMMARY:Brian R. Williams (University of Edinburgh) DTSTART;VALUE=DATE-TIME:20211117T170000Z DTEND;VALUE=DATE-TIME:20211117T180000Z DTSTAMP;VALUE=DATE-TIME:20240423T102923Z UID:TQFT/46 DESCRIPTION:Title: Ex ceptional super Lie algebras in twisted M-theory\nby Brian R. Williams (University of Edinburgh) as part of Topological Quantum Field Theory Clu b (IST\, Lisbon)\n\n\nAbstract\nWith Saberi and Raghavendran we constructe d\, in the BV formalism\, the minimal\, holomorphic\, twist of 11-dimensio nal supergravity. Amazingly\, on flat space\, the theory shares a close re lationship to an exceptional simple super Lie algebra called E(5\,10). Mot ivated by holographic duality\, I’ll turn attention to symmetries of the theories on M2 and M5 branes. In the twisted setting\, we find that the s uperconformal algebra enhances to other infinite-dimensional exceptional s uper Lie algebras. I will discuss further extensions of these exceptional algebras to factorization algebras and applications to pinning down correl ation functions in M-theory.\n LOCATION:https://researchseminars.org/talk/TQFT/46/ END:VEVENT BEGIN:VEVENT SUMMARY:Vladimir M. Stojanovic (TU Darmstadt) DTSTART;VALUE=DATE-TIME:20211215T170000Z DTEND;VALUE=DATE-TIME:20211215T180000Z DTSTAMP;VALUE=DATE-TIME:20240423T102923Z UID:TQFT/47 DESCRIPTION:Title: Li e-algebraic aspects of quantum control: gate realization and W-to-GHZ stat e conversion\nby Vladimir M. Stojanovic (TU Darmstadt) as part of Topo logical Quantum Field Theory Club (IST\, Lisbon)\n\n\nAbstract\nIn this ta lk I will try to demonstrate the use of Lie-algebraic concepts in the quan tum control of interacting qubit arrays\, with examples from both operator (gate)- and state control. I will start from the basics of quantum contro l and briefly review the Lie-algebraic underpinnings of the concept of com plete controllability. I will then specialize to qubit arrays with Heisenb erg-type interactions\, summarizing the conditions for their complete cont rollability and showing a few examples of gate realization. The second par t of the talk will be devoted to a rather unconventional use of Lie-algebr aic concepts within a dynamical-symmetry-based approach to the determinist ic conversion between W- and Greenberger-Horne-Zeilinger (three-qubit) sta tes. The underlying physical system consists of three neutral atoms subjec t to several external laser pulses\, where the atomic ground- and a highly -excited Rydberg state play the role of the two relevant logical qubit sta tes.\n LOCATION:https://researchseminars.org/talk/TQFT/47/ END:VEVENT BEGIN:VEVENT SUMMARY:Giorgio Trentinaglia (Instituto Superior Técnico\,Lisbon) DTSTART;VALUE=DATE-TIME:20220119T170000Z DTEND;VALUE=DATE-TIME:20220119T180000Z DTSTAMP;VALUE=DATE-TIME:20240423T102923Z UID:TQFT/48 DESCRIPTION:Title: Si mplicial vector bundles and representations up to homotopy\nby Giorgio Trentinaglia (Instituto Superior Técnico\,Lisbon) as part of Topological Quantum Field Theory Club (IST\, Lisbon)\n\n\nAbstract\nThe classical Dol d–Kan correspondence for simplicial objects in an abelian category is on e of the cornerstones of homological algebra. When the abelian category is that of vector spaces\, it gives a full identification between simplicial vector spaces and chain complexes of vector spaces vanishing in negative degrees. The Grothendieck construction for fibered categories\, on the oth er hand\, is a cornerstone of category theory. It relates the fibered cate gory point of view with the pseudo-functor point of view and lies at the h eart of the theory of stacks. Our main result can be understood as a far-r eaching simultaneous generalization of both ideas within the contexts of l inear algebra and differential geometry. In our result\, simplicial vector spaces and chain complexes of vector spaces are replaced respectively by vector fibrations over a given (higher) Lie groupoid G and by representati ons up to homotopy of G. (Joint work with Matias del Hoyo.)\n LOCATION:https://researchseminars.org/talk/TQFT/48/ END:VEVENT BEGIN:VEVENT SUMMARY:Jeffrey C. Morton (SUNY Buffalo State) DTSTART;VALUE=DATE-TIME:20220126T170000Z DTEND;VALUE=DATE-TIME:20220126T180000Z DTSTAMP;VALUE=DATE-TIME:20240423T102923Z UID:TQFT/49 DESCRIPTION:Title: Th e Fock Pseudomonad: Groupoidifying Second Quantization\nby Jeffrey C. Morton (SUNY Buffalo State) as part of Topological Quantum Field Theory Cl ub (IST\, Lisbon)\n\n\nAbstract\nEdward Nelson said "First quantization is a mystery\, but second quantization is a functor". This functor takes the Hilbert space H representing a quantum mechanical system\, and gives i ts Fock space F(H)\, representing a multi-particle system with any number of indistinguishable copies of the original system as in quantum field the ory (I am considering the bosonic case). In a categorical analysis of the harmonic oscillator\, Vicary revised Nelson's slogan to say "second quanti zation is a monad" - that is\, the functor in question is equipped with so me extra algebraic structure\, making it the "Fock Monad" (F\,$\\eta$\,$\\ epsilon$).

\n\nGroupoidification is one of a number of approaches to "categorifying" quantum-mechanical systems: finding higher-categorical an alogs of those systems. It uses a 2-category Span(Gpd) whose objects are g roupoids\, and whose morphisms are "spans". This has had some success in d escribing extensions of topological field theory to systems with boundary\ , with the "categorified" theory describing the evolution of open systems\ , which can be composed along their boundaries\, over time. In this talk\, I will use this framework to describe a categorification of F to the "Foc k Pseudomonad" which can be defined in any suitable 2-category\, and the c ompatibility of this pseudomonad in Span(Gpd) with that in 2-Hilbert space s\, and\, under the "degroupoidification" map\, with the usual Fock constr uction on Hilbert spaces.

\n LOCATION:https://researchseminars.org/talk/TQFT/49/ END:VEVENT BEGIN:VEVENT SUMMARY:Emily Cliff (University of Sherbrooke) DTSTART;VALUE=DATE-TIME:20220202T170000Z DTEND;VALUE=DATE-TIME:20220202T180000Z DTSTAMP;VALUE=DATE-TIME:20240423T102923Z UID:TQFT/50 DESCRIPTION:Title: Mo duli spaces of principal 2-group bundles and a categorification of the Fre ed–Quinn line bundle\nby Emily Cliff (University of Sherbrooke) as p art of Topological Quantum Field Theory Club (IST\, Lisbon)\n\n\nAbstract\ nA 2-group is a higher categorical analogue of a group\, while a smooth 2- group is a higher categorical analogue of a Lie group. An important exampl e is the string 2-group in the sense of Schommer-Pries. We study the notio n of principal bundles for smooth 2-groups\, and investigate the moduli "s pace" of such objects.\n\nIn particular in the case of flat principal bund les for a finite 2-group over a Riemann surface\, we prove that the moduli space gives a categorification of the Freed–Quinn line bundle. This lin e bundle has as its global sections the state space of Chern–Simons theo ry for the underlying finite group. We can also use our results to better understand the notion of geometric string structures (as previously studie d by Waldorf and Stolz–Teichner).\n\n\nThis is based on joint work with Dan Berwick-Evans\, Laura Murray\, Apurva Nakade\, and Emma Phillips.\n LOCATION:https://researchseminars.org/talk/TQFT/50/ END:VEVENT BEGIN:VEVENT SUMMARY:Daniel Grady (Texas Tech University) DTSTART;VALUE=DATE-TIME:20220216T170000Z DTEND;VALUE=DATE-TIME:20220216T180000Z DTSTAMP;VALUE=DATE-TIME:20240423T102923Z UID:TQFT/51 DESCRIPTION:Title: De formation classes of invertible field theories and the Freed-Hopkins conje cture\nby Daniel Grady (Texas Tech University) as part of Topological Quantum Field Theory Club (IST\, Lisbon)\n\nLecture held in Room 3.10 (3rd floor\, Mathematics Department\, Instituto Superior Técnico.\n\nAbstract \nIn this talk\, I will discuss a recent result which provides an affirmat ive answer to a conjecture by Freed and Hopkins. The conjecture concerns a classification of reflection positive invertible field theories. I will b egin by reviewing motivation and background on reflection positive theorie s. Then I will state the conjecture and sketch of the proof\n LOCATION:https://researchseminars.org/talk/TQFT/51/ END:VEVENT BEGIN:VEVENT SUMMARY:Dmitri Pavlov (Texas Tech University) DTSTART;VALUE=DATE-TIME:20220330T160000Z DTEND;VALUE=DATE-TIME:20220330T170000Z DTSTAMP;VALUE=DATE-TIME:20240423T102923Z UID:TQFT/52 DESCRIPTION:Title: Th e geometric cobordism hypothesis\nby Dmitri Pavlov (Texas Tech Univers ity) as part of Topological Quantum Field Theory Club (IST\, Lisbon)\n\nLe cture held in Room 3.10 (3rd floor\, Mathematics Department\, Instituto Su perior Técnico).\n\nAbstract\nI will explain my recent work with Danie l Grady on the locality of functorial field theories (arXiv:2011.01208) an d the geometric cobordism hypothesis (arXiv:2111.01095). The latter genera lizes the Baez–Dolan cobordism hypothesis to nontopological field theori es\, in which bordisms can be equipped with geometric structure\, such as smooth maps to a fixed target manifold\, Riemannian metrics\, conformal st ructures\, principal bundles with connection\, or geometric string structu res.

\n\nApplications include

\n\n- \n
- a generalization of the Galatius–Madsen–Tillmann–Weiss theorem\; \n
- a solution t o a conjecture of Stolz and Teichner on representability of concordance cl asses of functorial field theories\; \n
- a construction of power op erations on the level of field theories (extending the recent work of Bart hel–Berwick-Evans–Stapleton)\; \n
- and a recent solution by Gra dy of a conjecture by Freed and Hopkins on deformation classes of reflecti on positive invertible field theories. \n

\nIf time p
ermits\, I will talk about planned work on the nonperturbative quantizatio
n of functorial field theories and generalized Atiyah–Singer-style index
theorems.