MathematicsEvents calendar events
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Events related to MathematicsFeb 22 -
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Media Centre<br/>TBA<br/>TBA<br/><br/><br/>2018-2-22Feb 22 -
http://www.imsc.res.in/cgi-bin/CalciumShyam/Calcium40.pl?CalendarName=MathematicsEvents&EventID=902&Date=2018%2F2%2F22
Lounge<br/>Building works and Tender Committee<br/>Building works and Tender Committee<br/>Building works and Tender Committee<br/>Building works and Tender Committee<br/>2018-2-22Feb 22 -
http://www.imsc.res.in/cgi-bin/CalciumShyam/Calcium40.pl?CalendarName=MathematicsEvents&EventID=3331&Date=2018%2F2%2F22
Alladi Ramakrishnan Hall<br/>Shifted products of Fourier coefficients of cusp forms<br/>W. Kohnen<br/>University of Heidelberg<br/>We will discuss non-negativity results of a shifted product <br>of Fourier coefficients of<br>cusp forms, both in the classical case of elliptic modular forms as well <br>as in the case of Siegel<br>modular forms of degree two.<br/>2018-2-22Feb 22 -
http://www.imsc.res.in/cgi-bin/CalciumShyam/Calcium40.pl?CalendarName=MathematicsEvents&EventID=267&Date=2018%2F2%2F22
Media Centre<br/>Combinatorics and Quadratic Algebras<br/>Xavier Viennot<br/>CNRS, France<br/>This series of lectures is at the crossroad of algebra, combinatorics and theoretical physics. I shall expose a new theory I call "cellular Ansatz", which offers a framework for a fruitful relation between quadratic algebras and combinatorics, extending some very classical theories such as the Robinson-Schensted-Knuth (RSK) correspondence between permutations and pairs of standard Young tableaux.<br><br>In a first step, the cellular Ansatz is a methodology which allows, with a cellular approach on a square lattice, to associate some combinatorial objects (called Q-tableaux) to certain quadratic algebra Q defined by relations and generators. This notion is equivalent to the new notions (with a background from theoretical computer science) of "planar automata" or "planar rewriting rules". A Q-tableau is a Young (Ferrers) diagram, with some kind of labels in the cells of the diagram.<br><br>The first basic example is the algebra defined by the relation UD=qDU+Id (or Weyl-Heisenberg algebra, well known in quantum physics). The associated Q-tableaux are the permutations, or more generally placements of towers on a Young diagram (Ferrers diagram). The second basic example is the algebra defined by the relation DE=qED+E+D which is fundamental for the resolution of the PASEP model ("partially asymmetric exclusion process"), a toy model in the physics of dynamical systems far from equilibrium, with the calculus of the stationary probabilities. Many recent researches have been made for the associated Q-tableaux, under the name or "alternative tableaux", "tree-like tableaux" or "permutations tableaux". Note that these last tableaux were introduced ("Le-diagrams") by Postnikov in relation with some positivity problems in algebraic geometry. Another algebra is the so-called XYZ algebra, related to the 8-vertex model in statistical physics, and is related to the famous alternating sign matrices, plane partitions, non-crossing configurations of paths and tilings on a planar lattice, such as the well-known Aztec tilings.<br><br>In a second step, the cellular Ansatz allows a "guided" construction of bijections between Q-tableaux and other combinatorial objects, as soon one has a "representation" by combinatorial operators of the quadratic algebra Q. The basic example is the Weyl-Heisenberg algebra and we can recover the RSK correspondence with the Fomin formulation of "local rules" and "growth diagrams", from a representation of Q by operators acting on Ferrers diagrams (i.e. the theory of differential posets for the Young lattice). For the PASEP algebra, we get a bijection between alternative tableaux (or permutations tableaux) and permutations. The combinatorial theory of orthogonal polynomials (Flajolet, Viennot) plays an important role, in particular with the moments of q-Hermite, q-Laguerre and Askey-Wilson polynomials. Particular case is the TASEP (q=0) and is related to Catalan numbers and the Loday-Ronco Hopf algebra of binary trees.<br><br>I shall finish by considering a third step in the cellular Ansatz, with the introduction of the idea of "demultiplication" of the relations defining the quadratic algebra Q, leading to guess a combinatorial representation of the algebra Q. In the case of the Weyl-Heisenberg algebra, we get back again the RSK correspondence, which translated in terms of planar automata leads to the new notion of bilateral RSK automata, related to the notion of duality in Young tableaux. We will also apply this methodology to the 8-vertex algebra, where many open problems remain, in particular for the alternating sign matrices.<br/>2018-2-22Feb 22 -
http://www.imsc.res.in/cgi-bin/CalciumShyam/Calcium40.pl?CalendarName=MathematicsEvents&EventID=3309&Date=2018%2F2%2F22
Alladi Ramakrishnan Hall<br/>Stability of Parabolic Poincare bundle<br/>Suratno Basu<br/>IMSc<br/>Let X be a smooth, irreducible, projective curve (defined over complex numbers) and S be a finite set of points on X. We denote by M_{\alpha}(\Lambda), the moduli space of rank r at least 2, semi-stable parabolic bundles with complete flags at each point of S and of fixed determinant \Lambda. In this case, it is known that there exists a `Parabolic universal bundle' on X\times M_{\alpha}(\Lambda), with parabolic structure over the divisor S\times M_{\alpha}(\Lambda). In this talk we shall discuss the (slope) stability of these bundles as parabolic bundles. This question was motivated by an earlier work of Balaji, Brambila-Paz and Newstead. We begin with a brief exposition of their work.<br/>2018-2-22Feb 23 -
http://www.imsc.res.in/cgi-bin/CalciumShyam/Calcium40.pl?CalendarName=MathematicsEvents&EventID=286&Date=2018%2F2%2F23
Media Centre<br/>TB A<br/>TBA<br/><br/><br/>2018-2-23Feb 23 -
http://www.imsc.res.in/cgi-bin/CalciumShyam/Calcium40.pl?CalendarName=MathematicsEvents&EventID=900&Date=2018%2F2%2F23
Lounge<br/>Purchase committee meeting<br/>CCM<br/><br/><br/>2018-2-23Feb 23 -
http://www.imsc.res.in/cgi-bin/CalciumShyam/Calcium40.pl?CalendarName=MathematicsEvents&EventID=396&Date=2018%2F2%2F23
Room 217<br/>Topology 2<br/>-<br/><br/><br/>2018-2-23Feb 26 -
http://www.imsc.res.in/cgi-bin/CalciumShyam/Calcium40.pl?CalendarName=MathematicsEvents&EventID=288&Date=2018%2F2%2F26
Media Centre<br/>The lattice Tamari(v) is a maule<br/>Xavier Viennot<br/>CNRS, France<br/>First I will introduce a new family of posets called "maule" and recall the definition of the classical Tamari lattice defined by a<br>"rotation" on binary trees. By translating the rotation of binary trees in the context of Dyck paths, the Tamari lattice has been extended by F. Bergeron to m-Tamari (m integer) in relation with diagonal coinvariant spaces of the symmetric group. The problem of extending this lattice to any rational number m was solved by L.-F. Préville-Ratelle and the speaker. In fact, we defined a much more general extension: for any path v with elementary steps East and North we defined a lattice Tamari(v). I will prove that this lattice Tamari(v) is also a maule, which gives a new and more simple definition of this lattice (and thus also of the usual Tamari lattice). Again, as in the first lecture seminar about maules, Catalan alternative tableaux related to the TASEP model in physics play a crucial role, but with totally different bijections. These tableaux allow to relate this work with the recent work of C. Ceballos, A. Padrol et C. Sarmiento giving a geometric realization of Tamari(v), analogue to theclassical associahedron for the usual Tamari lattice.<br/>2018-2-26Feb 27 -
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Alladi Ramakrishnan Hall<br/>A Representation Theorem of Line Arrangements and its Generalization to Hyperplane Arrangements via Convex Positive Bijections<br/>C P Anil Kumar<br/>Center for Science,Technology and Policy (CSTEP) Bengaluru<br/>In this talk we first show that any line arrangement over a field with 1-ad structure can be isomorphically represented by a set of lines of same cardinality with a given set of distinct slopes. Then we generalize this theorem to higher dimensional hyperplane arrangements over a field with 1-ad structure. We prove, using a certain observation on the theme of central points, that, any two hyperplane arrangements are isomorphic modulo translations of any hyperplane if and only if there is a convex positive bijection between the corresponding associated normal systems. Finally we exhibit two normal systems in three dimensions of cardinality six which are not isomorphic.<br/>2018-2-27Feb 27 -
http://www.imsc.res.in/cgi-bin/CalciumShyam/Calcium40.pl?CalendarName=MathematicsEvents&EventID=3343&Date=2018%2F2%2F27
Alladi Ramakrishnan Hall<br/>Some combinatorial invariants for a finite abelian group<br/>Eshita Mazumdar<br/>-<br/>For a fnite abelian group $G$, the Davenport Constant $D_A(G)$ is defined to<br>be the least positive integer $k$ such that any sequence $S$ with length $k$ over $G$ has a non-empty $A$ weighted zero-sum subsequence. Similarly, the invariant<br>$s_A(G)$ is defined to be the least positive integer $k$ such that any sequence $S$<br>with length $k$ over $G$ has a non-empty $A$ weighted zero-sum subsequence of<br>length $exp(G)$. The precise value of these invariants for the cyclic group for<br>certain set $A$ is known but the general case is still an open question. In this<br>talk, I will present the results which we found towards this direction and also<br>discuss an extremal problem related to these combinatorial invariant. This is<br>a joint work with Prof. Niranjan Balachandran.<br/>2018-2-27Mar 1 -
http://www.imsc.res.in/cgi-bin/CalciumShyam/Calcium40.pl?CalendarName=MathematicsEvents&EventID=267&Date=2018%2F3%2F1
Media Centre<br/>Combinatorics and Quadratic Algebras<br/>Xavier Viennot<br/>CNRS, France<br/>This series of lectures is at the crossroad of algebra, combinatorics and theoretical physics. I shall expose a new theory I call "cellular Ansatz", which offers a framework for a fruitful relation between quadratic algebras and combinatorics, extending some very classical theories such as the Robinson-Schensted-Knuth (RSK) correspondence between permutations and pairs of standard Young tableaux.<br><br>In a first step, the cellular Ansatz is a methodology which allows, with a cellular approach on a square lattice, to associate some combinatorial objects (called Q-tableaux) to certain quadratic algebra Q defined by relations and generators. This notion is equivalent to the new notions (with a background from theoretical computer science) of "planar automata" or "planar rewriting rules". A Q-tableau is a Young (Ferrers) diagram, with some kind of labels in the cells of the diagram.<br><br>The first basic example is the algebra defined by the relation UD=qDU+Id (or Weyl-Heisenberg algebra, well known in quantum physics). The associated Q-tableaux are the permutations, or more generally placements of towers on a Young diagram (Ferrers diagram). The second basic example is the algebra defined by the relation DE=qED+E+D which is fundamental for the resolution of the PASEP model ("partially asymmetric exclusion process"), a toy model in the physics of dynamical systems far from equilibrium, with the calculus of the stationary probabilities. Many recent researches have been made for the associated Q-tableaux, under the name or "alternative tableaux", "tree-like tableaux" or "permutations tableaux". Note that these last tableaux were introduced ("Le-diagrams") by Postnikov in relation with some positivity problems in algebraic geometry. Another algebra is the so-called XYZ algebra, related to the 8-vertex model in statistical physics, and is related to the famous alternating sign matrices, plane partitions, non-crossing configurations of paths and tilings on a planar lattice, such as the well-known Aztec tilings.<br><br>In a second step, the cellular Ansatz allows a "guided" construction of bijections between Q-tableaux and other combinatorial objects, as soon one has a "representation" by combinatorial operators of the quadratic algebra Q. The basic example is the Weyl-Heisenberg algebra and we can recover the RSK correspondence with the Fomin formulation of "local rules" and "growth diagrams", from a representation of Q by operators acting on Ferrers diagrams (i.e. the theory of differential posets for the Young lattice). For the PASEP algebra, we get a bijection between alternative tableaux (or permutations tableaux) and permutations. The combinatorial theory of orthogonal polynomials (Flajolet, Viennot) plays an important role, in particular with the moments of q-Hermite, q-Laguerre and Askey-Wilson polynomials. Particular case is the TASEP (q=0) and is related to Catalan numbers and the Loday-Ronco Hopf algebra of binary trees.<br><br>I shall finish by considering a third step in the cellular Ansatz, with the introduction of the idea of "demultiplication" of the relations defining the quadratic algebra Q, leading to guess a combinatorial representation of the algebra Q. In the case of the Weyl-Heisenberg algebra, we get back again the RSK correspondence, which translated in terms of planar automata leads to the new notion of bilateral RSK automata, related to the notion of duality in Young tableaux. We will also apply this methodology to the 8-vertex algebra, where many open problems remain, in particular for the alternating sign matrices.<br/>2018-3-1Mar 1 -
http://www.imsc.res.in/cgi-bin/CalciumShyam/Calcium40.pl?CalendarName=MathematicsEvents&EventID=3324&Date=2018%2F3%2F1
Alladi Ramakrishnan Hall<br/>On Gross-Stark conjecture<br/>Mahesh Kakde<br/>King's college London<br/>In 1980, Gross conjectured a formula for the expected <br>leading term at s=0 of the p-adic L-function associated to <br>characters of totally real. number fields. The conjecture states a precise relation between this leading term and p-adic regulator of p-units in an abelian extension. In the talk I will present a precise formulation of the conjecture and describe its relevance <br>for Hilbert’s 12th problem. I will then sketch proof of this conjecture of Gross. This is a joint work with Samit Dasgupta and Kevin Ventullo.<br/>2018-3-1Mar 2 -
http://www.imsc.res.in/cgi-bin/CalciumShyam/Calcium40.pl?CalendarName=MathematicsEvents&EventID=396&Date=2018%2F3%2F2
Room 217<br/>Topology 2<br/>-<br/><br/><br/>2018-3-2Mar 3 -
http://www.imsc.res.in/cgi-bin/CalciumShyam/Calcium40.pl?CalendarName=MathematicsEvents&EventID=505&Date=2018%2F3%2F3
Ramanujan Auditorium<br/>ICA event<br/>-<br/>-<br/>TBA<br/>2018-3-3Mar 5 -
http://www.imsc.res.in/cgi-bin/CalciumShyam/Calcium40.pl?CalendarName=MathematicsEvents&EventID=259&Date=2018%2F3%2F5
Media Centre<br/>Combinatorics and Quadratic Algebras<br/>Xavier Viennot<br/>CNRS, France<br/>This series of lectures is at the crossroad of algebra, combinatorics and theoretical physics. I shall expose a new theory I call "cellular Ansatz", which offers a framework for a fruitful relation between quadratic algebras and combinatorics, extending some very classical theories such as the Robinson-Schensted-Knuth (RSK) correspondence between permutations and pairs of standard Young tableaux.<br><br>In a first step, the cellular Ansatz is a methodology which allows, with a cellular approach on a square lattice, to associate some combinatorial objects (called Q-tableaux) to certain quadratic algebra Q defined by relations and generators. This notion is equivalent to the new notions (with a background from theoretical computer science) of "planar automata" or "planar rewriting rules". A Q-tableau is a Young (Ferrers) diagram, with some kind of labels in the cells of the diagram.<br><br>The first basic example is the algebra defined by the relation UD=qDU+Id (or Weyl-Heisenberg algebra, well known in quantum physics). The associated Q-tableaux are the permutations, or more generally placements of towers on a Young diagram (Ferrers diagram). The second basic example is the algebra defined by the relation DE=qED+E+D which is fundamental for the resolution of the PASEP model ("partially asymmetric exclusion process"), a toy model in the physics of dynamical systems far from equilibrium, with the calculus of the stationary probabilities. Many recent researches have been made for the associated Q-tableaux, under the name or "alternative tableaux", "tree-like tableaux" or "permutations tableaux". Note that these last tableaux were introduced ("Le-diagrams") by Postnikov in relation with some positivity problems in algebraic geometry. Another algebra is the so-called XYZ algebra, related to the 8-vertex model in statistical physics, and is related to the famous alternating sign matrices, plane partitions, non-crossing configurations of paths and tilings on a planar lattice, such as the well-known Aztec tilings.<br><br>In a second step, the cellular Ansatz allows a "guided" construction of bijections between Q-tableaux and other combinatorial objects, as soon one has a "representation" by combinatorial operators of the quadratic algebra Q. The basic example is the Weyl-Heisenberg algebra and we can recover the RSK correspondence with the Fomin formulation of "local rules" and "growth diagrams", from a representation of Q by operators acting on Ferrers diagrams (i.e. the theory of differential posets for the Young lattice). For the PASEP algebra, we get a bijection between alternative tableaux (or permutations tableaux) and permutations. The combinatorial theory of orthogonal polynomials (Flajolet, Viennot) plays an important role, in particular with the moments of q-Hermite, q-Laguerre and Askey-Wilson polynomials. Particular case is the TASEP (q=0) and is related to Catalan numbers and the Loday-Ronco Hopf algebra of binary trees.<br><br>I shall finish by considering a third step in the cellular Ansatz, with the introduction of the idea of "demultiplication" of the relations defining the quadratic algebra Q, leading to guess a combinatorial representation of the algebra Q. In the case of the Weyl-Heisenberg algebra, we get back again the RSK correspondence, which translated in terms of planar automata leads to the new notion of bilateral RSK automata, related to the notion of duality in Young tableaux. We will also apply this methodology to the 8-vertex algebra, where many open problems remain, in particular for the alternating sign matrices.<br/>2018-3-5Mar 8 -
http://www.imsc.res.in/cgi-bin/CalciumShyam/Calcium40.pl?CalendarName=MathematicsEvents&EventID=267&Date=2018%2F3%2F8
Media Centre<br/>Combinatorics and Quadratic Algebras<br/>Xavier Viennot<br/>CNRS, France<br/>This series of lectures is at the crossroad of algebra, combinatorics and theoretical physics. I shall expose a new theory I call "cellular Ansatz", which offers a framework for a fruitful relation between quadratic algebras and combinatorics, extending some very classical theories such as the Robinson-Schensted-Knuth (RSK) correspondence between permutations and pairs of standard Young tableaux.<br><br>In a first step, the cellular Ansatz is a methodology which allows, with a cellular approach on a square lattice, to associate some combinatorial objects (called Q-tableaux) to certain quadratic algebra Q defined by relations and generators. This notion is equivalent to the new notions (with a background from theoretical computer science) of "planar automata" or "planar rewriting rules". A Q-tableau is a Young (Ferrers) diagram, with some kind of labels in the cells of the diagram.<br><br>The first basic example is the algebra defined by the relation UD=qDU+Id (or Weyl-Heisenberg algebra, well known in quantum physics). The associated Q-tableaux are the permutations, or more generally placements of towers on a Young diagram (Ferrers diagram). The second basic example is the algebra defined by the relation DE=qED+E+D which is fundamental for the resolution of the PASEP model ("partially asymmetric exclusion process"), a toy model in the physics of dynamical systems far from equilibrium, with the calculus of the stationary probabilities. Many recent researches have been made for the associated Q-tableaux, under the name or "alternative tableaux", "tree-like tableaux" or "permutations tableaux". Note that these last tableaux were introduced ("Le-diagrams") by Postnikov in relation with some positivity problems in algebraic geometry. Another algebra is the so-called XYZ algebra, related to the 8-vertex model in statistical physics, and is related to the famous alternating sign matrices, plane partitions, non-crossing configurations of paths and tilings on a planar lattice, such as the well-known Aztec tilings.<br><br>In a second step, the cellular Ansatz allows a "guided" construction of bijections between Q-tableaux and other combinatorial objects, as soon one has a "representation" by combinatorial operators of the quadratic algebra Q. The basic example is the Weyl-Heisenberg algebra and we can recover the RSK correspondence with the Fomin formulation of "local rules" and "growth diagrams", from a representation of Q by operators acting on Ferrers diagrams (i.e. the theory of differential posets for the Young lattice). For the PASEP algebra, we get a bijection between alternative tableaux (or permutations tableaux) and permutations. The combinatorial theory of orthogonal polynomials (Flajolet, Viennot) plays an important role, in particular with the moments of q-Hermite, q-Laguerre and Askey-Wilson polynomials. Particular case is the TASEP (q=0) and is related to Catalan numbers and the Loday-Ronco Hopf algebra of binary trees.<br><br>I shall finish by considering a third step in the cellular Ansatz, with the introduction of the idea of "demultiplication" of the relations defining the quadratic algebra Q, leading to guess a combinatorial representation of the algebra Q. In the case of the Weyl-Heisenberg algebra, we get back again the RSK correspondence, which translated in terms of planar automata leads to the new notion of bilateral RSK automata, related to the notion of duality in Young tableaux. We will also apply this methodology to the 8-vertex algebra, where many open problems remain, in particular for the alternating sign matrices.<br/>2018-3-8Mar 9 -
http://www.imsc.res.in/cgi-bin/CalciumShyam/Calcium40.pl?CalendarName=MathematicsEvents&EventID=396&Date=2018%2F3%2F9
Room 217<br/>Topology 2<br/>-<br/><br/><br/>2018-3-9Mar 12 -
http://www.imsc.res.in/cgi-bin/CalciumShyam/Calcium40.pl?CalendarName=MathematicsEvents&EventID=259&Date=2018%2F3%2F12
Media Centre<br/>Combinatorics and Quadratic Algebras<br/>Xavier Viennot<br/>CNRS, France<br/>This series of lectures is at the crossroad of algebra, combinatorics and theoretical physics. I shall expose a new theory I call "cellular Ansatz", which offers a framework for a fruitful relation between quadratic algebras and combinatorics, extending some very classical theories such as the Robinson-Schensted-Knuth (RSK) correspondence between permutations and pairs of standard Young tableaux.<br><br>In a first step, the cellular Ansatz is a methodology which allows, with a cellular approach on a square lattice, to associate some combinatorial objects (called Q-tableaux) to certain quadratic algebra Q defined by relations and generators. This notion is equivalent to the new notions (with a background from theoretical computer science) of "planar automata" or "planar rewriting rules". A Q-tableau is a Young (Ferrers) diagram, with some kind of labels in the cells of the diagram.<br><br>The first basic example is the algebra defined by the relation UD=qDU+Id (or Weyl-Heisenberg algebra, well known in quantum physics). The associated Q-tableaux are the permutations, or more generally placements of towers on a Young diagram (Ferrers diagram). The second basic example is the algebra defined by the relation DE=qED+E+D which is fundamental for the resolution of the PASEP model ("partially asymmetric exclusion process"), a toy model in the physics of dynamical systems far from equilibrium, with the calculus of the stationary probabilities. Many recent researches have been made for the associated Q-tableaux, under the name or "alternative tableaux", "tree-like tableaux" or "permutations tableaux". Note that these last tableaux were introduced ("Le-diagrams") by Postnikov in relation with some positivity problems in algebraic geometry. Another algebra is the so-called XYZ algebra, related to the 8-vertex model in statistical physics, and is related to the famous alternating sign matrices, plane partitions, non-crossing configurations of paths and tilings on a planar lattice, such as the well-known Aztec tilings.<br><br>In a second step, the cellular Ansatz allows a "guided" construction of bijections between Q-tableaux and other combinatorial objects, as soon one has a "representation" by combinatorial operators of the quadratic algebra Q. The basic example is the Weyl-Heisenberg algebra and we can recover the RSK correspondence with the Fomin formulation of "local rules" and "growth diagrams", from a representation of Q by operators acting on Ferrers diagrams (i.e. the theory of differential posets for the Young lattice). For the PASEP algebra, we get a bijection between alternative tableaux (or permutations tableaux) and permutations. The combinatorial theory of orthogonal polynomials (Flajolet, Viennot) plays an important role, in particular with the moments of q-Hermite, q-Laguerre and Askey-Wilson polynomials. Particular case is the TASEP (q=0) and is related to Catalan numbers and the Loday-Ronco Hopf algebra of binary trees.<br><br>I shall finish by considering a third step in the cellular Ansatz, with the introduction of the idea of "demultiplication" of the relations defining the quadratic algebra Q, leading to guess a combinatorial representation of the algebra Q. In the case of the Weyl-Heisenberg algebra, we get back again the RSK correspondence, which translated in terms of planar automata leads to the new notion of bilateral RSK automata, related to the notion of duality in Young tableaux. We will also apply this methodology to the 8-vertex algebra, where many open problems remain, in particular for the alternating sign matrices.<br/>2018-3-12Mar 15 -
http://www.imsc.res.in/cgi-bin/CalciumShyam/Calcium40.pl?CalendarName=MathematicsEvents&EventID=267&Date=2018%2F3%2F15
Media Centre<br/>Combinatorics and Quadratic Algebras<br/>Xavier Viennot<br/>CNRS, France<br/>This series of lectures is at the crossroad of algebra, combinatorics and theoretical physics. I shall expose a new theory I call "cellular Ansatz", which offers a framework for a fruitful relation between quadratic algebras and combinatorics, extending some very classical theories such as the Robinson-Schensted-Knuth (RSK) correspondence between permutations and pairs of standard Young tableaux.<br><br>In a first step, the cellular Ansatz is a methodology which allows, with a cellular approach on a square lattice, to associate some combinatorial objects (called Q-tableaux) to certain quadratic algebra Q defined by relations and generators. This notion is equivalent to the new notions (with a background from theoretical computer science) of "planar automata" or "planar rewriting rules". A Q-tableau is a Young (Ferrers) diagram, with some kind of labels in the cells of the diagram.<br><br>The first basic example is the algebra defined by the relation UD=qDU+Id (or Weyl-Heisenberg algebra, well known in quantum physics). The associated Q-tableaux are the permutations, or more generally placements of towers on a Young diagram (Ferrers diagram). The second basic example is the algebra defined by the relation DE=qED+E+D which is fundamental for the resolution of the PASEP model ("partially asymmetric exclusion process"), a toy model in the physics of dynamical systems far from equilibrium, with the calculus of the stationary probabilities. Many recent researches have been made for the associated Q-tableaux, under the name or "alternative tableaux", "tree-like tableaux" or "permutations tableaux". Note that these last tableaux were introduced ("Le-diagrams") by Postnikov in relation with some positivity problems in algebraic geometry. Another algebra is the so-called XYZ algebra, related to the 8-vertex model in statistical physics, and is related to the famous alternating sign matrices, plane partitions, non-crossing configurations of paths and tilings on a planar lattice, such as the well-known Aztec tilings.<br><br>In a second step, the cellular Ansatz allows a "guided" construction of bijections between Q-tableaux and other combinatorial objects, as soon one has a "representation" by combinatorial operators of the quadratic algebra Q. The basic example is the Weyl-Heisenberg algebra and we can recover the RSK correspondence with the Fomin formulation of "local rules" and "growth diagrams", from a representation of Q by operators acting on Ferrers diagrams (i.e. the theory of differential posets for the Young lattice). For the PASEP algebra, we get a bijection between alternative tableaux (or permutations tableaux) and permutations. The combinatorial theory of orthogonal polynomials (Flajolet, Viennot) plays an important role, in particular with the moments of q-Hermite, q-Laguerre and Askey-Wilson polynomials. Particular case is the TASEP (q=0) and is related to Catalan numbers and the Loday-Ronco Hopf algebra of binary trees.<br><br>I shall finish by considering a third step in the cellular Ansatz, with the introduction of the idea of "demultiplication" of the relations defining the quadratic algebra Q, leading to guess a combinatorial representation of the algebra Q. In the case of the Weyl-Heisenberg algebra, we get back again the RSK correspondence, which translated in terms of planar automata leads to the new notion of bilateral RSK automata, related to the notion of duality in Young tableaux. We will also apply this methodology to the 8-vertex algebra, where many open problems remain, in particular for the alternating sign matrices.<br/>2018-3-15Mar 15 -
http://www.imsc.res.in/cgi-bin/CalciumShyam/Calcium40.pl?CalendarName=MathematicsEvents&EventID=3342&Date=2018%2F3%2F15
Alladi Ramakrishnan Hall<br/>tba<br/>Rekha Biswal<br/>Universite Laval, Canada<br/>tba<br/>2018-3-15Mar 16 -
http://www.imsc.res.in/cgi-bin/CalciumShyam/Calcium40.pl?CalendarName=MathematicsEvents&EventID=396&Date=2018%2F3%2F16
Room 217<br/>Topology 2<br/>-<br/><br/><br/>2018-3-16Mar 23 -
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Room 217<br/>Topology 2<br/>-<br/><br/><br/>2018-3-23