 |
Talks Arun Ram |
Department of Mathematics and Statistics University of Melbourne Parkville VIC 3010 Australia aram@unimelb.edu.au |
I have, too slowly, been coming to the realisation that what I understand in mathematics is far ahead of what I can write up in a polished form and that much of my contribution to the progress of mathematics comes from my lectures. Hopefully, putting as many of these lectures on the web as I can manage will help mathematics move along as quickly as possible. (This is not a complete list. There are some years where I seem to have no records at all, and others for which I have records only of colloquium and invited conference talks.)
Some of available from the links below is based upon work supported by the Australian Research Council ARC grants DP0986774 and DP087995 and the US National Science Foundation under Grant No. 0353038 and earlier awards. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of these agencies.
Talks of Arun Ram in 2012
- Schubert calculus for the affine Grassmannian, Seminar, University of Melbourne, 11 May April 2012.
Abstract: This talk will be a summary of a recent paper of Lam and Shimozono. arXiv:1105.2170, k-Double Schur functions and equivariant (co)homology of the affine Grassmannian. - Views from Castalia, Colloquium, University of Southern California, 27 April 2012.
Abstract: There are three mountains (the preprojective variety, the quiver variety and the loop Grassmannian). These three mountains feed the spring at their base (the quantum group). At the spring live three nymphs, very similar, but different (the semicanonical basis, the canonical basis and the MV basis). Each of these is nourished by the special nutrients of the corresponding source (the preprojective variety, the quiver variety and the loop Grassmannian). But the shadow of the muses is the same (the MV polytope). And, as we shall see, there is a Glass Bead Game in this story as well. - Elliptic Schubert Calculus, Colloquium, University of California, Los Angeles, 26 April 2012.
Abstract: Traditional Schubert calculus is the combinatorial study of intersections of Schubert varieties inside the flag variety. This is usually done by computing cup products in cohomology or equivariant cohomology. This talk is a summary of our study of how to extend the story to other equivariant cohomology theories: K-theory, elliptic cohomology and cobordism. The talk is intended to be a general audience survey: to give a feel for what flag varieties "look like", what cohomology theories do for us, and the combinatorial structure (reflection group symmetry) that makes the game go. The subject is fascinating in the confluence of different parts of mathematics: Lie groups, loop groups, symmetries of regular polytopes, and algebraic topology. This talk is based on ongoing joint work with Nora Ganter. - Generalized equivariant cohomology of flag varieties, Geometry/Physics seminar, Northwestern University, 24 April 2012.
Abstract: I will review some of the Kac-Peterson paper on affine Lie algebras and modular forms in order to set up a framework for the equivariant elliptic cohomology version of Schubert polynomials. I will also discuss some parts of an analogous story for equivariant cobordism Schubert polynomials and compare and contrast four cases: cohomology, K-theory, elliptic cohomology and cobordism. This is work in progress with Nora Ganter. - Views of Castalia, Geometry/Physics seminar, Northwestern University, 24 April 2012.
Abstract: There are three mountains (the preprojective variety, the quiver variety and the loop Grassmannian). These three mountains feed the spring at their base (the quantum group). At the spring live three nymphs, very similar, but different (the semicanonical basis, the canonical basis and the MV basis). Each of these is nourished by the special nutrients of the corresponding source (the preprojective variety, the quiver variety and the loop Grassmannian). At three particular moments of each day (dawn, noon, dusk) the shadow of the muses on the valley below is the same (the MV polytope). And, as we shall see, there is a Glass Bead Game in this story as well. - Views from Castalia, Special Geometry seminar, University of Texas at Austin, 23 April 2012.
Abstract: There are three mountains (the preprojective variety, the quiver variety and the loop Grassmannian). These three mountains feed the spring at their base (the quantum group). At the spring live three nymphs, very similar, but different (the semicanonical basis, the canonical basis and the MV basis). Each of these is nourished by the special nutrients of the corresponding source (the preprojective variety, the quiver variety and the loop Grassmannian). But the shadow of the muses is the same (the MV polytope). And, as we shall see, there is a Glass Bead Game in this story as well. - Generalized equivariant cohomology of flag varieties, Infinite dimensional algebra seminar, MIT, 20 April 2012.
Abstract: I will review some of the Kac-Peterson paper on affine Lie algebras and modular forms in order to set up a framework for the equivariant elliptic cohomology version of Schubert polynomials. I will also discuss some parts of an analogous story for equivariant cobordism Schubert polynomials and compare and contrast four cases: cohomology, K-theory, elliptic cohomology and cobordism. This is work in progress with Nora Ganter. - Schubert Calculus I, II and III, Seminar, University of Melbourne, 9, 11 and 13 April 2012.
Abstract: Schubert calculus I -- Examples of the Borel picture for generalised cohomologies of flag varieties, Schubert calculus II -- Generalised cohomology of the Bott-Samelson map, Schubert calculus III -- BGG/Demazure operators for generalized cohomology theories: These talks are intended as a (hopefully) leisurely tour of the generalised Schubert calculus that we are setting up in our join work with Nora Ganter. In each case I will try to indicate how to work with the appropriate rings and maps, both in terms of generators and relations, and via moment graphs. Traditional Schubert calculus is the combinatorial study of intersections of Schubert varieties inside the flag variety. This is usually done by computing cup products in cohomology or equivariant cohomology. This talk is a summary of our study of how to extend the story to other equivariant cohomology theories: K-theory, elliptic cohomology and cobordism. - Combinatorics and growth in Chevalley groups and their representations, Connecting finite and infinite mathematics through symmetry, 1-3 February 2012, University of Wollongong, 2 February 2012.
Abstract: I will endeavour to illustrate the ‘path model’, a powerful tool for labeling points in Chevalley groups in a way which is consistent with all finite dimensional representations (‘finite’ quotients of the group) at once. The main tools are to have a set of generators and relations which interacts in a controllable way with the reflection group geometry of the group (coming from automorphisms) and a powerful combinatorics for manipulating computations using these generators and relations. - Views from 20 years trekking on the LS path, CMI-IMSc Mathematics Colloquium, in celebration of the 80 birthday of CS Seshadri, 23-27 January 2012, Chennai Mathematical Institute, Chennai, 23 January 2012.
Abstract: My travels on the LS (Lakshmibai-Seshadri) path have brought many personal realizations and its vistas have allowed me to look out over many beautiful structures. As best I can, I shall give a brief summary (1 hour instead of 20 years) including: Kashiwara crystals, combinatorial aspects of Schubert calculus, affine Hecke algebras and structure of Chevalley groups.
Talks of Arun Ram in 2011
- RTT algebras and Algebraic Bethe ansatz, working seminar at University of Melbourne, 6 December 2011.
Abstract: This is a review of RTT algebras (as found in Drinfeld's ICM paper in section 10), spectral subalgebras (following Drinfeld's ICM paper section 11), lattice models (following deGier's thesis chapter 5), the source of the Yangian (following Drinfeld and Chari-Pressley) and a formulation of the algebraic Bethe ansatz (following Takhtajan and Faddeev, Kulish and Reshitikhin, Kirillov and Reshetikhin, and Nazarov and Tarasov). If time permits I will formulate the L-matrices in the Drinfeld double context (following Reshetikhin-Takhtajan-Faddeev). - Schubert Calculus, working seminar at University of Melbourne, 28 October 2011.
Abstract: In connection with my work with Nora on the Elliptic Schubert Calculus I have found that I need to read two recent papers on Complex Cobordism and Schubert calculus: Calmes-Petrov-Zainoulline arXiv:0905.1341 and Kritichecko-Krishna arXiv:1104.1089 Viewing this talk as a working seminar, I will try to present some survey of the contents of these papers from my perspective. - An introduction to the Knizhnik-Zamolodchikov equations, working seminar at University of Melbourne, 19 September 2011.
Abstract: The Knizhnik-Zamolodchikov connection provides a remarkable passage between representation theory, differential equations and conformal field theory. I will try to give a brief introduction to this fascinating correspondence. - Elliptic Schubert calculus, invited talk at Perspectives in Algebraic Lie Theory, at the Isaac Newton Institute, Cambridge, 12-16 September 2011.
Abstract: Bernstein-Gelfand-Gelfand and Demazure operators (also called divided difference operators) are the foundation of the Schubert calculus, the study of the cohomology and K-theory of the flag variety in terms of the natural basis coming from the Schubert varieties. In this work, joint with Nora Ganter, we define elliptic (double) Schubert polynomials by using rings of theta functions. The elliptic Schubert polynomials are combinatorial realisations of the classes of Schubert varieties in equivariant elliptic cohomology. We set up the BGG calculus and the corresponding moment graph perspective for elliptic cohomology of flag varieties and make the connection to the representation theory of affine Lie algebras (following Kac-Peterson, Grojnowski, Ando). - Elliptic Schubert calculus, invited talk at Algebraic Cycles and the Geometry of Group Orbits, A
Conference on the occasion of the 60th birthday of Peter O'Sullivan, at the Australian National University, Canberra, 2-4 September 2011.
Abstract: Bernstein-Gelfand-Gelfand and Demazure operators (also called divided difference operators) are the foundation of the Schubert calculus, the study of the cohomology and K-theory of the flag variety in terms of the natural basis coming from the Schubert varieties. In this work, joint with Nora Ganter, we produce an analogous study of the elliptic cohomology of the flag varieties, by using rings of theta functions and the appropriate elliptic cohomology versions of the Thom isomorphism and localisation at T-fixed points. Additionally, we make the connection to the representation theory of affine Lie algebras (following Kac-Peterson, Grojnowski, Ando). - Cohomology of Grassmannians and isotropic Grassmanians, working seminar talk at University of Melbourne, 2 June 2011.
Abstract: The classical example of the Schubert calculus is the case of the Grassmannian where the Schubert classes can be represented as Schur functions, and the problem is then solved with representation theory. The case of the isotropic Grassmanians (orthogonal and symplectic groups) was treated by Pragacz, making contact with the Schur Q-functions that arise in the projective representation theory of the symmetric group (and Nora's recent work). I recently understood how this generalises to p-compact groups, what the generalizations of the Schur Q-functions are and note, following Morris, an interesting connection to the evaluation of characters of the symmetric groups at r-regular conjugacy classes (the type of phenomena that Jamie is noting ought to happen in the p-compact setting). A future goal here is to get the whole picture clear also in K-theory and T-equivariant K-theory. - Affine Weyl group, Heisenberg groups and classifying complex reflection groups, working seminar talk at University of Melbourne, 26 May 2011.
Abstract: I will try to outline the generalised Cartan matrix approach to the affine Weyl group, as found in Kac's book on Infinite dimensional Lie algebras, and explain its relation to the Heisenberg group, and an attempt to extend it so that it might have a chance of providing a new way to approach the classification of complex reflection groups. This is some meld of current work with Nora on elliptic stuff, and what Don Taylor and I came up with during his visit in April. - Heisenberg groups, abelian varieties and theta functions, working seminar talk at University of Melbourne, 12 May 2011.
- Rank 1 reductive Lie groups, in perspective, working seminar talk at University of Melbourne, 5 May 2011.
Abstract: I will discuss the Lie groups SU(2), Spin(3), Sp(1), SL(2), PGL(2) their relationships, and their adjoint representations in the context of the other reductive Lie groups of Lie types A,B, C and D. I'll derive the adjoint representation in terms of Hamiltonians=Quaternions, Pauli matrices and Chevalley generators. - The Borel-Weil-Bott Theorem, working seminar talk at University of Melbourne, 29 April 2011.
Abstract: The lecture will essentially be a version of a lecture I gave at the WinterSchool on The interaction of Geometry and Combinatorics in Representation Theory at the Hausdorff Institute in Bonn in January. Understanding this is, hopefully, a path to a full understanding of the Weyl character formula, cohomology and K-theory for compact and p-compact groups, the elliptic cohomology of flag varieties, theta functions, Looijenga line bundles, and modular forms from abelian varieties. - Polytopes, shuffles, quivers and flags, seminar talk at University of Melbourne, 8 March 2011.
Abstract: There are two geometries that show remarkable similarities: that of quiver varieties and that of affine flag varieties. By work of Braverman-Gaitsgory and Gaussent-Littelmann and Kashiwara-Saito and Kamnitzer-Baumann one sees the crystals, in the sense of Kashiwara, coming from both quivers and flags. In the picture of Leclerc-Geiss-Schroer one sees how elements of the shuffle algebra come from quiver varieties. In joint work with A. Ghitza and S. Kannan we are seeing shuffle elements coming from affine flag varieties. - A probabilistic interpretation of Macdonald polynomials , invited speaker at the Combinatorial Representation Theory day at the Leibniz Universität Hannover, 18 February, 2011.
Abstract: P. Hanlon studied a random walk on partitions that has the Jack polynomials as eigenvectors. This random walk arises from a Markov chain on permutations by "lumping". In this work we generalise this process to a much more vigorous walk which has eigenvectors, the Macdonald polynomials. - Polytopes, shuffles, quivers and flags, invited speaker at the La troisième du séminaire de combinatoire énumérative et analytique at the Institut Henri Poincaré, Paris, 3 February, 2011.
Abstract: There are two geometries that show remarkable similarities: that of quiver varieties and that of affine flag varieties. By work of Braverman-Gaitsgory and Gaussent-Littelmann and Kashiwara-Saito and Kamnitzer-Baumann one sees the crystals, in the sense of Kashiwara, coming from both quivers and flags. In the picture of Leclerc-Geiss-Schroer one sees how elements of the shuffle algebra come from quiver varieties. In joint work with A. Ghitza and S. Kannan we are seeing shuffle elements coming from affine flag varieties. Following my recent joint work Ghitza and S. Kannan, I will explain the purely combinatorial approach for seeing the moment polytopes and the shuffle elements. - Combinatorics of the flag variety: Minicourse of three lectures: Chevalley groups and Hecke algebras,
Cohomology, and The Borel-Weil-Bott theorem, Hausdorff Insitute of Mathematics, special Trimester on "The interaction of geometry and combinatorics in Representation Theory", Winterschool, 10-14 January 2011.
Abstract: This was a review of the cohomology and K-theory of G/B, following a readign of the paper of Lam, Schilling and Shimozono, arXiv 0901.1506.
Talks of Arun Ram in 2010
- On the cohomology of G/B, Working seminar, University of Melbourne, December 15, 2010.
Abstract: This was a review of the cohomology and K-theory of G/B, following a readign of the paper of Lam, Schilling and Shimozono, arXiv 0901.1506. - Elliptic cohomology and Weyl character formulas, Invited speaker at the IGA/AMSI Workshop
"Dirac Operators in Geometry, Topology, Representation Theory, and Physics" at University of Adelaide October 18-22, 2010.
Abstract: In this work, joint with Nora Ganter, we establish an elliptic cohomology version of the Atiyah-Segal-Lefschetz fixed point formula and apply it to the flag variety of a compact Lie group. We make contact with the work of Looijenga on Root systems and Elliptic Curves and the work of Kac and Peterson on Affine Lie algebras and Modular Forms and obtain Weyl characters for the loop group as push forwards in elliptic cohomology. -
The Glass Bead Game, Colloquium,
University of Queensland, Brisbane, 11 October 2010.
Abstract: This title is taken from the novel of Hermann Hesse. In joint work with A. Kleshchev, we were amused to discover a glass bead game for constructing representations of quiver Hecke algebras (algebras recently defined by Khovanov-Lauda and Rouquier whose representation theory categorifies quantum groups of Kac-Moody Lie algebras). In fact, the glass bead game is tantalizingly simple, and may soon be marketed in your local toy store. I will explain how this game works, and some of the fascinating numerology that appears in the scoring of the plays. - What are KLRBMW algebras?,Invited speaker at the International Conference on the Non-Commutative Rings and Combinatorial Representation Theory" at Pondicherry University September 2-3, 2010.
Abstract: In 2008, Khovanov-Lauda and Rouquier defined a family of diagram algebras whose representations have the property that their
characters are elements of the quantum group. The characters of the simple modules of the KLR algebras are the canonical basis elements of the quantum group. In the type A case, the KLR algebras are a graded version of the affine Hecke algebra. This talk will be a survey,
with the question of the definition of the KLR Birman-Wenzl-Murakami and Brauer algebras, as the motivation. - Towards elliptic Chevalley groups and flag varieties,Seminar at Chennai Mathematical Institute, India, September 1, 2010.
Abstract: Assuming that the double affine Hecke algebra is a shadow of a double loop group or an "elliptic" Chevalley group we learn about the structure of the elliptic flag variety. Though we are not yet ready to make a proper definition we can see many properties of the object which should be the analogue of the flag variety for the elliptic case. This talk is based on joint work with Martha Yip on the combinatorics of the double affine Hecke algebra and with Nora Ganter on obtaining Weyl character formulas from elliptic cohomology using an elliptic cohomology analogue of the Atiyah-Segal Lefschetz fixed point formula and localization. - Musings towards elliptic buildings,Invited speaker at the ICM Satellite conference "Buildings, Finite Geometries and Groups" at the Indian Statistical Institute, Bangalore, India, during August 29 - 31, 2010.
Abstract: Assuming that the double affine Hecke algebra is a shadow of a double loop group or an "elliptic" Chevalley group we learn about the structure of the elliptic building. Though we are not yet ready to make a Tits style definition we can see many properties of the object which should be the analogue of the Tits building for the double loop and the elliptic cases. This talk is based on joint work with Martha Yip on the combinatorics of the double affine Hecke algebra and with Nora Ganter on obtaining Weyl character formulas from elliptic cohomology. - Symmetry and identities, Melbourne University Mathematics
Society (MUMS) seminar, University of Melbourne, 20 August 2010.
Abstract: I will explain some relationship between classical partition identities of Euler, Gauss and Jacobi are related to the symmetries of rigid polyhedra and the way that they fill up space. - On affine BMW algebras,Invited speaker at the International Conference on Representation Theory, Xian China August 9 -August 14, 2010.
- The Glass Bead Game, Colloquium, University of Adelaide, 25 June 2010.
Abstract: This title is taken from the novel of Hermann Hesse. In joint work with A. Kleshchev, we were amused to discover a glass bead game for constructing representations of quiver Hecke algebras (algebras recently defined by Khovanov-Lauda and Rouquier whose representation theory categorifies quantum groups of Kac-Moody Lie algebras). In fact, the glass bead game is tantalizingly simple, and may soon be marketed in your local toy store. I will explain how this game works, and some of the fascinating numerology that appears in the scoring of the plays. - Affine BMW algebras,
Pure Mathematics Seminar,
University of Adelaide, 25 June 2010.
Abstract: I will describe a family of algebras of tangles (which give rise to link invariants following the methods of Reshetikhin-Turaev and Jones) and describe some aspects of their structure and their representation theory. The main goal will be to explain how to use universal Verma modules for the symplectic group to compute the representation theory of affine BMW (Birman-Murakami-Wenzl) algebras. - Combinatorics and Spherical functions, invited talk at the BIRS Workshop 10w5096, Whittaker Functions, Crystal Bases, and Quantum Groups, Banff Canada
June 6-11, 2010.
- What is a line bundle?,
Informal Seminar,
University of Melbourne, 4 June 2010.
- Three examples when the space of global sections of line bundles
is interesting (Borel-Weil-Bott, modular forms, and toric varieties from polytopes),
Informal Seminar,
University of Melbourne, 2 June 2010.
- Lecture 1: Quantum groups and Lyndon words, Lecture 2: Graded quiver algerbas and their representations, Lecture 3: Indexings of canonical bases: Lyndon words, MV polytopes and the path model, Invited speaker at the 64th Séminaire Lotharingien de Combinatoire, Program, Institut Camille Jordan - Bâtiment Braconnier, Lyon, Sunday, March 28th, 2010 (evening) to Wednesday March 31st, 2010.
Talks of Arun Ram in 2009
-
The Glass Bead Game,
Short talk to the Vacation Scholars, University of Melbourne, 18 December 2009.
This was a short presentation of the bead game in recent work with A. Kleshchev on homogeneous representations of Quiver Hecke algebras. -
Moment maps on flag varieties and piecewise linear functions,
Seminar/Introduction, University of Melbourne, 26 November 2009.
We discuss the theorem of Borel-Bott-Weil and the Weyl character formula via localization. -
Introduction to equivariant cohomology,
Seminar/Introduction, University of Melbourne, 19 November 2009.
We will introduce foundational material on Brion-Vergne lattice point counting via the Jeffrey-Kirwan localization formula. -
Introduction to categories,
Seminar/Introduction, University of Melbourne, 12 November 2009.
We introduce the notions of chain complexes, categories, natural transformations, totalization, homotopy and derived categories. - Universal Verma modules and Translation, at the
53rd Annual meeting of the Australian Mathematical Society, Special session in Algebra and Number Theory,
University of South Australia, Adelaide 28 Spet.- 1 October, 2009.
Abstract: We will introduce a framework for studying the combinatorics of translation functors in a "universally integral" framework and explain a unified perspective on Gabber-Joseph's approach to the Kazhdan-Lusztig conjectures, Kleshchev and Brundan's approach to modular branching rules, and the Misra-Miwa Fock space. This talk is based on joint work with Peter Tingley. - Universal Verma modules and Translation, at the
International workshop on combinatorial and geometric approach to representation
theory,
Seoul National University, Korea, 21-24 September, 2009.
Abstract: We will introduce a framework for studying the combinatorics of translation functors in a "universally integral" framework and explain a unified perspective on Gabber-Joseph's approach to the Kazhdan-Lusztig conjectures, Kleshchev and Brundan's approach to modular branching rules, and the Misra-Miwa Fock space. This talk is based on joint work with Peter Tingley. - Why I care about p-compact groups, Reading seminar, University of Melbourne, 21 August 2009.
Abstract: A survey of symmetric functions, Schur functions, Weyl characters, the Borel-Weil-Bott theorem, the cohomology and K-theory of flag varieties, the classification of p-compact groups and the Clark-Ewing formula. - Poles, strings, braids and lattices, Colloquium, La Trobe University, 1 May 2009.
Abstract: The double affine braid group has important applications to Macdonald polynomials, group representations, mathematical physics and combinatorics. The classical type double affine braid groups have nice pictorial presentations which exhibit the tantalizing symmetries at play. In this talk I'll draw some of these pictures and explain their role in topology, harmonic analysis, combinatorics and the study of symmetry. - Lyndon Bases, "blackboard seminar", University of Melbourne, 31 March 2009.
Abstract: I will define Lyndon words and good Lyndon words and explain how we associate standard and simple quiver Hecke algebra modules to these words. I will not assume any memory of last week's talk. - Quiver Hecke alagebras, "blackboard seminar", University of Melbourne, 24 March 2009.
Abstract: Quiver Hecke algebras were recently defined by Khovanov-Lauda and, independently, by Rouquier. The importance of these algebras is that the category of graded modules for the quiver Hecke algebras is a categorfication of the Drinfeld-Jimbo quantum group. I will give a survey of this exciting new subject, perhaps highlighting some of our recent results joint with Kleshchev. - A path model formula for Macdonald polynomials,Séminaire sur les Algèbres Enveloppantes et Théorie des Représentations, Paris Jussieu, 6 March 2009.
Abstract: The path model of Littelmann provides a combinatorial formula for Weyl characters. In this talk we shall explain the generalization of the Littelmann formula to Macdonald polynomials. - A path model formula for Macdonald polynomials, Seminar Algebra and Topologie, University of Basel, 20 February 2009.
Abstract: The path model of Littelmann provides a combinatorial formula for Weyl characters. In this talk we shall explain the generalization of the Littelmann formula to Macdonald polynomials. - Two boundary Hecke algebras and tantalizer algebras,
Algebra seminar, Maxwell Institute for the mathematical sciences, University of Edinburgh, 17 February 2009.
Abstract: The double affine Hecke algebra (DAHA) of type C has special properties (6 parameters!) and distinguished quotients. The Macdonald polynomials for this Hecke algebra are the Koornwinder polynomials and the Askey-Wilson polynomials. One interesting quotient of the DAHA is the two boundary Temperley-Lieb algebra. The 2 boundary Temperley-Lieb algebra points the way to a family of centralizer algebras which includes the 2 boundary BMW (Birman-Murakami-Wenzl) algebras. This talk will be a medley of vignettes around double affine type C braid groups and quotient algebras. - Two boundary Hecke algebras and tantalizer algebras, Algebra seminar at Cambridge University, 28 January 2009.
Abstract: The double affine Hecke algebra (DAHA) of type C has special properties (6 parameters!) and distinguished quotients. The Macdonald polynomials for this Hecke algebra are the Koornwinder polynomials and the Askey-Wilson polynomials. One interesting quotient of the DAHA is the two boundary Temperley-Lieb algebra. The 2 boundary Temperley-Lieb algebra points the way to a family of centralizer algebras which includes the 2 boundary BMW (Birman-Murakami-Wenzl) algebras. This talk will be a medley of vignettes around double affine type C braid groups and quotient algebras. - Symmetry, Polynomials and quantisation Lecture1 Lecture2 Lecture3 Lecture4,
Minicourse of four lectures at the program Algebraic Lie Theory, Isaac Newton Institute, 12-23 January 2009.
Abstract: These talks will provide a pictorial approach to Weyl groups, braid groups and their Hecke algebras. With the pictures in hand, we can use them to study orthogonal polynomials, representations of braid groups, solutions of difference and differential equations, integrable systems and the quantisations that produce them.
Talks of Arun Ram in 2008
- Beads on runners, Invited talk at the special session Group actions and Representation Theory at the 7th Australia-New Zealand Mathematics Convention, Christchurch, New Zealand, 8-12 December 2008.
Abstract: Khovanov-Lauda algebras are a family of algebras whose representation theory provides a categorification of quantum groups. In this work we classify and construct homogeneous representations of these algebras. The construction generalises the construction of irreducible representations of the symmetric groups and the notions of partitions, skew shapes, and abaci. - The mysteries of symmetry, Colloquium, Australian National University, 20 November 2008.
Abstract: In recent joint work with Martha Yip we gave a combinatorial formula for Macdonald polynomials. The formula is a weighted sum of paths and the construction of the paths is completely elementary. The mystery is that these paths are describing subtle information about fancier objects: loop groups, integrable hierarchies of differential equations, representation theory and cohomology theories. I will try to formulate some of my speculations about how these objects are related. The underlying symmetry is certainly touching many parts of modern mathematics and it is all the more amazing that the elementary combinatorics of paths has something deep to say about it all. - Beads on runners, Colloquium, Monash University, 6 November 2008.
Abstract: We think of beads on runners like an abacus, or like one of those games for toddlers where the children slide the beads on the runners (these games are sometimes found in waiting rooms of the offices of pediatricians). In joint work with A. Kleshchev we have shown this is a perfect model for representations of Khovanov-Lauda algebras, the recently discovered algebras whose representations categorify quantum groups. I shall explain the bead and runner model and how to have your toddlers compute representations of Khovanov-Lauda algebras while waiting for the doctor at the medical centre. The model generalizes partitions and their classical connection to the symmetric group. At the end of the talk I will explain how these algebras are related to Lie algebras and quantum groups and why they are considered a great new advance in the art of "categorification". - Short lecture at the University of Melbourne/BHP Billiton School Mathematics competition, 11 October 2008.
Abstract: This was a 10 minute talk to school students -- maths competition winners. I told them that I went into mathematics for the lifestyle and pointed out the existence of a coffee shop/restaurant on the lakefront in Lugano on Lago Como in Swizerland. Then we looked at the wonderful Bratelli diagram on Tom Halverson's web page, and finally I told them that Persi Diaconis has a knack for finding uses of pure maths in other arenas and will be visiting Melbourne in 2010. - A combinatorial formula for Macdonald polynomials, Victorian Algebra Conference, RMIT Melbourne, 2-3 October 2008.
- Generalising Pascal's triangle, Melbourne University Mathematics and Statistics Society (MUMS), lunchtime seminar, 12 September 2008.
- Two boundary Hecke algebras and tantalizer algebras, Invited speaker at the International conference on
Combinatorics and Representation Theory, Graduate School of Mathematics,
Nagoya University, 1-5 September 2008.
Abstract: The double affine Hecke algebra (DAHA) of type C has special properties (6 parameters!) and distinguished quotients. One interesting quotient is the two boundary Temperley-Lieb algebra. The 2 boundary Temperley-Lieb algebra points the way to a family of centralizer algebras which includes the 2 boundary BMW (Birman-Murakami-Wenzl) algebras. This talk will survey this family of algebras. -
Introduction to Buildings,
Algebra-Geometry-Topology Discussion session, University of Melbourne, 21 August 2008.
Abstract: This is a brief introduction to buildings in hopes of strengthening the analogy to the curve complex and Fenchel-Nielsen coordinates. -
Mg,n and Fock space,
Algebra-Geometry-Topology Processing seminar, University of Melbourne, 18 August 2008.
Abstract: In his January lectures at MSRI, Okounkov outlined how to use partition combinatorics and Fock space to give formulas for the coefficients of the M_{g,n} volume polynomial. I shall try to explain this combinatorics and summarize the results of Okounkov-Pandharipande. -
Mg,n, counting and recursions,
Algebra-Geometry-Topology Processing seminar, University of Melbourne, 4 August 2008.
Abstract: I will attempt an elementarification/bigpicturification of the topic of Paul Norbury's talk last week. -
Introduction to the Path Model,
Seminar/Introduction, University of Melbourne, 2 July 2008.
About the path model: I mean the model of P. Littelmann which generalises Dyck paths to give combinatorial models for representations of compact Lie groups. I am interested in using it to determine the polytope whose integer points describe the places where Littlewood-Richardson coefficients (also called Clebsh-Gordon or tensor product coefficients) are nonzero. - Type C Hecke and Temperley-Lieb algebras, Seminar/Introduction, University of Melbourne, 27 June 2008.
-
A combinatorial formula for Macdonald polynomials,
Stanford Combinatorics and Geometry seminar,
Stanford University, April 30, 2008.
Abstract: We will explain a common generalization of Littelmann's formula for Weyl characters and Schwer's formula for spherical functions for a p-adic group. These formulas hold for arbitrary Lie type. -
Symmetric functions d'après Macdonald,
Keynote presentation at the 4th annual
Graduate Student
Combinatorics Conference, UC Davis, April 12-13, 2008.
Abstract: This talk will be a road map to Macdonald's classic book on Symmetric functions, highlighting the combinatorics, representation theory and geometry coded by symmetric function identities. -
Path Models
(pdf
notes), invited talk at
Topics in Combinatorial Representation Theory,
MSRI, Berkeley, March 17-21, 2008.
Abstract: Recent years have seen big developments in the theory and applications of path models. The new applications are in understanding the combinatorics of the affine Hecke algebra, spherical functions, and the geometry of points in affine flag varieties. This talk will survey some of these recent results. -
Tantalizer algebras,
Colloquium
University of Utah, March 6, 2008.
Abstract: Abstract: Tantalizer is short for tensor power centralizer. These algebras often come as algebras of diagrams or of tangles, and so working with them requires drawing lots of pictures. Their structure and representation theory contains and immense amount of information about the representation theory of groups and quantum groups of types GL, SO, Sp, and they can be used to construct corresponding link polynomials and 3-manifold invariants. This talk will be a survey of some recent developments in tantalizer algebras. -
Path Models,
Representation Theory seminar,
University of Utah, March 7, 2008.
Abstract: This talk will be a survey of applications of path models: The Weyl character formula, Schubert calculus, spherical functions, normal forms in Chevalley groups, and indexing of points in affine flag varieties and Mirkovic-Vilonen cycles. -
Tantalizers, invited talk at
University of California Lie Theory Workshop,
a conference in honor of Georgia Benkart,
University of California, San Diego, February 16-18, 2008.
Abstract: A tensor power centralizer algebra (tantalizer) is the algebra of commuting operators for a Lie group or quantum group action on tensor space. The favourite examples are the group algebra of the symmetric group and the Brauer algebra. This talk will survey some recent work on tantalizers: giving definitions and recent results for affine and graded BMW algebras and some two boundary tantalizers. - Minicourse: Combinatorics of Lie Type, three lectures at the Introductory Workshop on Combinatorial Representation Theory at MSRI, January 22-25, 2008.
Talks of Arun Ram in 2007
- Combinatorial Representation Theory 2008-2018,
Colloquium,
University of Minnesota, November 15, 2007.
Abstract: The 1997 survey article of Barcelo-Ram entitled Combinatorial Representation Theory “defined” the field and set out its structure. In 2007 this field is thriving and vibrant. In Spring 2008 there will be a full semester program at MSRI entitled Combinatorial Representation Theory. Where is the field now? What has happened in the interim 1997-2007? More importantly, what will happen in Combinatorial Representation Theory in 2008-2018? - Generalizing partitions and standard tableaux,
combinatorics seminar, University of Minnesota, November 15, 2007.
Abstract: The irreducible representations of the symmetric group are indexed by partitions and bases of these representations are indexed by standard tableaux. The representation theory of the affine Hecke algebras provides a generalization of partitions and standard tableaux. I will explain these combinatorial indexings and how they arise. - Two row partitions and the Temperley-Lieb algebra, Combinatorics seminar, University of Wisconsin, Madison, October 15, 2007.
Abstract: Following a good idea of V. Rittenberg, two boundary diagram algebras are getting more and more attention, with two boundary Temperley-Lieb algebras being a fundamental example. This talk will begin to answer the question: Which Type C affine Hecke algebra representations are two boundary Temperley-Lieb representations and what is a good combinatorial set for indexing these representations? - Boundary diagram algebras,
Representation theory seminar, University of Wisconsin, Madison,
October 12, 2007.
This talk was a repeat of a talk given at University of Koln on 19 November 2005. - Centers of tantalizers, Representation theory seminar, University of Wisconsin, Madison, September 14, 2007.
Abstract: Many diagram algebras arise as tantalizers. The Schur-Weyl duality makes it possible to steal most of the center of the tantalizer from the corresponding dual object in the duality. I will outline this process and explain how combinatorial results pop out of the picture. This talk is based on joint work with Zajj Daugherty and Rahbar Virk. -
Today I feel like a mathematician - personality, music and geometry, The 21st Behrend Memorial Lecture, a public lecture at the University of Melbourne, August 21, 2007.
Abstract: What does it feel like to be a mathematician? Who are the people who discovered and proved the Weil conjectures (one of the great human achievements of the 20th century)? Are they artists, musicians, or scientists? So, what does it feel like to be a mathematician, really? - Combinatorics in affine flag varieties, 6 July 2007; invited talk
at GL07, Geometry and Lie Theory, a
conference in honor of Gus Lehrer's 60th birthday,
University of Sydney, July 2-6 and July 9-13, 2007.
Abstract: This talk is about the combinatorics of indexing points in affine flag varieties. It is possible to make choices so that the points are indexed by a refinement of Littelmann's path model in such a way that the Schubert cell and the Mirkovic-Vilonen slice are easily read off the "path" indexing of the point. From this, the relations for the affine Hecke algebra can be derived, both in the Iwahori-Matsumoto and in the Bernstein generators. If time permits I will discuss the action of the "root operators" on points, and/or the relation to the Kamnitzer and Baumann-Gaussent indexings of Mirkovic-Vilonen cycles. - What is a Weyl group?, Summer Representation Theory seminar, University of Wisconsin-Madison, 14 June 2007.
- Level l Fock spaces and the polynomial representation of Cherednik's double affine Hecke algebra,American Institute of Mathematics workshop: Arithmetic harmonic analysis on character and quiver varieties, American Institute of Mathematics, Palo Alto, June 4-8, 2007.
- Introduction to Buildings and Combinatorial Representation Theory, American Institute of Mathematics workshop on Buildings and Combinatorial Representation Theory, Palo Alto, March 26, 2007.
- Introduction to moment maps on flag varieties, Lie Theory seminar, University of Wisconsin, Madison, 21 March, 2007.
Talks of Arun Ram in 2006
- Plenary lecture at the 2006 Fall American Mathematical Society Southeastern Section Meeting, University of Arkansas, Fayetteville, Arkansas, 3-4 November 2006.
- Lecture at the workshop Modern Math: An Introduction to 2007-08 Programs at MSRI, at the Society for the Advancement of Chicanos and Native Americans in Science , National Conference, Tampa, Florida, October 24-25, 2006.
- Combinatorial Hopf algebras: An outsider's survey, Minicourse at the conference Hopf algebras, Combinatorics and Quantum field Theory, Max-Planck Institute for Mathematics in the Sciences, Leipzig, Germany, 25-28 September 2006.
- Path models and Chevalley groups, Oberwolfach meeting on
Finite groups and representation theory, March 25-31, 2006.
- Representations and translation, Special lecture in Quantum groups course, KdV Institut, Amsterdam, March 22, 2006.
- Row reduction and loop groups, Lie Theory seminar, University of Wisconsin, Madison, February 28, 2006.
- The Schur Hopf algebra, Combinatorics seminar, University of Wisconsin, Madison, February 27, 2006.
- Alcove walks and reductive groups over local fields, Lie group and Representation Theory seminar,
University of Maryland, February 3, 2006.
In this talk I presented the precise combinatorial construction of the generalized MV cycles by labeled alcove walks. - Examples of groups: Lecture 1 -
Reflection groups and braid groups,
Lecture 2 -
Matrix groups and Lie groups,
Special minicourse,
University of Rome "La Sapienza",
January 23-24, 2006.
These lectures were for advanced undergraduates in mathematics in Italy: introductory lectures on braid groups, reflection groups, matrix groups and Lie groups. - Hecke algebras and spherical functions,
Harmonic analysis seminar, University of Rome "La Sapienza",
January 18, 2006.
Talks of Arun Ram in 2005
-
Boundary diagram algebras,
Seminar on Transformation groups & mathematical physics,
a joint seminar of the Universities of Koln, Hamburg, Bochum,
Bremen and Darmstadt, University of Koln, November 19, 2005.
This talk is about diagram algebras which come from the two-boundary braid group (braids with two poles). This is a generalization of recent work (from statistical mechanics) on two-boundary Temperley-Lieb algebras. The generalized setting naturally includes two boundary Hecke algebras and two-boundary BMW algebras. These algebras are like affine Hecke algebras (of type A) and affine BMW algebras except with two poles. - p-compact groups,
Oberseminar Geometrie, University of Fribourg,
Switzerland, October 26, 2005
This talk was the first time I realised that applying Milnor's construction of the classifying space to the p-compact groups of Clark-Ewing gives the space where path models (such as the Littelmann path models) live. - Random walks, spherical functions and representations, Colloquium, University of Fribourg, Switzerland, October 25, 2005.
- p-compact groups,
Algebra seminar, University of Rome "La Sapienza",
October 20, 2005.
This talk was my first attempt to learn something about p-compact groups. - Diagram algebras as tantalizers,
Colloquium, University of Rome "Tor Vergata",
October 19, 2005.
This talk was the first time I realized and defined the graded version of the group algebra of the affine braid group which has, as quotients, the graded BMW algebras (also called cyclotomic Nazarov-Wenzl algebras), and the graded Hecke algebras. - Representations of affine Hecke algebras,
Algebra seminar, University of Rome "La Sapienza",
October 13, 2005.
This talk was a survey on the representations of affine Hecke algebras. - Alcove walks and Iwahori cosets Algebra seminar, University of Rome "La Sapienza",
October 6, 2005.
This talk was where I first worked out the generalization of the MV cycles to G/I, in the example SL_2. In other words, coset representatives for the cosets in U^+vI\cap IwI, coset representatives for the cosets in U^+vI\cap IwI, where I is an Iwahori, and v and w are elements of the affine Weyl group. - Picturing Hecke algebras and loop groups,
Algebra seminar, University of Rome "La Sapienza",
September 29, 2005.
This talk was an attempt to explain the alcove walk method of looking at affine Hecke algebras and loop groups. - Random walks, spherical functions and representations,
Colloquium, University of Stuttgart, July 4, 2005.
- q-crystals , Invited speaker in the special session in honor of Adriano Garsia at the conference Formal Power Series and Algebraic Combinatorics 2005, June 24-25, 2005.
- Commuting elements in diagram algebras, Algebra seminar, University of Wuppertal, June 7, 2005.
- Verma crystals, Algebra seminar, University of Lyon 1, May 27, 2005.
- Random walks, spherical functions and representations, Colloquium, University of Lyon 1, May 26, 2005.
- Walks, crystals and polytopes, Algebra seminar, University of Caen, May 24, 2005.
- Random walks, spherical functions and representations, Colloquium, University of Freiburg, May 13, 2005.
- Murphy elements in diagram algebras, Plenary speaker at the conference Cellular and diagram algebras and their applications in mathematics and physics, University of Leicester, England, April 3-10, 2005.
- Combinatorial Representation Theory, Colloquium, Max-Planck-Institut fur Mathematik, March 24, 2005.
- Combiantorial Representation theory II-Crystals, Talk 2 of a lecture series at University of Zaragoza, Spain, February 24, 2005.
- Combinatorial Representation theory I-Towers and Centralizers, Talk 1 of a lecture series at University of Zaragoza, Spain, February 22, 2005.
- Representations of Reflection groups, Seminar, Bernoulli Centre, EPFL, Lausanne,
January 19, 2005.
Talks of Arun Ram in 2004
Talks of Arun Ram in 2003
Talks of Arun Ram in 2002
- Plenary speaker for the 46th Annual conference of the Australian Mathematical Society, Newcastle, Australia, September 30-October 3, 2002.
- Plenary speaker for the Quantum groups day at the XXIV International Colloquium on Group Theortical Methods in Physics, Paris, France, July 15-20, 2002.
- Geometry seminar at University of Wuppertal, Wuppertal, Germany, July 2, 2002.
- Invited speaker at the conference on Computational Lie theory, at the Centre Recherches Mathematiques of the University of Montreal, May 27-June 7, 2002.
- Colloquium, University of Massachusetts, Amherst, March 28, 2002.
- Plenary speaker, two one hour lectures, at the Mid-Atlantic Algebra Conference, Wake Forest University, March 16-17, 2002.
Talks of Arun Ram in 2001
- February 1, 2001, Symmetric functions seminar, Isaac Newton Institute, Cambridge, England.
- February 16, 2001, Colloquium, University of Southampton, England.
- February 21, 2001, Algebra seminar, Cambridge University, England.
- February 26, 2001, Speaker, Isaac Newton Insitute Colloquium for General Scientific Audience.
- March 8, 2001, Colloquium, City University, London, England.
- March 21, 2001, Colloquium, University of Birmingham, England.
- March 25-31, 2001, Invited speaker at the conference, ``Representations of Finite groups'' Oberwolfach, Germany.
- April 30, 2001, Algebra seminar, University of Copenhagen, Denmark.
- May 2, 2001, Algebra semianr, University of Aarhus, Denmark.
- May 11, 2001, Colloquium, University of Warwick, England.
- May 21, 2001, Algebra seminar, University of Leicester.
- May 23, 2001, Algebra seminar, University of Glasgow, Scotland.
- May 31, 2001, Seminaire Chevalley, Institut Henri Poincare, Paris, France.
- May 31, 2001, Colloquium, Ecole Normale Superieur, Paris, France.
- June 2-5, 2001, Invited plenary speaker at ``The Heritage of Schur's 1901 dissertation: a conference in honor of J.A. Green''.
- June 12, 2001, Algebra seminar, University of Oxford.
- June 12, 2001, Representation theory seminar, University of Oxford.
- June 18, 2001, Speaker, Symmetric functions seminar, Isaac Newton Institute, Cambridge, England.
- October 6-7, 2001, Invited speaker at the conference ``Midwest Lie algebras and Related Topics'' conference, DePaul University.
- October 22-26, 2001, Invited speaker at the conference ``Combinatorial and Geometric Representation Theory'', Seoul, Korea.
- December 7-16, 2001, Invited speaker at the conference ``Algebra and Geometry'', University of Hyderabad, India.
- December 16-22, 2001, Invited speaker at the special year on ``Recent results and conjectures on Hilbert functions'', IIT Bombay, India.
Talks of Arun Ram in 2000
- December 1, 2000, Colloquium, University of Wisconsin--Milwaukee.
- November 22-23, 2000, Series of two talks, algebra seminar, Instituto de Matematica, UNAM, Morelia,
- November 21, 2000, Colloquium, Instituto de Matematica, UNAM, Morelia, Mexico.
- October 16-20, 2000, Invited speaker at ``Algebres de Hecke affines et groupes reductifs'', Luminy-Marseille, France.
- Algebra seminar, University of Sydney, Sydney Australia, August 11, 2000.
- Colloquium, Macquarie University, Sydney Australia, August 7, 2000.
- Algebra seminar, Mathmatisches Institut B, Universität Stuttgart, June 8, 2000.
- Algebra seminar, University of Strasbourg, France, June 7, 2000.
- Basel-Freiburg-Strasbourg joint Algebraic Groups Seminar, May 30, 2000.
- Combinatorics seminar, University of Michigan, Ann Arbor, March 31, 2000.
- Algebra seminar, University of Oregon, Eugene, March 7, 2000.
- Colloquium, University of Oregon, Eugene, March 6, 2000.
Talks of Arun Ram in 1999
- Invited speaker at the KIAS Lie Theory Conference at the Korea Institute for Advanced Study (KIAS), Seoul, Korea, October 5-8, 1999.
- Invited speaker at the conference Quantum groups and knot theory, at L'Institut de Recherche Mathématique Avancée, Strasbourg, France, September 27-29, 1999.
- Invited lecturer at Seoul National University Math Camp, Chunan, Korea, June 21, 1999.
- Invited lecture series (Minicourse on Hecke algebra representations) at the Korea Institute for Advanced Study (KIAS), Seoul, Korea, June 15-17, 1999.
- Tableaux, hyperplanes and representations, plenary talk at the 11th Conference on Formal Power Series and Algebraic Combinatorics, Barcelona, June 7-11, 1999.
- Tableaux, hyperplanes and representations, Colloquium, Center for Communications Research, Princeton, May 18, 1999.
- Affine braids, quantum groups, and Jantzen filtrations, Invited speaker at the special session on Representations of Lie algebras at the American Math. Society meeting, Buffalo, NY, April 24-25, 1999.
- Lie Theory seminar, MIT, April 21, 1999.
- Combinatorics seminar, MIT, April 19, 1999.
- Algebra and Geometry seminar, Stanford University, April 8, 1999.
- Lie group seminar, Rutgers University, March 5, 1999.
- Tableaux, hyperplanes and constructing representations, a lecture series at the Institute of Advanced Study, February 1999.