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Talks Arun Ram |
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Department of Mathematics and Statistics University of Melbourne Parkville VIC 3010 Australia aram@unimelb.edu.au |
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Department of Mathematics University of Wisconsin-Madison 480 Lincoln Drive Madison, WI 53706 ram@math.wisc.edu |
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 this material is based upon work supported by the National Science Foundation under Grant No. 0353038. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.
Talks of Arun Ram in 2010
- Invited speaker at the ICM Satellite conference "Buildings, Finite Geometries and Groups" at the Indian Statistical Institute, Bangalore, India, during August 29 - 31, 2010.
- Invited speaker at the International Conference on Representation Theory, Xian China
August 9 -August 14, 2010.
- Invited participant at the BIRS Workshop 10w5096, Whittaker Functions, Crystal Bases, and Quantum Groups, Banff Canada
June 6-11, 2010.
- Lyndon words, MV polytopes and Macdonald polynomials, Invited speaker at the 64th Séminaire Lotharingien de Combinatoire, 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.