Title of proposal

Subfactors and fusion categories

Organizer information

David Penneys, University of Toronto, dpenneys@math.toronto.edu (contact organizer)

Scott Morrison, The Australian National University, scott@tqft.net

Emily Peters, Northwestern University, eep@math.northwestern.edu

Noah Snyder, Indiana University, nsnyder@gmail.com

Vaughan Jones, Vanderbilt University, vaughan.f.jones@vanderbilt.edu

Other details

Subject area: "Operator theory / algebras"

18D10, 46L37, 16T05, 57R56


Overview

This workshop will bring together experts in the overlapping fields of subfactors and fusion categories.  Although the former has its original home in operator algebras and the latter in representation theory, the two subjects turn out to be very closely related.  This workshop will further collaboration between experts in the two subjects.  Fusion categories generalize finite groups, and the long term goal is to understand fusion categories as well as we understand finite groups.  We hope that a BIRS meeting will provide a focal point for the classification project, building on strong recent activity in conferences at AIM, Vanderbilt and Maui, much as a series of conferences in the last half of the 20th century did for the classification of finite simple groups.

Fusion categories are a natural generalization of the representation categories of finite groups.  Further examples include representation categories of finite-dimensional semisimple quasi-Hopf algebras, the subcategory of N-N bimodules which are submodules of a tensor power of M for a subfactor N of M (where both are type II_1 factors), and idempotent completions of tangle categories associated with quantum link-invariants (at roots of unity).  As the diversity of constructions indicates, fusion categories are of interest in several distinct research areas: representation theory, operator algebras and quantum topology.  A major recent impetus for attacking the classification of fusion categories is the application to topological phases and quantum computation.  

We have written to invite 42 mathematicians whose participation in this conference will make it a great success.  We’ve already had enthusiastic replies from 41 of them. In fact, 20 of these replies came within the first few hours, reflecting the strong interest in holding a workshop on this subject!  We've received positive responses from Fields medallist Michael Freedman and Zhenghan Wang, both from Microsoft station Q, who work on topological quantum computing (where fusion categories have remarkable practical applications).  Others, like Fields medallist Vaughan Jones (Vanderbilt), Alice Guionnet (MIT), and Dmitri Shlyakhtenko (UCLA), have recently made exciting connections between subfactors and free probability theory.   Still others, like David Evans (Cardiff University), Terry Gannon (University of Alberta), and Hans Wenzl (UCSD), bring together the study of quantum groups and subfactors.  We’ve also received enthusiastic responses from experts on deformation rigidity for II_1 factors including Sorin Popa (UCLA) and Adrian Ioana (UCSD). There will also be specialists in quantum field theory and Hopf algebras.  Though many of our participants are established mathematicians with long and diverse bodies of work, we’ve also made sure to include younger mathematicians, such as United States NSF fellows Stephen Curran and David Jordan.

Particular examples of fusion categories have been studied since the 80's. Only in the past decade, however, has a coherent theory of fusion categories been developed.  There has recently been great progress in this direction coming from a purely algebraic point of view, for example in the work of Etingof, Nikshych and Ostrik (Nikshych and Ostrik will be attending).  On the other hand, since the early 90’s most of the interesting new examples of fusion categories have come from subfactors and von Neumann algebras (via constructions due to Izumi, Haagerup, Asaeda and others).  These interesting constructions are poorly understood theoretically.  In order to make progress it is critical that there be direct communication between the two fields, so that the theoretical study of fusion categories can be broadened to give a better understand of the key examples and ideas coming from subfactors.  A conference at BIRS would give a boost to this interchange of ideas at a critical time for these fields.

We're confident the proposed workshop will lead to new collaborations and exchange of ideas across sometimes separate mathematical fields. The workshop will bring together experts from fusion categories, modular tensor categories, subfactors and von Neumann algebras. (In particular, we have positive responses from some of the top researchers in each of these fields!) In recent years strong interactions between these subjects have been discovered, and we anticipate bringing together a cross-disciplinary conference on these topics will result in new discoveries (applying ideas from one field to another) as well as new collaborations, allowing separate expertises to be applied to the hardest problems in the field.  There is active research in these topics taking place in Canada, the U.S., Europe, Israel, Japan, and Australia, so this workshop could play a special role in promoting collaborations which would otherwise not begin for geographical reasons.

Statement of objectives

Before turning to the major problems in fusion categories and subfactors, let us first mention a few results about finite groups which we would like to emulate.  The most famous is the classification of finite simple groups.  Another classic problem in finite groups is the classification of multiply transitive group actions.  The third result is more modern, the McKay correspondence and the classification of groups with a 2-dimensional representation (or more generally, the finite subgroups of another compact Lie group).

Note that in all of these cases the classification results come hand-in-hand with interesting examples: the sporadic finite simple groups, doubly transitive Frobenius groups, and the Platonic solid groups.  We anticipate that in the study of fusion categories and subfactors the discovery of new examples should be closely related to classification programs.  Indeed, most of the new examples of subfactors and fusion categories are “multiply transitive” in an appropriate sense, or come from analogues of classifying the finite subgroups of Lie groups.

A close analogue of a finite group with a representation of small dimension (as in the McKay correspondence) is a fusion category with an object of small dimension or a subfactor of small index.  In fact, that classification of subfactors of index less than 4 has a McKay-like ADE classification, proved in the 80s.  One major program in subfactor theory has been to push this classification to higher indices.  This is analogous to finding finite subgroups of other small Lie groups, for example the higher Platonic solid groups in dimension 4.   The current state of the art is a complete classification up to index 5.  This classification has yielded three subfactors which were not known by other means (the Haagerup, Asaeda-Haagerup, and extended Haagerup subfactors).  These examples are all constructed by direct calculation, using either the technology of planar algebras or connections for bimodules. One major goal is finding a natural theoretical home for these exotic examples.  There has been recent progress in this direction in work of Izumi, Evans-Gannon, and Grossman-Snyder.

One analogue of transitivity for group actions is supertransitivity of subfactors.  Recall that the only 6- or more transitive group actions are the symmetric and alternating families, the only 4- or 5- transitive group actions are the Mathieu groups, and there are several interesting families of 2- and 3- transitive group actions.  A fusion category is supertransitive if there is an object whose small tensor powers are as simple as possible. We have few highly transitive examples with dimension greater than 2. The exotic Asaeda-Haagerup and extended Haagerup subfactors have supertransitivity 6 and 8 respectively, and examples of higher supertransitivity are not known.

Over the last year, there has been significant progress studying "near-group" subfactors and fusion categories. A near-group fusion category has one more object than a finite group representation category. (There is also a related class where there is a finite group sub-category constituting half the simple objects.) Izumi has given a prescription for constructing near-group categories.  The construction requires solving certain polynomial equations whose variables are indexed by a finite group, and he gave explicit solutions for 3-supertransitive subfactors based on Z_3 (which is the Haagerup subfactor), Z_4, Z_2xZ_2, and Z_5. Recently, Evans and Gannon found exact and numeric solutions for more groups, and suggested that the Haagerup subfactor thus lies in an infinite family. They also deal with a different class of 2-supertransitive near-group examples, and explicitly produce 40 new subfactors!

One of the major theoretical deficits in our current understanding of fusion categories is the lack of a good extension theory. Recent beautiful results of Etingof, Nikshych and Ostrik give a homotopy-theoretic approach to finding graded extensions; that is, categories graded by a finite group with a fixed fusion category in the trivial grading. The general case, however, remains very poorly understood and is a major goal in the field. Similarly, we do not yet have a good notion of a fusion category being "simple". The near-group examples described above, and others recently constructed by Izumi, appear to be excellent test cases --- they certainly look like they should count as extensions of simpler categories, and the challenge is to find a framework which explains this.

One of the key tools for studying a group is understanding its subgroups.  One analogue in subfactor theory is the study of intermediate subfactors or lattices thereof.  This has been an active area of research in subfactors, with a paper of Bisch-Jones in Inventiones and another of Grossman-Jones in JAMS.  Nonetheless these ideas have not been explored in detail on the fusion category side.  One unexplained and tantalizing observation of Grossman-Izumi is that the Haagerup and Asaeda-Haagerup subfactors appear naturally in certain classes of quadrilaterals of subfactors.

Another interpretation of “subgroup” is a module category over a fusion category.  This leads to the maximal atlas, and Brauer-Picard groupoid.  Etingof-Nikshych-Ostrik use this to give a homotopy-theoretic description of some extension problems in fusion categories. Grossman-Snyder explore the maximal atlas of the Haagerup and Asaeda-Haagerup subfactors, but there is much more work to be done in this direction.

As an example of the interactions between the ideas above, at a recent conference it was suggested that an Izumi near-group subfactor based on the group Z_2xZ_4 might provide the crucial missing case of Grossman and Snyder's recent analysis of the maximal atlas of the highly supertransitive Asaeda-Haagerup subfactor. If this could be achieved it would provide a uniform explanation for what is currently one of the most exotic objects in subfactor theory.

A significant remaining piece of work in the subject is to extend the current classification results, in several directions. We would really like to have the classification of small index subfactors extended out to index 3+\sqrt{5}, and potentially even to 6. Similarly, there are very good prospects for classifying fusion categories with small global dimension (ambitiously out to 60); inconclusive previous attempts in this direction already produced a number of potential new examples. The case of modular categories is especially interesting from the point of view of topological quantum computing; the classification is understood up to rank 4, and it seems likely that we can go further. There are powerful number theoretic approaches for understanding integral modular tensor categories, for example by restricting the prime factors of the global dimension.

The proposed subjects, subfactors and fusion categories, allow many fruitful interactions with other subjects.  Three of the most prominent are number theory, free probability and conformal field theory.  The main connection with number theory is that dimensions of objects in fusion categories (and thus the indices of subfactors) must be algebraic integers in cyclotomic fields.  This result has been surprisingly useful in the study of subfactors.  Ng has made some exciting progress recently, using the rotation operator in planar algebras to give an alternative proof of this result.  Free probability and subfactors have been brought into contact recently via the work of Guionnet-Jones-Shlyakhtenko.  These ideas have not yet spread much to the tensor category community, although free products already play an important role there.  Finally, many known examples of fusion categories can be enriched to give conformal field theories.  Since the subject of conformal field theories is so rich it will be important to discover whether all unitary fusion categories are related to conformal field theories.  Recent work of Evans-Gannon gives tantalizing clues that the Haagerup subfactor could come from a conformal field theory, while work of Kawahigashi-Longo has proved classification results for small conformal field theories analogous to the small index subfactor classification.

Press release

Symmetry has long played a key role in mathematics and the sciences.  Classically, the symmetries of an object form a group.  In the past few decades it has become clear that in certain “quantum” settings the notion of group doesn’t capture all possible symmetries.  One very early place that these more general symmetries were observed was in the study of von Neumann factors.  These quantum analogues of finite groups are called fusion categories.  It turns out that these kinds of quantum symmetries turn up in other places in mathematics, like knot theory and representation theory.  Surprisingly, fusion categories also crop up in solid state physics, where they govern the behavior of certain exotic materials which may be useful in quantum computing.  The aim of this workshop is to bring together experts working on subfactors and fusion categories.  With both of these fields developing in new and interesting directions, our goal is to build stronger connections and bridge the gaps between the languages and techniques used by the two communities.  We will work towards gaining as rich an understanding of these finite quantum groups as we have of ordinary finite groups.

Dates

The BIRS workshop schedule is created using a Monte-Carlo algorithm that satisfies the greatest number of preferred workshop dates, given the constraints of impossible dates and 48 workshops with often competing preferred dates. In the interest of fairness, you must select at least 3 preferred dates, and can have no more than 5 preferred dates and 2 impossible dates.

Please use the following form to select your preferred dates, in the order of most preferred to least preferred.

Please provide at least one off-season (outside of the 2nd week of May to the end of August) alternative.

To continue to the final step, press the "Next step" button at the bottom of the page.

From the invite:

We realize that 2014 is still a long way off. Ideally we would hold the workshop in the early summer, and otherwise sometime during the spring (we will have to be flexible to fit the Banff calendar)

(Yasu requests not the week following Feb 26, and not containing April 1)

Participants

This is just a tentative list. BIRS encourages organizers to contact proposed participants about their interest ahead of time, and to indicate having done this in the proposal. We should also explicitly address inviting women and minorities to the extent we can.

A full workshop is 42 people, a half workshop is 21 people.

Indicated interest:

Hagge, Tobias, The University of Texas at Dallas -- Mathematics Department

Snyder, Noah, Indiana University -- Mathematics Department

Peters, Emily, Northwestern University -- Mathematics Department

Morrison, Scott, Australian National University -- Mathematics Department

Jones, Vaughan, Vanderbilt University -- Mathematics Department

Penneys, David, University of Toronto -- Mathematics Department

Wang, Zhenghan, Microsoft Station Q

Kashina, Yevgenia, DePaul University -- Mathematics Department

Plavnik, Julia, Universidad Nacional de Córdoba --  Facultad de Matemática, Astronomía y Física

Bisch, Dietmar, Vanderbilt University -- Mathematics Department

Tener, James, UC Berkeley -- Mathematics Department

Nikshych, Dmitri, University of New Hampshire -- Mathematics Department

Kawahigashi, Yasu, University of Tokyo -- Mathematics Department

Guionnet, Alice, MIT -- Mathematics Department

Rowell, Eric, Texas A&M University -- Mathematics Department

Gelaki, Shlomo, Technion - IIT -- Mathematics Department

Peterson, Jesse, Vanderbilt University -- Mathematics Department

Shlyakhtenko, Dima, UCLA -- Mathematics Department

Hartglass, Michael, UC Berkeley -- Mathematics Department

Brothier, Arnaud, KU Leuven -- Mathematics Department

Evans, David, Cardiff University -- Mathematics Department

Galindo-Martinez, Cesar, Universidad de los Andes -- Mathematics Department

Jordan, David, UT Austin -- Mathematics Department

Curran, Stephen, UCLA -- Mathematics Department

Naidu, Deepak, Northern Illinois University -- Mathematics Department

Grossman, Pinhas, University of New South Wales -- Mathematics Department

Natale, Sonia, Universidad Nacional de Córdoba --  Facultad de Matemática, Astronomía y Física

Davydov, Alexei, University of New Hampshire -- Mathematics Department

Izumi, Masaki, Kyoto University -- Mathematics Department

Hong, Seung-Moon, University of Toledo -- Mathematics Department

Gannon, Terry, University of Alberta -- Mathematics Department

Walker, Kevin, Microsoft Station Q

Wenzl, Hans, UC San Diego -- Mathematics Department

Jenkins, Evan, University of Chicago -- Mathematics Department

Ng, Richard, Iowa State University -- Mathematics Department

Bruillard, Paul, Texas A&M University -- Mathematics Department

Bigelow, Stephen, UCSB -- Mathematics Department

Ioana, Adrian, UCSD -- Mathematics Department

Freedman, Michael, Microsoft Station Q

Liu, Zhengwei, Vanderbilt University -- Mathematics Department

Popa, Sorin, UCLA -- Mathematics Department

Xu, Feng, UC Riverside -- Mathematics Department

Tobias Hagge (The University of Texas at Dallas) <hagge@utdallas.edu>

Noah Snyder (Columbia University) <nsnyder@gmail.com>

Emily Peters (Northwestern University) <eep@math.mit.edu>

Scott Morrison (Australian National University) <scott@tqft.net>

Vaughan Jones (Vanderbilt University) <vfr@math.berkeley.edu>

David Penneys (University of Toronto) <dpenneys@math.berkeley.edu>

Zhenghan Wang (Microsoft Station Q) <zhenghwa@microsoft.com>

Yevgenia Kashina (DePaul University) <ykashina@depaul.edu>

Julia Plavnik (FaMAF - National University of Córdoba) <juliaplavnik@gmail.com>

Dietmar Bisch (Vanderbilt University) <dietmar.bisch@vanderbilt.edu>

James Tener (UC Berkeley) <jtener@math.berkeley.edu>

Dmitri Nikshych (University of New Hampshire) <nikshych@math.unh.edu>

Yasu Kawahigashi (University of Tokyo) <yasuyuki@ms.u-tokyo.ac.jp>

Alice Guionnet (MIT) <Alice.Guionnet@ens-lyon.fr>

Eric Rowell (Texas A&M University) <rowell@math.tamu.edu>

Shlomo Gelaki (Technion - IIT) <gelaki@fermat.technion.ac.il>

Jesse Peterson (Vanderbilt University) <jesse.d.peterson@vanderbilt.edu>

Dima Shlyakhtenko (UCLA) <shlyakht@math.ucla.edu>

Michael Hartglass (UC Berkeley) <mhartgla@math.berkeley.edu>

Arnaud Brothier (KU Leuven) <Arnaud.Brothier@wis.kuleuven.be>

David Evans (Cardiff University) <EvansDE@cf.ac.uk>

Cesar Galindo Martinez (Universidad de los Andes) <cesarneyit@gmail.com>

David Jordan (UT Austin) <djordan@math.mit.edu>

Stephen Curran (UCLA) <curransr@math.ucla.edu>

Deepak Naidu (Northern Illinois University) <dnaidu@math.niu.edu>

Pinhas Grossman (University of New South Wales) <pinhas@gmail.com>

Sonia Natale (Universidad Nacional de Córdoba. CIEM-CONICET) <natale@famaf.unc.edu.ar>

Alexei Davydov (University of New Hampshire) <alexei1davydov@gmail.com>

Masaki Izumi (Kyoto University) <izumi@math.kyoto-u.ac.jp>

Seung-Moon Hong (University of Toledo) <SeungMoon.Hong@utoledo.edu>

Terry Gannon (University of Alberta) <tgannon@math.ualberta.ca>

Kevin Walker (Microsoft Station Q) <kevin@canyon23.net>

Hans Wenzl (UC San Diego) <hwenzl@ucsd.edu>

Evan Jenkins (University of Chicago) <ejenkins@math.uchicago.edu>

Richard Ng (Iowa State University) <rng@iastate.edu>

Paul Bruillard (Texas A&M University) <pjb2357@gmail.com>

Stephen Bigelow (UCSB) <bigelow@math.ucsb.edu>

Adrian Ioana (UCSD) <aioana@math.ucsd.edu>

Michael Freedman (Microsoft Station Q) <michaelf@microsoft.com>

Zhengwei Liu (Vanderbilt University) <zhengwei.liu@vanderbilt.edu>

Sorin Popa (UCLA) <popa@math.ucla.edu>

Feng Xu (UC Riverside) <xufeng@math.ucr.edu>

Outstanding invites:

Liang Chang (UC Santa Barbara) <liangchang@math.ucsb.edu>

Uffe Haagerup (University of Copenhagen) <haagerup@math.ku.dk>

Michael Müger (Radboud Universiteit Nijmegen IMAPP) <mueger@math.ru.nl>

Marta Asaeda (UC Riverside) <marta@math.ucr.edu>

Victor Ostrik (University of Oregon) <vostrik@darkwing.uoregon.edu>

People to invite:

Brianna Riepel (working with Dmitri Nikshych), briepel@wildcats.unh.edu

Palcoux Sebastien <sebastienpalcoux@yahoo.fr>

Invitation email:

Dear everyone,

We are hoping to propose a workshop at Banff <http://www.birs.ca/> on subfactors and fusion categories, for 2014. As part of the proposal, they would like to have an indication that the intended participants are actually interested in coming.

We realize that 2014 is still a long way off. Ideally we would hold the workshop in the early summer, and otherwise sometime during the spring (we will have to be flexible to fit the Banff calendar). It would be very helpful for our proposal if you could reply briefly saying that you would be interested in attending such a workshop!

best regards,

Vaughan Jones, Scott Morrison, David Penneys, Emily Peters and Noah Snyder.

To solicit grad student recommendations:

Yevgenia Kashina (DePaul University) <ykashina@depaul.edu>

Dmitri Nikshych (University of New Hampshire) <nikshych@math.unh.edu>

Alice Guionnet <Alice.Guionnet@ens-lyon.fr>

Eric Rowell (Texas A&M University) <rowell@math.tamu.edu>

Shlomo Gelaki (Technion - IIT) <gelaki@fermat.technion.ac.il>

Dima Shlyakhtenko (UCLA) <shlyakht@math.ucla.edu>

Deepak Naidu (Northern Illinois University) <dnaidu@math.niu.edu>

Sonia Natale (Universidad Nacional de Córdoba. CIEM-CONICET) <natale@famaf.unc.edu.ar>

Terry Gannon <tgannon@math.ualberta.ca>

Richard Ng (Iowa State University) <rng@iastate.edu>

Paul Bruillard (Texas A&M University) <pjb2357@gmail.com>

Stephen Bigelow <bigelow@math.ucsb.edu>

Dear everyone,

Thanks for your enthusiastic response to our email about a workshop on subfactors and fusion categories at Banff!  BIRS encourages participation in their conferences of mathematicians who are women and/or ethnic minorities, and we'd like to be sure we haven’t missed anyone who might be interested, but who we might not know personally.  If you have any suggestions of graduate students or postdocs who we should invite, please let us know.

best regards,

Vaughan Jones, Scott Morrison, David Penneys, Emily Peters and Noah Snyder.

Invites in format required by Banff:

Wang, Zhenghan, zhenghwa@microsoft.com, Microsoft Station Q, M

Plavnik, Julia, juliaplavnik@gmail.com, FaMAF - National University of Córdoba, F

Jordan, David, djordan@math.mit.edu, UT Austin, M

Grossman, Pinhas, pinhas@gmail.com, University of New South Wales, M

Riepel, Brianna, <briepel@wildcats.unh.edu>, University of New Hampshire, F

Palcoux, Sebastien, <sebastienpalcoux@yahoo.fr>, Institut de Mathématiques de Luminy, M

Bisch, Dietmar, dietmar.bisch@vanderbilt.edu, Vanderbilt University, M

Tener, James, jtener@math.berkeley.edu, UC Berkeley, M

Nikshych, Dmitri, nikshych@math.unh.edu, University of New Hampshire, M

Kawahigashi, Yasuyuki, yasuyuki@ms.u-tokyo.ac.jp, University of Tokyo, M

Guionnet, Alice, Alice.Guionnet@ens-lyon.fr, MIT, F

Rowell, Eric, rowell@math.tamu.edu, Texas A&M University, M

Gelaki, Shlomo, gelaki@fermat.technion.ac.il, Technion - IIT, M

Peterson, Jesse, jesse.d.peterson@vanderbilt.edu, Vanderbilt University, M

Shlyakhtenko, Dima, shlyakht@math.ucla.edu, UCLA, M

Hartglass, Michael, mhartgla@math.berkeley.edu, UC Berkeley, M

Brothier, Arnaud, Arnaud.Brothier@wis.kuleuven.be, KU Leuven, M

Evans, David, EvansDE@cf.ac.uk, Cardiff University, M

Curran, Stephen, curransr@math.ucla.edu, UCLA, M

Naidu, Deepak, dnaidu@math.niu.edu, Northern Illinois University, M

Natale, Sonia, natale@famaf.unc.edu.ar, Universidad Nacional de Córdoba. CIEM-CONICET, F

Davydov, Alexei, alexei1davydov@gmail.com, University of New Hampshire, M

Izumi, Masaki, izumi@math.kyoto-u.ac.jp, Kyoto University, M

Gannon, Terry, tgannon@math.ualberta.ca, University of Alberta, M

Walker, Kevin, kevin@canyon23.net, Microsoft Station Q, M

Wenzl, Hans, hwenzl@ucsd.edu, UC San Diego, M

Jenkins, Evan, ejenkins@math.uchicago.edu, University of Chicago, M

Ng, Richard, rng@iastate.edu, Iowa State University, M

Bigelow, Stephen, bigelow@math.ucsb.edu, UCSB, M

Ioana, Adrian, aioana@math.ucsd.edu, UCSD, M

Freedman, Michael, michaelf@microsoft.com, Microsoft Station Q, M

Liu, Zhengwei, zhengwei.liu@vanderbilt.edu, Vanderbilt University, M

Popa, Sorin, popa@math.ucla.edu, UCLA, M

Xu, Feng, xufeng@math.ucr.edu, UC Riverside, M

Haagerup, Uffe, haagerup@math.ku.dk, University of Copenhagen, M

Müger, Michael, mueger@math.ru.nl, Radboud Universiteit Nijmegen IMAPP, M

Ostrik, Victor, vostrik@darkwing.uoregon.edu, University of Oregon, M

backup invites:

Kashina, Yevgenia, ykashina@depaul.edu, DePaul University, F, Backup

Hagge, Tobias, hagge@utdallas.edu, The University of Texas at Dallas, M, Backup

Asaeda, Marta, marta@math.ucr.edu, UC Riverside, F, Backup

Hong, Seung-Moon, SeungMoon.Hong@utoledo.edu, University of Toledo, M, Backup

Galindo, César, cesarneyit@gmail.com, Universidad de los Andes, M, Backup

Bruillard, Paul, pjb2357@gmail.com, Texas A&M University, M, Backup

Chang, Liang, liangchang@math.ucsb.edu, UC Santa Barbara, M, Backup