"Operator Algebras, Quantum Information Theory and the Grothendieck Program"

I will discuss recently emerged interconnections between operator algebras and quantum information theory, leading to reformulations of one of the most important open questions in operator algebras: the Connes embedding problem. The lectures are based on joint work with Uffe Haagerup, and are dedicated to his memory.

Date | Time | Room | Title |
---|---|---|---|

Tuesday, April 5, 2016 |
5:00-6:00pm | HLMS 199 |
Classical and Quantum Correlations, Operator Algebras, and Asymptotic Behaviour of Quantum ChannelsIn 1980, Tsirelson showed that Bell’s inequalities—that have played an important role in distinguishing classical correlations from quantum ones, and that were used to test, and ultimately disprove the Einstein-Podolski-Rosen postulate of “hidden variables", coincide with Grothendieck’s famous inequalities from functional analysis. Tsirelson further studied sets of quantum correlations arising under two different assumptions of commutativity of observables. While he showed that they are the same in the finite dimensional case, the equality of these sets was later proven to be equivalent to the most famous still open question in operator algebras theory: the Connes embedding problem. In recent joint work with Haagerup, we establish a different reformulation of the Connes embedding problem in terms of an asymptotic property of quantum channels possessing a certain factorizability property (that originates in operator algebras). Several concrete examples will be discussed. Following Tuesday's lecture, there will be a reception in honor of Professor Musat at the Koenig Alumni Center, 1202 University Avenue (the SE corner of Broadway and University). |

Wednesday, April 6, 2016 |
5:00-6:00pm | BESC 185 |
Grothendieck Inequalities — From Classical to NoncommutativeThe highlight of Grothendieck’s celebrated “Résumé", published in 1956, is a highly nontrivial factorization result for bounded bilinear forms on C(K1) C(K2), where K1 and K2 are compact sets, which is now referred to as the Grothendieck Theorem (or, Grothendieck Inequality). The “Résumé" contains several equivalent formulations of it, all describing fundamental relationships between Hilbert spaces (e.g., L2), and the Banach spaces L1, respectively, C(K), and L1. It ends with a remarkable list of six problems, one of which is the conjecture that an analogue factorization for bounded bilinear forms on the product of (noncommutative) C -algebras holds. This was later proven by Pisier (under an approximability assumption), and by Haagerup (in full generality). I will survey Grothendieck’s Inequalities, from classical to noncommutative, including extensions to the setting of completely bounded bilinear forms on C -algebras and operator spaces, due to Pisier-Shlyakhtenko, and joint work of Haagerup and myself. I will also point out connections with quantum information theory and reformulations of the Connes embedding problem. |

## Magdalena Musat |
Magdalena Musat is an Associate Professor in Mathematics at the University of Copenhagen, Denmark. A native of Romania, she earned her undergraduate degree from the University of Bucharest in 1993, and her Ph.D. from the University of Illinois at Urbana-Champaign in 2002, under the joint supervision of Donald Burkholder and Marius Junge. Before her appointment in 2009 at the University of Copenhagen, she was an S.E.Warschawski Visiting Assistant Professor at the University of California, San Diego, an Assistant Professor at the University of Memphis, and a Lektor (Associate Professor) at the Southern Denmark University (SDU). Musat’s mathematical interests lie at the interface between operator algebras, noncommutative probability (in particular, Lp-theory of noncommutative martingales), and some of the analytic, geometric and probabilistic aspects of group theory related to operator algebras. A postdoctoral position at SDU in 2006 marked the beginning of a fruitful collaboration with Uffe Haagerup. Musat has regularly been an invited speaker at international conferences and workshops, and enjoys sharing her enthusiasm for mathematics with colleagues and her many students. She is the organizer of the Distinguished Harald Bohr Lecture Series in Mathematics at the University of Copenhagen. |

This Lecture Series is funded by an endowment given by Professor Ira M. DeLong, who came to the University of Colorado in 1888 at the age of 33. Professor DeLong essentially became the mathematics department by teaching not only the college subjects but also the preparatory mathematics courses. Professor DeLong was a prominent citizen of the community of Boulder as well as president of the Mercantile Bank and Trust Company, organizer of the Colorado Education Association, and president of the charter convention that gave Boulder the city manager form of government in 1917. After his death in 1942 it was decided that the bequest he made to the mathematics department would accumulate interest until income became available to fund DeLong prizes for undergraduates and DeLong Lectureships to bring outstanding mathematicians to campus each year. The first DeLong Lectures were delivered in the 1962-63 academic year.

1962-1963 Paul Halmos

1963-1964 Marshall Hall Jr.

1964-1965 Edwin Hewitt

1965-1966 George Polya

1966-1967 Alfred Tarski

1967-1968 John Milnor

1968-1969 Paul Cohen

1969-1970 Jurgen Moser

1970-1971 Mark Kac, Irving Kaplansky

1971-1972 Abraham Robinson

1972-1973 George Mackey

1973-1974 Olga Taussky Todd

1974-1975 Andrew Gleason

1975-1976 Tosio Kato

1976-1977 Hugh Montgomery

1977-1978 Elias Stein

1978-1979 Raoul Bott

1979-1980 Alan Weinstein

1980-1981 Enrico Bombieri

1981-1982 Richard S. Varga

1963-1964 Marshall Hall Jr.

1964-1965 Edwin Hewitt

1965-1966 George Polya

1966-1967 Alfred Tarski

1967-1968 John Milnor

1968-1969 Paul Cohen

1969-1970 Jurgen Moser

1970-1971 Mark Kac, Irving Kaplansky

1971-1972 Abraham Robinson

1972-1973 George Mackey

1973-1974 Olga Taussky Todd

1974-1975 Andrew Gleason

1975-1976 Tosio Kato

1976-1977 Hugh Montgomery

1977-1978 Elias Stein

1978-1979 Raoul Bott

1979-1980 Alan Weinstein

1980-1981 Enrico Bombieri

1981-1982 Richard S. Varga

1982-1983 Charles Fefferman

1983-1984 S.S. Chern

1984-1985 Robert Zimmer

1985-1986 Gerd Faltings

1986-1987 Dennis Sullivan

1987-1988 Stephen Smale

1988-1989 Branko Grunbaum

1989-1990 Ronald Graham

1990-1991 Kenneth Ribet

1991-1992 Michael Atiyah

1992-1993 John H. Conway

1993-1994 John Tate

1994-1995 Vladimir Arnold

1996-1997 Alain Connes

1997-1998 Barry Mazur

1999-2000 Nigel Higson

2000-2001 Jeff Cheeger

2001-2002 Vaughan F. R. Jones

2002-2003 Richard Taylor

2003-2004 Phillip A. Griffiths

1983-1984 S.S. Chern

1984-1985 Robert Zimmer

1985-1986 Gerd Faltings

1986-1987 Dennis Sullivan

1987-1988 Stephen Smale

1988-1989 Branko Grunbaum

1989-1990 Ronald Graham

1990-1991 Kenneth Ribet

1991-1992 Michael Atiyah

1992-1993 John H. Conway

1993-1994 John Tate

1994-1995 Vladimir Arnold

1996-1997 Alain Connes

1997-1998 Barry Mazur

1999-2000 Nigel Higson

2000-2001 Jeff Cheeger

2001-2002 Vaughan F. R. Jones

2002-2003 Richard Taylor

2003-2004 Phillip A. Griffiths

2004-2005 Paul Baum

2005-2006 Isadore M. Singer

2006-2007 Sir Roger Penrose

2007-2008 Maxim Kontsevich

2008-2009 Persi Diaconis

2009-2010 Ieke Moerdijk

2010-2011 Endre Szemerédi

2011-2012 Vitaly Bergelson

2012-2013 Yuval Peres

2013-2014 Benedict H. Gross

2014-2015 Robert Bryant

2015-2016 Magdalena Musat

2017-2018 Michael J. Hopkins

2005-2006 Isadore M. Singer

2006-2007 Sir Roger Penrose

2007-2008 Maxim Kontsevich

2008-2009 Persi Diaconis

2009-2010 Ieke Moerdijk

2010-2011 Endre Szemerédi

2011-2012 Vitaly Bergelson

2012-2013 Yuval Peres

2013-2014 Benedict H. Gross

2014-2015 Robert Bryant

2015-2016 Magdalena Musat

2017-2018 Michael J. Hopkins

If you have any questions concerning this lecture series, please contact
Mathematics.