*On November 9, 2020, the Simons Center for Geometry and Physics (SCGP) celebrated its 10th anniversary with a special symposium and luncheon at the Center. The SCGP was officially founded in 2008 and the physical building’s inauguration was on November 9, 2010. Thus, this special commemoration marked ten years from the opening of the brick-and-mortar building. It was a memorable day with insightful reflections on the scientific and historical events that led to the creation of a Center for advancement at the nexus of geometry and theoretical physics.*

The decennial celebration took place in the Della Pietra Family Auditorium as well as in the Center’s atrium and gallery. Moderated by Luis Álvarez-Gaumé, Director of the SCGP, it was in tune with our times as a hybrid event with some guests in attendance at the Center and most participants attending via Zoom. Álvarez-Gaumé introduced Stony Brook University President Maurie McInnis, and Simons Foundation President Marilyn Simons, both providing tributes to the University and the SCGP. President McInnis spoke about collaboration as one of Stony Brook’s defining characteristics and honored the Center’s success in bringing geometry and physics together. She stated: “Collaboration is deep in the core of what we do—from our interdisciplinary perspective to our incredibly special relationship with our community, to our integrated, cutting-edge research. We want to be on the forefront of breakthrough scientific discovery. We want to not only respond, but help shape and define the world around us. […]The Simons Center for Geometry and Physics is a shining example of our success in doing just that.”

Marilyn Simons spoke next and commended the Simons Center for “bringing together the greatest minds in mathematics and physics to expand our understanding of the fundamental nature of our universe. The beauty of the ideas here are so well reflected in the beauty of the building itself; the architectural design, the umbilic torus, the iconic wall—all over, it’s stunning.” She graciously praised the Center for engaing visitors “not only with its art, but its newsletter and its events, and people are invited here […]to come, and to and contemplate the creativity and the wonder of the human mind.” Accolades and candid remarks were also provided by the Chair of the SCGP Board of Trustees Yakov Eliashberg (Stanford) and by Cumrun Vafa (Harvard), former Chair of the Board.

Historical and scientific talks followed remarking upon the seeds planted for the Center long ago in the many collaborative and intellectual exchanges that took place over the decades between Stony Brook’s mathematicians and theoretical physicists. A thorough review of the highlights of mathematics research at the SCGP was presented by John Morgan, founding Director of the SCGP. Nathan Seiberg (IAS) provided an outside perspective highlighting the theoretical physics research at the SCGP. Martin Roček (YITP), one of the founders and organizer of the Simons Summer Workshops, talked about the Workshops’ history and role in the creation and development of the Center. And Alexander (Sasha) Abanov, SCGP Deputy Director, spoke about the history and structure of the SCGP’s multitude of programs and workshops.

Nigel Hitchin (Oxford), an emeritus member of the SCGP Board of Trustees, reminisced about the SCGP’s official opening and its early planning stages. Regarding the role of Research Assistant Professors, he recollected how it was agreed to create “an atmosphere of talking and listening, benefiting from both sides and sets of intuition,” as this is where ideas between mathematics and physics come together. “This is an important role the Center plays.” He also mentioned the life of Galileo, who was forced to quarantine for weeks due to the plague of the 1630s before being put on trial for heresy by the Inquisition. We are reminded of the COVID pandemic of 2020.

George Sterman (Director of the YITP and a member of the SCGP Board of Trustees) discussed the role of the Center as a research institute, comparing it to “an ivory tower with open discussions and open doors,” which, after 10 years, “under the guidance of John, and of Luis, is among Stony Brook’s signature contributions to scientific research.”

Scientific talks were given by the four SCGP permanent faculty members: Simon Donaldson (mathematics), Kenji Fukaya (mathematics), Zohar Komargodski (physics), and Nikita Nekrasov (physics). Other distinctive lectures were provided by Juan Maldacena (IAS), and Andrei Okounkov (Columbia), both members of the SCGP Board of Trustees. Their outlook talks presented exciting discoveries and the leading role the Simons Center plays in geometry and physics that evidences a promising future.

Luis Álvarez-Gaumé concluded the celebration by announcing the Center’s many plans for the next ten years. “A decade from now, we will have a better perspective to assess the achievement fostered by the SCGP. I am quite certain we will be looking back in awe and grateful for the success story of this collaboration between the Simons Center and Stony Brook University.”

Jim Simons’ closing remarks succinctly summed up a life’s experience for the realization of a Center that fuses mathematics, physics, and all the permeations thereof, for a space where discovery and imagination ignite: “When I came to Stony Brook to be the Chair of the Math Department…I began to realize that physics, or at least a lot of physics, and mathematics, were inextricably bound together…and could flourish, even better perhaps, if there were a place where mathematicians and physicists were working together.”

*The Simons Center for Geometry and Physics was founded with a generous gift from Jim and Marilyn Simons. For more on the Center’s founding and history, please, see SCGP News Volume XV: Special 10th Anniversary Edition, pages 34–47.*

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]]>*Since 1978, the Wolf Foundation awards the acclaimed, international Wolf Prize to outstanding scientists and artists from around the world. In 2020, the Wolf Prize in Mathematics was awarded jointly to Yakov Eliashberg (Stanford) and Simon Donaldson (SCGP). In this article, Kenji Fukaya shares his thoughts on the awardees esteemed contributions to differential geometry and topology.*

As one of the founders of global symplectic geometry, Yakov Eliashberg has been a leader in this field for the past 40 years. The field of symplectic geometry studies mechanics (Hamiltonian mechanics) from the geometric point of view. This area itself is very classical, but recently people found that non-trivial global symplectic geometry also exists. Importantly, among many different kinds of geometries, global geometry is rarely non-trivial

The existence of global symplectic geometry was established via a certain remarkable result by Eliashberg. Soon after this was determined, Mikhael Gromov introduced the method of pseudo-holomorphic curve. Since then, global symplectic geometry has been one of the most active areas of mathematics.

Eliashberg also obtained many basic results in contact geometry—an odd dimensional cousin of symplectic geometry. Together with Helmut Hofer and Alexander Givental, Eliashberg also proposed Symplectic Field Theory, which associates highly sophisticated algebraic structure to a contact manifold. This conjecture is now being established by mathematicians of a younger generation such as Bao-Honda, Pardon and Ishikawa.

Eliashberg’s recent research is on the flexibility of symplectic geometry. The existence of global symplectic geometry means that certain objects cannot exist in a symplectic world. The flexibility means that certain objects do exist in a symplectic world. It is ideal that we know exactly the borderline between a rigid world and a flexible world. For a long time this was a dream far from reality. New work by Eliashberg (together with his younger collaborators such as Murphy) shows that the world of symplectic geometry is more flexible than we thought before. So now we are closer to the dream.

Sir Simon Donaldson is one of the most accomplished researchers of geometry in the last 50 years. When he was just in his 20s, Donaldson found an astonishing result on 4-dimensional topology. Topologists had used linear differential equations for a long time in their research. Donaldson found that by using a certain non-linear differential equation—Yang-Mills equation—we can prove that 4-dimensional Euclidean space has an ‘exotic’ smooth structure. When this result appeared many topologists wondered whether this was a sporadic result or not. Donaldson then solved many fundamental open questions on differential topology of 4-manifolds by using Yang-Mills equation. Since then Gauge theory (such as Yang-Mills equation) plays the dominant role in the study of differential topology of 4-dimensional manifo

Donaldson also gave a fundamental contribution to symplectic geometry. Gromov discovered that one can use a complex analytic map from complex one-dimensional space to a symplectic manifold as a tool to study symplectic manifolds. On the other hand, complex analytic function on a symplectic manifold does not exist generically. Donaldson nevertheless showed we can still study symplectic manifolds by using ‘almost’ complex analytic function.

Donaldson’s recent important achievement focuses on the existence of Einstein metric on complex manifolds. One of the most important results in geometric analysis (by Yau) is the existence of Kähler metric with Ricci curvature 0 (that is, a solution of Einstein’s equation of gravity) under a certain easily checkable assumption. There is a similar result by Audin and Yau in the case of negative Ricci curvature. The case of positive Ricci curvature has long been a standing open question. Donaldson, together with his collaborators Sun and Chen, finally solved this problem.

*Congratulations to Yakov Eliashberg and Simon Donaldson on their ground-breaking contributions in geometry and topology.*