SIAGA: A New Window for Algebra and Geometry

By Bernd Sturmfels

The SIAM Journal on Applied Algebra and Geometry (SIAGA) is the latest member in the outstanding family of journals published by SIAM. SIAM News readers are already familiar with the “storyboard” behind the new journal, thanks to Anna Seigal’s inspiring three-part article from last year, “A SIAGA of Seven Pictures.” 

SIAGA offers a new home for exciting themes in the core of mathematics and their emerging applications. The journal’s creation was the result of a thorough planning process that dates back several years. Its impetus came from the SIAM Activity Group on Algebraic Geometry (SIAG/AG), which held its inaugural conference in Raleigh, NC, in October 2011. Following the second meeting in Fort Collins, CO, in August 2013, SIAG/AG members formed a committee to discuss the possibility of a new journal. The committee wrote a proposal under the leadership of Frank Sottile and Thorsten Theobald, and with strong support from the publications committee and SIAM’s board and council, the journal received its final approval in December 2015.

SIAGA’s mission is to publish “research articles of exceptional quality on the development of algebraic, geometric, and topological methods with strong connection to applications.” The journal covers mathematical subjects such as algebraic geometry, algebraic topology, algebraic and topological combinatorics, differential geometry, convex and discrete geometry, commutative and noncommutative algebra, multilinear and tensor algebra, number theory, representation theory, and symbolic and numerical computation. Areas of application include biology, data science, coding theory, complexity theory, computer graphics, computer vision, control theory, cryptography, machine learning, game theory and economics, geometric design, optimization, quantum computing, robotics, statistics, and social choice.

Douglas Arnold, former president of SIAM and director of the Institute for Mathematics and its Applications, played a decisive role by supporting these developments. “Just a decade ago, algebra and geometry would have seemed strange directions for SIAM, and the title of the journal something of an oxymoron,” he writes. “But now, after the formation of the algebraic geometry activity group in 2009, the establishment of a biennial conference series, and the resulting influx of people, ideas, and interaction, applied algebra and geometry have become core areas for SIAM. This signals a change in scientific culture for which SIAM has been an important catalyst. The new SIAM Journal on Applied Algebra and Geometry is both a recognition and a natural outcome of this change. It’s great for math and science, and it’s great for SIAM.”

Jan Draisma, former chair of SIAG/AG and an associate editor of SIAGA, also attributes the journal’s birth to the two expanding disciplines. “The SIAM activity group in algebraic geometry unites the rapidly growing communities of algebraists and geometers fascinated by applications and scientists in need of new algebro-geometric techniques,” he says. “SIAGA is quickly becoming the journal of choice for the very best of their combined research.”

The new journal began taking submissions in March 2016. A team of three corresponding editors and 26 associate editors is handling the growing number of submissions in a timely and professional manner. Many referees are contributing excellent reports, ensuring high standards for acceptance. Sottile, inaugural chair of SIAG/AG and a corresponding editor for SIAGA, emphasizes the journal’s influence. “This journal, because of its focus and editorial board, can get quality reports for papers that are interdisciplinary and require refereeing from two or more perspectives,” he says.

By the end of 2016, authors had submitted 90 manuscripts for publication to SIAGA, and the journal was ready for its inaugural all-electronic volume. The first nine articles were published in February 2017. This initial batch touches upon a wide range of subjects, such as complexity theory, convex optimization, frame theory, graphical models, machine learning, numerical analysis, projective geometry, signal processing, and tensor methods. A focus on models governed by nonlinear algebraic constraints remains a common thread. “We are overjoyed to see so many high-quality submissions, especially from our junior colleagues,” says Alicia Dickenstein, vice president of the International Mathematical Union and a SIAGA corresponding editor.

To increase SIAGA readership well beyond the current community, the editorial board reaches out to everyone interested in studying nonlinear problems that arise in the aforementioned application areas. We hope that the new approaches and research presented in the journal will be of interest to many scientists, engineers, and industrial mathematicians.

Now we will briefly introduce the authors and articles featured in the first volume of SIAGA. The article “On the geometry of border rank decompositions for matrix multiplication and other tensors with symmetry,” by Joseph M. Landsberg and Mateusz Michalek, offers a new approach to tensors with symmetry, with focus on complexity lower bounds for matrix multiplication. Michael Kech and Felix Krahmer advance our understanding of inverse problems by deriving “Optimal injectivity conditions for bilinear inverse problems with applications to identifiability of deconvolution problems.” Jameson Cahill, Dustin Mixon, and Nate Strawn resolve a longstanding problem in applied harmonic analysis with “Connectivity and irreducibility of algebraic varieties of finite unit norm tight frames.” Diego Cifuentes and Pablo Parrilo achieve dramatic speed-ups when solving algebraic equations by using “Chordal networks of polynomial ideals.” Peter Bürgisser connects numerics and algebraic geometry in an article titled “Condition of intersecting a projective variety with a varying linear subspace.” In “On Fano schemes of toric varieties,” Nathan Ilten and Alexandre Zontine show how to solve binomial equations in terms of linear forms.

“The geometry of rank-one tensor completion” reveals the work of Thomas Kahle, Kaie Kubjas, Mario Kummer, and Zvi Rosen. In a mathematical contribution to deep learning, titled “Dimension of marginals of Kronecker product models,” Guido Montufar and Jason Morton prove that restricted Boltzmann machines are identifiable. Greg Blekherman, Rainer Sinn, and Mauricio Velasco advance convex geometry and polynomial optimization by addressing the intriguing question, “Do sums of squares dream of free resolutions?”

As the inaugural issue approached publication, the SIAGA team was also busy with articles to appear in the near future. For instance, next in line is an article on mathematical neuroscience that grew out of a 2014 Mathematics Research Communities program. Carina Curto, Elizabeth Gross, Jack Jeffries, Katherine Morrison, Mohamed Omar, Zvi Rosen, Anne Shiu, and Nora Youngs answer the question, “What makes a neural code convex?”

We are still hoping to branch out further and attract truly outstanding submissions from a wider range of communities. For instance, we’d like to receive more articles from fields such as applied topology, cryptography, geometric modeling, differential geometry, and mathematical biology. As the journal approaches its steady state, the future looks bright. SIAGA will serve as a window for first-rate research that transcends the historic division of mathematics into “pure” and “applied.” In the immediate future, it welcomes outward-looking authors and all readers with a taste for algebra, geometry, and topology.

Bernd Sturmfels is a professor of mathematics, statistics, and computer science at the University of California, Berkeley. Starting in summer 2017, he will be  director of the Max Planck Institute for Mathematics in the Sciences in Leipzig, Germany. He serves as the editor-in-chief of SIAGA.