First time director Antonio Campos's downbeat tale of teenage alienation in which a public school pupil confronts death in the digital age. High school loner Robert (Ezra Miller) spends his spare time surfing the net for hardcore pornography and random clips of unrelated items that appeal to him. Given a digital video camera to record footage for an audiovisual class, Robert happens to be present when two popular girl students accidentally die from a drugs overdose. With the school in mourning, Robert is given the job of producing the school's official memorial video. But as he becomes immersed in his task, he soon finds himself becoming even more alienated from those around him.

Worlds collide in The Flash when Barry Allen uses his superpowers to travel back in time in order to change the events of the past. But when his attempt to save his family inadvertently alters the future, Barry becomes trapped in a reality in which General Zod has returned, threatening annihilation, and there are no Super Heroes to turn to.

That is, unless Barry can coax a very different Batman out of retirement and rescue an imprisoned Kryptonian… albeit not the one he’s looking for.

Ultimately, to save the world that he is in and return to the future that he knows, Barry’s only hope is to race for his life. But will making the ultimate sacrifice be enough to reset the universe?

This volume contains research papers and surveys reflecting the topics discussed at the EMS Summer School on Multigraded Algebra and Applications held in Romania in August 2016. The school, which served as the 24th National School on Algebra, presented the main research directions of combinatorial commutative algebra with a strong focus on its applications in combinatorics, statistics, and biology. Recent progress in the field has led to new insights and suggested algebraic techniques for solving real-world data analysis problems. The summer school and resulting proceedings volume have raised numerous novel questions and encouraged a more interdisciplinary approach for young researchers when considering problems in pure and applied mathematical research. Featured topics in this volume include toric rings, binomial edge ideals, Betti numbers for numerical semigroup rings, and Waldschmidt constants. Researchers and graduate students interested in the developments of the field will find this book useful for their studies.

This volume contains research papers and surveys reflecting the topics discussed at the EMS Summer School on Multigraded Algebra and Applications held in Romania in August 2016. The school, which served as the 24th National School on Algebra, presented the main research directions of combinatorial commutative algebra with a strong focus on its applications in combinatorics, statistics, and biology. Recent progress in the field has led to new insights and suggested algebraic techniques for solving real-world data analysis problems. The summer school and resulting proceedings volume have raised numerous novel questions and encouraged a more interdisciplinary approach for young researchers when considering problems in pure and applied mathematical research. Featured topics in this volume include toric rings, binomial edge ideals, Betti numbers for numerical semigroup rings, and Waldschmidt constants. Researchers and graduate students interested in the developments of the field will find this book useful for their studies.

This book is aimed to provide an introduction to local cohomology which takes cognizance of the breadth of its interactions with other areas of mathematics. It covers topics such as the number of defining equations of algebraic sets, connectedness properties of algebraic sets, connections to sheaf cohomology and to de Rham cohomology, Grobner bases in the commutative setting as well as for $D$-modules, the Frobenius morphism and characteristic $p$ methods, finiteness properties of local cohomology modules, semigroup rings and polyhedral geometry, and hypergeometric systems arising from semigroups. The book begins with basic notions in geometry, sheaf theory, and homological algebra leading to the definition and basic properties of local cohomology. Then it develops the theory in a number of different directions, and draws connections with topology, geometry, combinatorics, and algorithmic aspects of the subject.

Recent developments are covered

Contains over 100 figures and 250 exercises

Includes complete proofs

Recent developments are covered

Contains over 100 figures and 250 exercises

Includes complete proofs