Algorithmic Advance: the Group Isomorphism Problem

Anonymous — Tue, 23 Jan 2024 07:00:00 +0000

Algorithmic Advance: the Group Isomorphism Problem Anonymous (not verified) Tue, 01/23/2024 - 00:00

Assistant Professor Joshua Grochow provides insight into a breakthrough theorem in group theory.

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Computer Scientists Inch Closer to Major Algorithmic Goal

Anonymous — Fri, 23 Jun 2023 06:00:00 +0000

Computer Scientists Inch Closer to Major Algorithmic Goal Anonymous (not verified) Fri, 06/23/2023 - 00:00

Assistant Professor Joshua Grochow helps provide insight into a new, faster method for determining when two mathematical groups are the same.

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How Big Data Carried Graph Theory Into New Dimensions

Anonymous — Thu, 19 Aug 2021 06:00:00 +0000

How Big Data Carried Graph Theory Into New Dimensions Anonymous (not verified) Thu, 08/19/2021 - 00:00

Computer science Assistant Professor Joshua Grochow and others explain how big data has benefited graph theory.

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Tensors: Connecting the dots of everything

Anonymous — Mon, 17 May 2021 19:32:57 +0000

Tensors: Connecting the dots of everything Anonymous (not verified) Mon, 05/17/2021 - 13:32

Grace Wilson

Joshua Grochow

Picture a suspension bridge. The river erodes the pillars, semi-trucks rumble across, blasts of gale-force wind whip between the cables. Through it all, the bridge must stand strong.

If we can't see the whole collection of forces acting on a bridge's supports, we could overload its limits and jeopardize the safety of all who use it. This same complexity applies to intricate chemical reactions, realistic physics simulations, even multi-company mergers and convoluted patent chains.

When different forces cannot be condensed into a single stress, we have to describe the overall system. An excellent way to do this is through tensors.

Elegance and power

Tensors are an elegant mathematical method to describe whole systems where there are many influencing factors. They have rules and properties that give them logical strength and consistency. This makes them powerful computational tools for organizing information with many dimensions.

Joshua Grochow, an assistant professor in the Department of Computer Science with a courtesy appointment to the Department of Mathematics at CU Boulder, has just received a CAREER awardーthe most prestigious early-career award given by the National Science Foundation.

The award allows Grochow to gain a deeper understanding of the computational properties of tensors and to develop a foundational theory of the mathematics and algorithmics for these complex interactions.

Through Grochow's examination beyond pure computational application, the strength of tensor mathematics could be more powerfully applied to day-to-day occurrences in big data, physics, chemistry and beyond.

Similarity and difference

Grochow is, in particular, fascinated by "isomorphism problems".

"I've always been fascinated by how you tell when two mathematical objects are secretly the same thing," Grochow said.

Isomorphism problems ask when two given objects―be they data sets, topological spaces, algebraic groups, or tensors―have the same structure, despite being presented differently.

When two tensors are isomorphic, they have some properties that are always identical, and these properties are also the most useful for understanding more about foundational tensor theory.

Because isomorphic tensors present differently, discovering that they're secretly the same can actually be quite difficult. Grochow's project will allow him to move back and forth between creating new tools to test whether tensors are isomorphic and building foundational computational theory as a result of those tools.

Theory and application

"You might wonder why an algorithm for solving isomorphism could help you understand tensors mathematically, but that's the way algorithms go,” he said. “With advanced mathematical topics, you only find an algorithm by understanding the math more deeply and vice versa."

The project is focusing on both tensor isomorphism and group isomorphism; these two problems already have implications for fields as diverse as material science, network analysis, and quantum information.

We need to develop new computational techniques for understanding tensors, as their applications continue to grow across engineering and the sciences. This opportunity gives Grochow a pathway to understanding some of the most fundamental questions in mathematics and computer science.

The award provides Grochow $600,000 over five years to support his project, Higher-Order Interactions in Tensors and Isomorphism Problems, as well as multidisciplinary workshops, and education and training at the undergraduate, graduate, and postdoctoral levels.

Joshua Grochow, an assistant professor in the Department of Computer Science, has just received $600,000 through an NSF CAREER award to advance fundamental computation theory with applications across big data, multi-company mergers, quantum entanglement and beyond.

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Cooler computing through statistical physics?

Anonymous — Tue, 24 Jul 2018 15:57:42 +0000

Cooler computing through statistical physics? Anonymous (not verified) Tue, 07/24/2018 - 09:57

CU Boulder researcher Josh Grochow co-authors study and wiki on recent advances and open questions regarding thermodynamics and computation.

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