Which groups are amenable to proving exponent two for matrix multiplication?

The Cohn-Umans group-theoretic approach to matrix multiplication suggests embedding matrix multiplication into group algebra multiplication, and bounding $\omega$ in terms of the representation theory of the host group. This framework is general enough to capture the best known upper bounds on $\omega$ and is conjectured to be powerful enough to prove $\omega = 2$, although finding a suitable group and constructing such an embedding has remained elusive. Recently it was shown, by a generalization of the proof of the Cap Set Conjecture, that abelian groups of bounded exponent cannot prove $\omega = 2$ in this framework, which ruled out a family of potential constructions in the literature. In this paper we study nonabelian groups as potential hosts for an embedding. We prove two main results: (1) We show that a large class of nonabelian groups---nilpotent groups of bounded exponent satisfying a mild additional condition---cannot prove $\omega = 2$ in this framework. We do this by showing that the shrinkage rate of powers of the augmentation ideal is similar to the shrinkage rate of the number of functions over $(\mathbb{Z}/p\mathbb{Z})^n$ that are degree $d$ polynomials; our proof technique can be seen as a generalization of the polynomial method used to resolve the Cap Set Conjecture. (2) We show that symmetric groups $S_n$ cannot prove nontrivial bounds on $\omega$ when the embedding is via three Young subgroups---subgroups of the form $S_{k_1} \times S_{k_2} \times \dotsb \times S_{k_\ell}$---which is a natural strategy that includes all known constructions in $S_n$. By developing techniques for negative results in this paper, we hope to catalyze a fruitful interplay between the search for constructions proving bounds on $\omega$ and methods for ruling them out.Comment: 23 pages, 1 figur

The Amino-Acid Mutational Spectrum of Human Genetic Disease

Background: Nonsynonymous mutations in the coding regions of human genes are responsible for phenotypic differences between humans and for susceptibility to genetic disease. Computational methods were recently used to predict deleterious effects of nonsynonymous human mutations and polymorphisms. Here we focus on understanding the amino-acid mutation spectrum of human genetic disease. We compare the disease spectrum to the spectra of mutual amino-acid mutation frequencies, non-disease polymorphisms in human genes, and substitutions fixed between species. Results: We find that the disease spectrum correlates well with the amino-acid mutation frequencies based on the genetic code. Normalized by the mutation frequencies, the spectrum can be rationalized in terms of chemical similarities between amino acids. The disease spectrum is almost identical for membrane and non-membrane proteins. Mutations at arginine and glycine residues are together responsible for about 30% of genetic diseases, whereas random mutations at tryptophan and cysteine have the highest probability of causing disease. Conclusions: The overall disease spectrum mainly reflects the mutability of the genetic code. We corroborate earlier results that the probability of a nonsynonymous mutation causing a genetic disease increases monotonically with an increase in the degree of evolutionary conservation of the mutation site and a decrease in the solvent-accessibility of the site; opposite trends are observed for non-disease polymorphisms. We estimate that the rate of nonsynonymous mutations with a negative impact on human health is less than one per diploid genome per generation

The Grizzly, January 26, 1999

A Work in Progress • Area Congressmen Discuss Leadership in the 21st Century • Montco Welcomes Clintons, Gores • Classes Held Martin Luther King Day • The Message of Dr. King Remembered • Students Honored for Fundraising • Spring/Summer Internships • Collegeville Squares is a Hit • Opinion: State of Surreality; Impeachment: Democracy on Trial? • Film Society: New & Improved! • Ursinus Gymnastics Back in Full Swing • First Conference Win for UC Swimming • Winter Sports Wrap-uphttps://digitalcommons.ursinus.edu/grizzlynews/1431/thumbnail.jp

On cap sets and the group-theoretic approach to matrix multiplication

In 2003, Cohn and Umans described a framework for proving upper bounds on the exponent ω of matrix multiplication by reducing matrix multiplication to group algebra multiplication, and in 2005 Cohn, Kleinberg, Szegedy, and Umans proposed specific conjectures for how to obtain ω = 2. In this paper we rule out obtaining ω = 2 in this framework from abelian groups of bounded exponent. To do this we bound the size of tricolored sum-free sets in such groups, extending the breakthrough results of Croot, Lev, Pach, Ellenberg, and Gijswijt on cap sets. As a byproduct of our proof, we show that a variant of tensor rank due to Tao gives a quantitative understanding of the notion of unstable tensor from geometric invariant theory

Using Digital Technology for Sexual and Reproductive Health: Are Programs Adequately Considering Risk?

Digital technologies provide opportunities for advancing sexual and reproductive health and services but also present potential risks. We propose 4 steps to reducing potential harms: (1) consider potential harms during intervention design, (2) mitigate or minimize potential harms during the design phase, (3) measure adverse outcomes during implementation, and (4) plan how to support those reporting adverse outcomes

The Grizzly, February 16, 1999

Date Auction Helps Ursinus Senior and Community Charity • Jordanian Student Reacts to Leader\u27s Death • Art History Minor Approved • Professors, Students Mark Bookstore Opening • Opinion: Life After Bill and Monica; Cuban Threat, Fact or Fiction? • History Through Theater • Cloud 9 at proTheatre • Men\u27s Basketball Falls to Second Place • Women\u27s Basketball Wearing Out • Basketball Players Named to Weekly Honor Roll • Gymnastics Team Bests School Record • U.C. Wrestling Sweeps Quad Meethttps://digitalcommons.ursinus.edu/grizzlynews/1434/thumbnail.jp

The Grizzly, April 27, 1999

USGA Calls Do-Over • Faculty to Prepare for Laptops • It Happened One Wednesday • IR Club Encourages Global Awareness, Student Involvement • Dance at UC • STAR Raises Awareness of Sexual Assault on Campus • Scholar Speaks on African Women Writers • Opinion: Free Laptops, What Could Possibly be Wrong With That?; Administration Unresponsive to Student Problem; Don\u27t Be Too Quick to Name This Devil; Blood Lessons: How America Responds to Tragedy and Why we Don\u27t Learn From it • Final Exam Schedule • Student Art Exhibit Opens, Prizes Awarded • Restaurant Review: The Perk, A Taste of History • Ursinus Softball Suffers Through Wild Week • Wayne Gretzky Retires • Golf Takes on Centennial Conference • Baseball Drops Two, but Remains Atop Centennial Conference • Lacrosse Reigns in Centennial Conferencehttps://digitalcommons.ursinus.edu/grizzlynews/1442/thumbnail.jp

The Grizzly, February 24, 1999

Moyer Named Most Outstanding Wrestler as UC Wins Centennial Conference Gold • Middle States Evaluation • Let\u27s Play the Greek Feud • Film Society Offers a Thursday Night Alternative • Opinion: Clinton\u27s Kosovo Policy a Step in the Right Direction; Real Menace of Substance Abuse; Is Apathy a Threat to the Quality of Life? Or Don\u27t You Care? • Biology Student Art in Berman • Cloud 9 • UC Swimming Improves at Championships • UC Basketball Drops Out of Playoffs With Heartbreak Loss • Women\u27s Basketball Finishes Season • Rough Week for UC Gymnastics • Wrestlers Win Conference Titlehttps://digitalcommons.ursinus.edu/grizzlynews/1435/thumbnail.jp

The Lantern Vol. 66, No. 1, Fall 1998

• Curled • Bruised • Accident Prone • Bob or Beatrice or Something • Burrow • The French Door • Before the War: On Earth as it is in Texas • Love Seat • Delusionus Prime • Delineate • The Color of Coffee • A Memory • August Snapshots • The Eulogy and Judgement of My Friend, the Philosophe • Haunted • Being • Playing Children • Her Name is in the Breeze • Aftermath • A Meal for the Brown Philadelphia Twilighthttps://digitalcommons.ursinus.edu/lantern/1153/thumbnail.jp

The Grizzly, April 7, 1999

History and English Departments Vandalized • RPR Fulfills $100,000 Pfahler Pledge • Administration to Consider Providing Laptops for Freshmen • Former CIA Agent Speaks at Ursinus • Emergency Care • Voices of Praise in Bomberger • Dawn of New Millennium Brings Fieldhouse to UC • Career Best for Junior O\u27Hara • Brackin, Erdosy Players of the Week • Softball Vies for Top Spot in Conference • Baseball Downs Stockton • UC Lax Claws Way to First Place • Baseball 2-1 in Conference Playhttps://digitalcommons.ursinus.edu/grizzlynews/1439/thumbnail.jp