This book gives an introduction to the very active field of combinatorics of affine Schubert calculus, explains the current state of the art, and states the current open problems. Affine Schubert calculus lies at the crossroads of combinatorics, geometry, and representation theory. Its modern development is motivated by two seemingly unrelated directions. One is the introduction of k-Schur functions in the study of Macdonald polynomial positivity, a mostly combinatorial branch of symmetric function theory. The other direction is the study of the Schubert bases of the (co)homology of the affine Grassmannian, an algebro-topological formulation of a problem in enumerative geometry. This is the first introductory text on this subject. It contains many examples in Sage, a free open source general purpose mathematical software system, to entice the reader to investigate the open problems. This book is written for advanced undergraduate and graduate students, as well as researchers, who want to become familiar with this fascinating new field.
The authors give a combinatorial expansion of a Schubert homology class in the affine Grassmannian $\mathrm{Gr}_{\mathrm{SL}_k}$ into Schubert homology classes in $\mathrm{Gr}_{\mathrm{SL}_{k+1}}$. This is achieved by studying the combinatorics of a new class of partitions called $k$-shapes, which interpolates between $k$-cores and $k+1$-cores. The authors define a symmetric function for each $k$-shape, and show that they expand positively in terms of dual $k$-Schur functions. They obtain an explicit combinatorial description of the expansion of an ungraded $k$-Schur function into $k+1$-Schur functions. As a corollary, they give a formula for the Schur expansion of an ungraded $k$-Schur function.
This book gives an introduction to the very active field of combinatorics of affine Schubert calculus, explains the current state of the art, and states the current open problems. Affine Schubert calculus lies at the crossroads of combinatorics, geometry, and representation theory. Its modern development is motivated by two seemingly unrelated directions. One is the introduction of k-Schur functions in the study of Macdonald polynomial positivity, a mostly combinatorial branch of symmetric function theory. The other direction is the study of the Schubert bases of the (co)homology of the affine Grassmannian, an algebro-topological formulation of a problem in enumerative geometry. This is the first introductory text on this subject. It contains many examples in Sage, a free open source general purpose mathematical software system, to entice the reader to investigate the open problems. This book is written for advanced undergraduate and graduate students, as well as researchers, who want to become familiar with this fascinating new field.
The authors give a combinatorial expansion of a Schubert homology class in the affine Grassmannian $\mathrm{Gr}_{\mathrm{SL}_k}$ into Schubert homology classes in $\mathrm{Gr}_{\mathrm{SL}_{k+1}}$. This is achieved by studying the combinatorics of a new class of partitions called $k$-shapes, which interpolates between $k$-cores and $k+1$-cores. The authors define a symmetric function for each $k$-shape, and show that they expand positively in terms of dual $k$-Schur functions. They obtain an explicit combinatorial description of the expansion of an ungraded $k$-Schur function into $k+1$-Schur functions. As a corollary, they give a formula for the Schur expansion of an ungraded $k$-Schur function.
We give a combinatorial expansion of a Schubert homology class in the affine Grassmannian GrSLk into Schubert homology classes in GrSLk+1. This is achieved by studying the combinatorics of a new class of partitions called k-shapes, which interpolates between k k-cores and kk+1-cores. We define a symmetric function for each k-shape, and show that they expand positively in terms of dual k-Schur functions. We obtain an explicit combinatorial description of the expansion of an ungraded k k-Schur function into k+1-Schur functions. As a corollary, we give a formula for the Schur expansion of an ungraded k-Schur function.
Practical and highly organized, The5-Minute Clinical Consult 2023 provides rapid access to the diagnosis, treatment, medications, follow-up, and associated conditions for more than 540 disease and condition topics to help you make accurate decisions at the point of care. Organized alphabetically by diagnosis, it presents brief, bulleted points in a templated format and contains more than 100 diagnostic and therapeutic algorithms. Edited by Frank J. Domino, Robert A. Baldor, Kathleen A. Barry, Jeremy Golding, and Mark B. Stephens, this up-to-date, bestselling reference delivers maximum clinical confidence as efficiently as possible, allowing you to focus your valuable time on providing high-quality care to your patients.
It is well known that isotopic metrics of positive scalar curvature are concordant. Whether or not the converse holds is an open question, at least in dimensions greater than four. The author shows that for a particular type of concordance, constructed using the surgery techniques of Gromov and Lawson, this converse holds in the case of closed simply connected manifolds of dimension at least five.
Practical and highly organized, The 5-Minute Clinical Consult 2022 provides rapid access to the diagnosis, treatment, medications, follow-up, and associated conditions for more than 540 disease and condition topics to help you make accurate decisions at the point of care. Organized alphabetically by diagnosis, it presents brief, bulleted points in a templated format, and contains more than 100 diagnostic and therapeutic algorithms. This up-to-date, bestselling reference delivers maximum clinical confidence as efficiently as possible, allowing you to focus your valuable time on providing high-quality care to your patients.
The author studies the interaction between the EHP sequence and the Goodwillie tower of the identity evaluated at spheres at the prime $2$. Both give rise to spectral sequences (the EHP spectral sequence and the Goodwillie spectral sequence, respectively) which compute the unstable homotopy groups of spheres. He relates the Goodwillie filtration to the $P$ map, and the Goodwillie differentials to the $H$ map. Furthermore, he studies an iterated Atiyah-Hirzebruch spectral sequence approach to the homotopy of the layers of the Goodwillie tower of the identity on spheres. He shows that differentials in these spectral sequences give rise to differentials in the EHP spectral sequence. He uses his theory to recompute the $2$-primary unstable stems through the Toda range (up to the $19$-stem). He also studies the homological behavior of the interaction between the EHP sequence and the Goodwillie tower of the identity. This homological analysis involves the introduction of Dyer-Lashof-like operations associated to M. Ching's operad structure on the derivatives of the identity. These operations act on the mod $2$ stable homology of the Goodwillie layers of any functor from spaces to spaces.
For nearly 50 years, Sleisenger & Fordtran’s Gastrointestinal and Liver Disease has been the go-to reference for gastroenterology and hepatology residents, fellows, physicians, and the entire GI caregiving team. Now in a fully revised 11th Edition, this two-volume masterwork brings together the knowledge and expertise of hundreds of global experts who keep you up to date with the newest techniques, technologies, and treatments for every clinical challenge you face in gastroenterology and hepatology. A logical organization, more than 1,100 full-color illustrations, and easy-to-use algorithms ensure that you’ll quickly and easily find the information you need. Features new and expanded discussions of chronic hepatitis B and C, Helicobacter pylori infection, colorectal cancer prevention through screening and surveillance, biologic agents and novel small molecules to treat and prevent recurrences of inflammatory bowel disease (IBD), gastrointestinal immune and autoimmune diseases, and more. Offers reliable coverage of key topics such as Barrett’s esophagus, gut microbiome, enteric microbiota and probiotics, fecal microbiota transplantation, and hepatic, pancreatic, and small bowel transplantation. Provides more quick-reference algorithms that summarize clinical decision making and practical approaches to patient management. Employs a consistent, templated, format throughout for quick retrieval of information. Includes monthly updates online, as well as more than 20 procedural videos.
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