Transactions of the American Mathematical Society, Series B最新文献

{"title":"Restricted shifted Yangians and restricted finite 𝑊-algebras","authors":"Simon M. Goodwin, L. Topley","doi":"10.1090/BTRAN/63","DOIUrl":"https://doi.org/10.1090/BTRAN/63","url":null,"abstract":"<p>We study the truncated shifted Yangian <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper Y Subscript n comma l Baseline left-parenthesis sigma right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mi>Y</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>n</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>l</mml:mi>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>σ<!-- σ --></mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">Y_{n,l}(sigma )</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> over an algebraically closed field <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck k\">\u0000 <mml:semantics>\u0000 <mml:mi mathvariant=\"double-struck\">k<!-- k --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">Bbbk</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> of characteristic <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p greater-than 0\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:mo>></mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">p >0</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, which is known to be isomorphic to the finite <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper W\">\u0000 <mml:semantics>\u0000 <mml:mi>W</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">W</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-algebra <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper U left-parenthesis German g comma e right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>U</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"fraktur\">g</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>e</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">U(mathfrak {g},e)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> associated to a corresponding nilpotent element <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"e element-of German g equals German g German l Subscript upper N Baseline left-parenthesis double-struck k right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>e</mml:mi>\u0000 <mml:mo>∈<!-- ∈ --></mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"fraktur\">g</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:msub>\u0000 <mml:mrow clas","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128984169","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Dimension and Trace of the Kauffman Bracket Skein Algebra","authors":"C. Frohman, J. Kania-Bartoszyńska, Thang T. Q. Lê","doi":"10.1090/BTRAN/69","DOIUrl":"https://doi.org/10.1090/BTRAN/69","url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F\">\u0000 <mml:semantics>\u0000 <mml:mi>F</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">F</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> be a finite type surface and <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"zeta\">\u0000 <mml:semantics>\u0000 <mml:mi>ζ<!-- ζ --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">zeta</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> a complex root of unity. The Kauffman bracket skein algebra <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K Subscript zeta Baseline left-parenthesis upper F right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:mi>ζ<!-- ζ --></mml:mi>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>F</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">K_zeta (F)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is an important object in both classical and quantum topology as it has relations to the character variety, the Teichmüller space, the Jones polynomial, and the Witten-Reshetikhin-Turaev Topological Quantum Field Theories. We compute the rank and trace of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K Subscript zeta Baseline left-parenthesis upper F right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:mi>ζ<!-- ζ --></mml:mi>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>F</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">K_zeta (F)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> over its center, and we extend a theorem of the first and second authors in [Math. Z. 289 (2018), pp. 889–920] which says the skein algebra has a splitting coming from two pants decompositions of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F\">\u0000 <mml:semantics>\u0000 <mml:mi>F</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">F</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>.</p>","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126984465","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Symmetric powers of algebraic and tropical curves: A non-Archimedean perspective","authors":"M. Brandt, Martin Ulirsch","doi":"10.1090/btran/113","DOIUrl":"https://doi.org/10.1090/btran/113","url":null,"abstract":"We show that the non-Archimedean skeleton of the \u0000\u0000 \u0000 d\u0000 d\u0000 \u0000\u0000-th symmetric power of a smooth projective algebraic curve \u0000\u0000 \u0000 X\u0000 X\u0000 \u0000\u0000 is naturally isomorphic to the \u0000\u0000 \u0000 d\u0000 d\u0000 \u0000\u0000-th symmetric power of the tropical curve that arises as the non-Archimedean skeleton of \u0000\u0000 \u0000 X\u0000 X\u0000 \u0000\u0000. The retraction to the skeleton is precisely the specialization map for divisors. Moreover, we show that the process of tropicalization naturally commutes with the diagonal morphisms and the Abel-Jacobi map and we exhibit a faithful tropicalization for symmetric powers of curves. Finally, we prove a version of the Bieri-Groves Theorem that allows us, under certain tropical genericity assumptions, to deduce a new tropical Riemann-Roch-Theorem for the tropicalization of linear systems.","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125813996","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On small values of indefinite diagonal quadratic forms at integer points in at least five variables","authors":"P. Buterus, F. Gotze, Thomas Hille","doi":"10.1090/btran/97","DOIUrl":"https://doi.org/10.1090/btran/97","url":null,"abstract":"<p>For any <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"epsilon greater-than 0\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>ε<!-- ε --></mml:mi>\u0000 <mml:mo>></mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">varepsilon > 0</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> we derive effective estimates for the size of a non-zero integral point <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"m element-of double-struck upper Z Superscript d minus StartSet 0 EndSet\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>m</mml:mi>\u0000 <mml:mo>∈<!-- ∈ --></mml:mo>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">Z</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi>d</mml:mi>\u0000 </mml:msup>\u0000 <mml:mo class=\"MJX-variant\">∖<!-- ∖ --></mml:mo>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">m in mathbb {Z}^d setminus {0}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> solving the Diophantine inequality <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartAbsoluteValue upper Q left-bracket m right-bracket EndAbsoluteValue greater-than epsilon\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">|<!-- | --></mml:mo>\u0000 <mml:mi>Q</mml:mi>\u0000 <mml:mo stretchy=\"false\">[</mml:mo>\u0000 <mml:mi>m</mml:mi>\u0000 <mml:mo stretchy=\"false\">]</mml:mo>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">|<!-- | --></mml:mo>\u0000 <mml:mo>></mml:mo>\u0000 <mml:mi>ε<!-- ε --></mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">lvert Q[m] rvert > varepsilon</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, where <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper Q left-bracket m right-bracket equals q 1 m 1 squared plus ellipsis plus q Subscript d Baseline m Subscript d Superscript 2\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>Q</mml:mi>\u0000 <mml:mo stretchy=\"false\">[</mml:mo>\u0000 <mml:mi>m</mml:mi>\u0000 <mml:mo stretchy=\"false\">]</mml:mo>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>q</mml:mi>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:msub>\u0000 <mml:msubsup>\u0000 <mml:mi>m</mml:mi>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:msubsup>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mo>…<!-- … --></mml:mo>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>q</mml:mi>\u0000 <mml:mi>d</mml:mi>\u0000 </mml:msub>\u0000 <mml:msubsup>\u0000 <mml:mi>m</mml:mi>\u0000 <mml:mi>d</mml:mi>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:msub","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"139 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132689328","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"NP–hard problems naturally arising in knot theory","authors":"Dale Koenig, A. Tsvietkova","doi":"10.1090/BTRAN/71","DOIUrl":"https://doi.org/10.1090/BTRAN/71","url":null,"abstract":"We prove that certain problems naturally arising in knot theory are NP–hard or NP–complete. These are the problems of obtaining one diagram from another one of a link in a bounded number of Reidemeister moves, determining whether a link has an unlinking or splitting number \u0000\u0000 \u0000 k\u0000 k\u0000 \u0000\u0000, finding a \u0000\u0000 \u0000 k\u0000 k\u0000 \u0000\u0000-component unlink as a sublink, and finding a \u0000\u0000 \u0000 k\u0000 k\u0000 \u0000\u0000-component alternating sublink.","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123959764","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Restrictions of higher derivatives of the Fourier transform","authors":"M. Goldberg, D. Stolyarov","doi":"10.1090/btran/45","DOIUrl":"https://doi.org/10.1090/btran/45","url":null,"abstract":"We consider several problems related to the restriction of $(nabla^k) hat{f}$ to a surface $Sigma subset mathbb R^d$ with nonvanishing Gauss curvature. While such restrictions clearly exist if $f$ is a Schwartz function, there are few bounds available that enable one to take limits with respect to the $L_p(mathbb R^d)$ norm of $f$. We establish three scenarios where it is possible to do so: \u0000$bullet$ When the restriction is measured according to a Sobolev space $H^{-s}(Sigma)$ of negative index. We determine the complete range of indices $(k, s, p)$ for which such a bound exists. \u0000$bullet$ Among functions where $hat{f}$ vanishes on $Sigma$ to order $k-1$, the restriction of $(nabla^k) hat{f}$ defines a bounded operator from (this subspace of) $L_p(mathbb R^d)$ to $L_2(Sigma)$ provided $1 leq p leq frac{2d+2}{d+3+4k}$. \u0000$bullet$ When there is _a priori_ control of $hat{f}|_Sigma$ in a space $H^{ell}(Sigma)$, $ell > 0$, this implies improved regularity for the restrictions of $(nabla^k)hat{f}$. If $ell$ is large enough then even $|nabla hat{f}|_{L_2(Sigma)}$ can be controlled in terms of $|hat{f}|_{H^ell(Sigma)}$ and $|f|_{L_p(mathbb R^d)}$ alone. \u0000The techniques underlying these results are inspired by the spectral synthesis work of Y. Domar, which provides a mechanism for $L_p$ approximation by \"convolving along surfaces\", and the Stein-Tomas restriction theorem. Our main inequality is a bilinear form bound with similar structure to the Stein--Tomas $T^*T$ operator, generalized to accommodate smoothing along $Sigma$ and derivatives transverse to it. It is used both to establish basic $H^{-s}(Sigma)$ bounds for derivatives of $hat{f}$ and to bootstrap from surface regularity of $hat{f}$ to regularity of its higher derivatives.","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"83 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116470251","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Newforms mod $p$ in squarefree level with applications to Monsky’s Hecke-stable filtration","authors":"Shaunak V. Deo, A. Medvedovsky, Alexandru Ghitza","doi":"10.1090/btran/35","DOIUrl":"https://doi.org/10.1090/btran/35","url":null,"abstract":"We propose an algebraic definition of the space of l-new mod-p modular forms for Gamma0(Nl) in the case that l is prime to N, which naturally generalizes to a notion of newforms modulo p in squarefree level. We use this notion of newforms to interpret the Hecke algebras on the graded pieces of the space of mod-2 level-3 modular forms described by Paul Monsky. Along the way, we describe a renormalized version of the Atkin-Lehner involution: no longer an involution, it is an automorphism of the algebra of modular forms, even in characteristic p.","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122919022","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Isomorphism of the cubical and categorical cohomology groups of a higher-rank graph","authors":"E. Gillaspy, Jianchao Wu","doi":"10.1090/BTRAN/38","DOIUrl":"https://doi.org/10.1090/BTRAN/38","url":null,"abstract":"We use category-theoretic techniques to provide two proofs showing that for a higher-rank graph \u0000\u0000 \u0000 Λ\u0000 Lambda\u0000 \u0000\u0000, its cubical (co-)homology and categorical (co-)homology groups are isomorphic in all degrees, thus answering a question of Kumjian, Pask and Sims in the positive. Our first proof uses the topological realization of a higher-rank graph, which was introduced by Kaliszewski, Kumjian, Quigg, and Sims. In our more combinatorial second proof, we construct, explicitly and in both directions, maps on the level of (co-)chain complexes that implement said isomorphism. Along the way, we extend the definition of cubical (co-)homology to allow arbitrary coefficient modules.","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121126824","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Bilateral identities of the Rogers–Ramanujan type","authors":"M. Schlosser","doi":"10.1090/btran/158","DOIUrl":"https://doi.org/10.1090/btran/158","url":null,"abstract":"We derive by analytic means a number of bilateral identities of the Rogers–Ramanujan type. Our results include bilateral extensions of the Rogers–Ramanujan and the Göllnitz–Gordon identities, and of related identities by Ramanujan, Jackson, and Slater. We give corresponding results for multisums including multilateral extensions of the Andrews–Gordon identities, of the Andrews–Bressoud generalization of the Göllnitz–Gordon identities, of Bressoud’s even modulus identities, and other identities. Our closed form bilateral and multilateral summations appear to be the very first of their kind.","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125015879","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"How Many Directions Determine a Shape and other Sufficiency Results for Two Topological Transforms","authors":"J. Curry, S. Mukherjee, Katharine Turner","doi":"10.1090/btran/122","DOIUrl":"https://doi.org/10.1090/btran/122","url":null,"abstract":"In this paper we consider two topological transforms that are popular in applied topology: the Persistent Homology Transform and the Euler Characteristic Transform. Both of these transforms are of interest for their mathematical properties as well as their applications to science and engineering, because they provide a way of summarizing shapes in a topological, yet quantitative, way. Both transforms take a shape, viewed as a tame subset \u0000\u0000 \u0000 M\u0000 M\u0000 \u0000\u0000 of \u0000\u0000 \u0000 \u0000 \u0000 R\u0000 \u0000 d\u0000 \u0000 mathbb { R}^d\u0000 \u0000\u0000, and associates to each direction \u0000\u0000 \u0000 \u0000 v\u0000 ∈\u0000 \u0000 S\u0000 \u0000 d\u0000 −\u0000 1\u0000 \u0000 \u0000 \u0000 vin S^{d-1}\u0000 \u0000\u0000 a shape summary obtained by scanning \u0000\u0000 \u0000 M\u0000 M\u0000 \u0000\u0000 in the direction \u0000\u0000 \u0000 v\u0000 v\u0000 \u0000\u0000. These shape summaries are either persistence diagrams or piecewise constant integer-valued functions called Euler curves. By using an inversion theorem of Schapira, we show that both transforms are injective on the space of shapes, i.e. each shape has a unique transform. Moreover, we prove that these transforms determine continuous maps from the sphere to the space of persistence diagrams, equipped with any Wasserstein \u0000\u0000 \u0000 p\u0000 p\u0000 \u0000\u0000-distance, or the space of Euler curves, equipped with certain \u0000\u0000 \u0000 \u0000 L\u0000 p\u0000 \u0000 L^p\u0000 \u0000\u0000 norms. By making use of a stratified space structure on the sphere, induced by hyperplane divisions, we prove additional uniqueness results in terms of distributions on the space of Euler curves. Finally, our main result proves that any shape in a certain uncountable space of PL embedded shapes with plausible geometric bounds can be uniquely determined using only finitely many directions.","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"97 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116360984","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}