We study the truncated shifted Yangian Yn,l(σ)Y_{n,l}(sigma ) over an algebraically closed field kBbbk of characteristic p>0p >0, which is known to be isomorphic to the finite WW-algebra U(g,e)U(mathfrak {g},e) associated to a corresponding nilpotent element e∈g=
我们研究了特征为p >0 p >0的代数闭域k Bbbk上截断移位的Yangian Y n,l (σ) Y_{n,l}(sigma)，已知它同构于有限的W W -代数U (g，e) U(mathfrak {g}，e)与对应的幂零元素e∈g = gl N(k) e in mathfrak {g} = mathfrak {gl}_N(Bbbk)相关联。我们得到了yn,l (σ) Y_{n,l}(sigma)的中心的显式描述，表明它是由它的Harish-Chandra中心和p -中心产生的。我们定义yn,l [p] (σ) Y_{n,l}^{[p]}(sigma)是yn,l (σ) Y_{n,l}(sigma)的商，这是由它的p -中心的平凡性质核生成的理想。我们的主要定理表明yn,l [p] (σ) Y_{n,l}^{[p]}(sigma)同构于有限的W -代数U [p] (g,e) U^{[p]}(mathfrak {g}，e)。因此，我们得到了这个受限ww -代数的显式表示。
{"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}
Let FF be a finite type surface and ζzeta a complex root of unity. The Kauffman bracket skein algebra Kζ(F)K_zeta (F) 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 Kζ(F)K_zeta (F) 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 FF.
{"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}
We show that the non-Archimedean skeleton of the d d -th symmetric power of a smooth projective algebraic curve X X is naturally isomorphic to the d d -th symmetric power of the tropical curve that arises as the non-Archimedean skeleton of X X . 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.
我们证明了光滑投影代数曲线X X的d d -对称幂的非阿基米德骨架与X X的d d -对称幂的热带曲线的d d -对称幂自然同构。对骨架的回缩正是除数的专门化映射。此外，我们还证明了热带化过程与对角态射和Abel-Jacobi映射的自然交换，并展示了对称幂曲线的忠实热带化。最后，我们证明了Bieri-Groves定理的一个版本，它允许我们在一定的热带一般性假设下，为线性系统的热带化推导出一个新的热带riemann - roch定理。
{"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}
For any ε>0varepsilon > 0 we derive effective estimates for the size of a non-zero integral point m∈Zd∖{0}m in mathbb {Z}^d setminus {0} solving the Diophantine inequality |Q[m]|>εlvert Q[m] rvert > varepsilon, where Q[m]=q1m12+…+qdmd2
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 k k , finding a k k -component unlink as a sublink, and finding a k k -component alternating sublink.
{"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}
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: $bullet$ 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. $bullet$ 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 $1 leq p leq frac{2d+2}{d+3+4k}$. $bullet$ 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. The 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.
{"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}
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.
{"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}
We use category-theoretic techniques to provide two proofs showing that for a higher-rank graph Λ Lambda , 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.
{"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}
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.
{"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}
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 M M of R d mathbb { R}^d , and associates to each direction v ∈ S d − 1 vin S^{d-1} a shape summary obtained by scanning M M in the direction v v . 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 p p -distance, or the space of Euler curves, equipped with certain L p L^p 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.
本文考虑了应用拓扑学中比较流行的两种拓扑变换:持久同调变换和欧拉特征变换。这两种变换的数学性质以及它们在科学和工程中的应用都很有趣，因为它们提供了一种以拓扑但定量的方式总结形状的方法。这两个变换都取一个形状，看作是R d mathbb {R}^d的一个温顺的子集M M，并且与S^{d-1}中的每个方向v∈S d-1 v相关联的是一个在v v方向上扫描M M得到的形状摘要。这些形状摘要要么是持久性图，要么是称为欧拉曲线的分段常整数值函数。利用Schapira的反演定理，证明了这两个变换在形状空间上是内射的，即每个形状都有一个唯一的变换。此外，我们证明了这些变换确定了从球到具有任意Wasserstein p -距离的持久图空间的连续映射，或具有一定的L p L^p范数的欧拉曲线空间的连续映射。利用球上由超平面划分引起的分层空间结构，证明了欧拉曲线空间上分布的唯一性结果。最后，我们的主要结果证明了在具有合理几何边界的PL嵌入形状的不可数空间中，任何形状都可以只用有限多个方向唯一确定。
{"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}
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