Locally compact flows on connected manifolds

In this paper, we completely characterize locally compact flows G G of homeomorphisms of connected manifolds M M by proving that they are either circle groups or real groups. For M = R m M = \mathbb R^m , we prove that every recurrent element in G G is periodic, and we obtain a generalization of the result of Yang [Hilbert’s fifth problem and related problems on transformation groups, American Mathematical Society, Providence, RI, 1976, pp. 142–146.] by proving that there is no nontrivial locally compact flow on R m \mathbb R^m in which all elements are recurrent.

Dr. Uri Treisman: Five Decades of Postsecondary Innovation

Philip Uri Treisman is a University Distinguished Teaching Professor, professor of mathematics, and professor of public affairs at The University of Texas at Austin. He is the founder and executive director of the Charles A. Dana Center, an organized research unit in the College of Natural Sciences that works to ensure that all students, regardless of their life circumstances, can access—and succeed—in rigorous mathematics and science education. Dr. Treisman is active in numerous organizations working to improve American mathematics education. He is a founder and member of the governing board of Transforming Post-Secondary Education in Mathematics (also known as TPSE-Math). He is a representative of the American Mathematical Society to the American Association for the Advancement of Science (Education, Section Q) and is a senior advisor to the Conference Board of the Mathematical Sciences Research Advisory Group. In addition, he is a member of the Roundtable on Data Science Postsecondary Education with the National Academies of Sciences, Engineering, and Medicine. Dr. Treisman has served as a Distinguished Senior Fellow at the Education Commission of the States since 2013. He is also chairman of the Strong Start to Finish Campaign (and its expert advisory board), a joint initiative of the Bill & Melinda Gates Foundation, The Kresge Foundation, and Ascendium Education Group that works nationally to ensure that all students get a strong start in their first year of college and finish with the skills they need to thrive. Treisman has served on the STEM working group of the President’s Council of Advisors on Science and Technology, the 21st-Century Commission on the Future of Community Colleges of the American Association of Community Colleges, and the Commission on Mathematics and Science Education of the Carnegie Corporation of New York and the Institute for Advanced Study. Treisman’s research and professional interests span mathematics and science education, education policy, social and developmental psychology, community service, and volunteerism.

Places of Near-Fields: To Heinrich Wefelscheid

Wefelscheid (Untersuchungen über Fastkörper und Fastbereiche, Habilitationsschrift, Hamburg, 1971) generalised the well-known Theorem of Artin/Schreier about the characterization of formally real fields and the fundamental result of Baer/Krull to near-fields. In the last fifty years arose from the Theorem of Baer/Krull a theory, which analyses the entirety of the orderings of a field (E. Becker, L. Bröcker, M. Marshall et al.), as presented e.g. in the book by Lam (Orderings, valuations and quadratic forms, American Mathematical Society, Providence, 1983). At the centre of this theory are preorders and their compatibility with valuations or places. We develop some essential results of this theory for the near-field case. In particular, we derive the Brown/Marshall’s inequalities and Bröcker’s Theorem on the trivialisation of fans in the near-field case.

Number of rational points of elliptic curves

In the modern theory of elliptic curves, one of the important problems is the determination of the number of rational points on an elliptic curve. The Mordel–Weil theorem [T. Shioda, On the Mordell–Weil lattices, Comment. Math. University St. Paul. 39(2) (1990) 211–240] points out that the elliptic curve defined above the rational points is generated by a finite group. Despite the knowledge that an elliptic curve has a final number of rational points, it is still difficult to determine their number and the way how to determine them. The greatest progress was achieved by Birch and Swinnerton–Dyer conjecture, which was included in the Millennium Prize Problems [A. Wiles, The Birch and Swinnerton–Dyer conjecture, The Millennium Prize Problems (American Mathematical Society, 2006), pp. 31–44]. This conjecture uses methods of the analytical theory of numbers, while the current knowledge corresponds to the assumptions of the conjecture but has not been proven to date. In this paper, we focus on using a tangent line and the osculating circle for characterizing the rational points of the elliptical curve, which is the greatest benefit of the contribution. We use a different view of elliptic curves by using Minkowki’s theory of number geometry [H. F. Blichfeldt, A new principle in the geometry of numbers, with some applications, Trans. Amer. Math. Soc. 15(3) (1914) 227–235; V. S. Miller, Use of elliptic curves in cryptography, in Proc. Advances in Cryptology — CRYPTO ’85, Lecture Notes in Computer Science, Vol. 218 (Springer, Berlin, Heidelberg, 1985), pp. 417–426; E. Bombieri and W. Gubler, Heights in Diophantine Geometry, Vol. 670, 1st edn. (Cambridge University Press, 2007)].

Subsystems of transitive subshifts with linear complexity

We bound the number of distinct minimal subsystems of a given transitive subshift of linear complexity, continuing work of Ormes and Pavlov [On the complexity function for sequences which are not uniformly recurrent. Dynamical Systems and Random Processes (Contemporary Mathematics, 736). American Mathematical Society, Providence, RI, 2019, pp. 125--137]. We also bound the number of generic measures such a subshift can support based on its complexity function. Our measure-theoretic bounds generalize those of Boshernitzan [A unique ergodicity of minimal symbolic flows with linear block growth. J. Anal. Math.44(1) (1984), 77–96] and are closely related to those of Cyr and Kra [Counting generic measures for a subshift of linear growth. J. Eur. Math. Soc.21(2) (2019), 355–380].

Sharp Hardy Identities and Inequalities on Carnot Groups

In this paper we establish general weighted Hardy identities for several subelliptic settings including Hardy identities on the Heisenberg group, Carnot groups with respect to a homogeneous gauge and Carnot–Carathéodory metric, general nilpotent groups, and certain families of Hörmander vector fields. We also introduce new weighted uncertainty principles in these settings. This is done by continuing the program initiated by [N. Lam, G. Lu and L. Zhang, Factorizations and Hardy’s-type identities and inequalities on upper half spaces, Calc. Var. Partial Differential Equations 58 2019, 6, Paper No. 183; N. Lam, G. Lu and L. Zhang, Geometric Hardy’s inequalities with general distance functions, J. Funct. Anal. 279 2020, 8, Article ID 108673] of using the Bessel pairs introduced by [N. Ghoussoub and A. Moradifam, Functional Inequalities: New Perspectives and New Applications, Math. Surveys Monogr. 187, American Mathematical Society, Providence, 2013] to obtain Hardy identities. Using these identities, we are able to improve significantly existing Hardy inequalities in the literature in the aforementioned subelliptic settings. In particular, we establish the Hardy identities and inequalities in the spirit of [H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid 10 1997, 443–469] and [H. Brezis and M. Marcus, Hardy’s inequalities revisited. Dedicated to Ennio De Giorgi, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 25 1997, 1–2, 217–237] in these settings.

Unknotting operations on knots and links

By considering unknotting operations, we obtain ways of measuring how knotted a knot is. Unknotting phenomena can be seen not only in knot theory, but also in various settings such as DNA knots, mind knots and so on ([C. C. Adams, The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots (American Mathematical Society, Providence, RI, 2004); A. Kawauchi, K. Kishimoto and A. Shimizu, Knot theory and game (in Japanese) (Asakura Publishing, Tokyo, 2013); L. Rudolph, Qualitative Mathematics for the Social Sciences (Routledge, London, 2013); K. Murasugi, Knot Theory and Its Applications, Translated from the 1993 Japanese original by Bohdan Kupita (Birkhauser, Boston, MA, 1996)], etc.). In this paper, we see how knots can be unknotted (and therefore how they are knotted) by considering various unknotting operations and their associated unknotting numbers.