A Symplectic Dynamics Proof of the Degree–Genus Formula

We classify global surfaces of section for the Reeb flow of the standard contact form on the 3-sphere (defining the Hopf fibration), with boundaries oriented positively by the flow. As an application, we prove the degree-genus formula for complex projective curves, using an elementary degeneration process inspired by the language of holomorphic buildings in symplectic field theory.

Rank 2 local systems and abelian varieties

Let $$X/\mathbb {F}_{q}$$ X / F q be a smooth, geometrically connected variety. For X projective, we prove a Lefschetz-style theorem for abelian schemes of $$\text {GL}_2$$ GL 2 -type on X, modeled after a theorem of Simpson. Inspired by work of Corlette-Simpson over $$\mathbb {C}$$ C , we formulate a conjecture that absolutely irreducible rank 2 local systems with infinite monodromy on X come from families of abelian varieties. We have the following application of our main result. If one assumes a strong form of Deligne’s (p-adic) companions conjecture from Weil II, then our conjecture for projective varieties reduces to the conjecture for projective curves. We also answer affirmitavely a question of Grothendieck on extending abelian schemes via their p-divisible groups.

Pediatric Surgical Reentry Strategy Following the COVID-19 Pandemic: A Tiered and Balanced Approach

Background In response to the COVID-19 pandemic, children’s hospitals across the country postponed elective surgery beginning in March 2020. As projective curves flattened, administrators and surgeons sought to develop strategies to safely resume non-emergent surgery. This article reviews challenges and solutions specific to a children’s hospital related to the resumption of elective pediatric surgeries. We present our tiered reentry approach for pediatric surgery as well as report early data for surgical volume and tracking COVID-19 cases during reentry. Methods The experience of shutdown, protocol development, and early reentry of elective pediatric surgery are reported from Levine’s Children’s Hospital (LCH), a free-leaning children’s hospital in Charlotte, North Carolina. Data reported were obtained from de-identified hospital databases. Results Pediatric surgery experienced a dramatic decrease in case volumes at LCH during the shutdown, variable by specialty. A tiered and balanced reentry strategy was implemented with steady resumption of elective surgery following strict pre-procedural screening and testing. Early outcomes showed a steady thorough fluctuating increase in elective case volumes without evidence of a surgery-associated positive spread through periprocedural tracking. Conclusion Reentry of non-emergent pediatric surgical care requires unique considerations including the impact of COVID-19 on children, each children hospital structure and resources, and preventing undue delay in intervention for age- and disease-specific pediatric conditions. A carefully balanced strategy has been critical for safe reentry following the anticipated surge. Ongoing tracking of resource utilization, operative volumes, and testing results will remain vital as community spread continues to fluctuate across the country.

On the unirationality of moduli spaces of pointed curves

We show that$$\mathcal {M}_{g,n}$$Mg,n, the moduli space of smooth curves of genusgtogether withnmarked points, is unirational for$$g=12$$g=12and$$2 \le n\le 4$$2≤n≤4and for$$g=13$$g=13and$$1 \le n \le 3$$1≤n≤3, by constructing suitable dominant families of projective curves in$$\mathbb {P}^1 \times \mathbb {P}^2$$P1×P2and$$\mathbb {P}^3$$P3respectively. We also exhibit several new unirationality results for moduli spaces of smooth curves of genusgtogether withnunordered points, establishing their unirationality for$$g=11, n=7$$g=11,n=7and$$g=12, n =5,6$$g=12,n=5,6.

Elliptic and Edwards Curves Order Counting Method

We consider the algebraic affine and projective curves of Edwards over the finite field Fpn. It is well known that many modern cryptosystems can be naturally transformed into elliptic curves. In this paper, we extend our previous research into those Edwards algebraic curves over a finite field. We propose a novel effective method of point counting for both Edwards and elliptic curves. In addition to finding a specific set of coefficients with corresponding field characteristics for which these curves are supersingular, we also find a general formula by which one can determine whether or not a curve Ed[Fp] is supersingular over this field. The method proposed has complexity O ( p log2 2 p ) . This is an improvement over both Schoof’s basic algorithm and the variant which makes use of fast arithmetic (suitable for only the Elkis or Atkin primes numbers) with complexities O(log8 2 pn) and O(log4 2 pn) respectively. The embedding degree of the supersingular curve of Edwards over Fpn in a finite field is additionally investigated. Due existing the birational isomorphism between twisted Edwards curve and elliptic curve in Weierstrass normal form the result about order of curve over finite field is extended on cubic in Weierstrass normal form.