Optimal bounds for a colorful Tverberg--Vrecica type problem

We prove the following optimal colorful Tverberg-Vrecica type transversal theorem: For prime r and for any k+1 colored collections of points C^l of size |C^l|=(r-1)(d-k+1)+1 in R^d, where each C^l is a union of subsets (color classes) C_i^l of size smaller than r, l=0,...,k, there are partition of the collections C^l into colorful sets F_1^l,...,F_r^l such that there is a k-plane that meets all the convex hulls conv(F_j^l), under the assumption that r(d-k) is even or k=0. Along the proof we obtain three results of independent interest: We present two alternative proofs for the special case k=0 (our optimal colored Tverberg theorem (2009)), calculate the cohomological index for joins of chessboard complexes, and establish a new Borsuk-Ulam type theorem for (Z_p)^m-equivariant bundles that generalizes results of Volovikov (1996) and Zivaljevic (1999).Comment: Substantially revised version: new notation, improved results, additional references; 12 pages, 2 figure

Joint distribution for the Selmer ranks of the congruent number curves

summary:We determine the distribution over square-free integers $n$ of the pair $(\dim _{\mathbb {F}_2}{\rm Sel}^\Phi (E_n/\mathbb {Q}),\dim _{\mathbb {F}_2} {\rm Sel}^{\widehat {\Phi }}(E_n'/\mathbb {Q}))$, where $E_n$ is a curve in the congruent number curve family, $E_n'\colon y^2=x^3+4n^2x$ is the image of isogeny $\Phi \colon E_n\rightarrow E_n'$, $\Phi (x,y)=(y^2/x^2,y(n^2-x^2)/x^2)$, and $\widehat {\Phi }$ is the isogeny dual to $\Phi$

A result on the size of iterated sumsets in Zd\mathbb{Z}^d

In this paper we give a different approach to determining the cardinality of $h$-fold sumsets $hA$ when $A\subset \mathbb{Z}^d$ has $d+2$ elements. This enables us to provide more general result with a shorter and simpler proof. We also obtain an upper bound for the value of $|hA|$ when $A\subset \mathbb{Z}^d$ is a set of $d+3$ elements with simplicial hull.Comment: It was noticed that the proof of the main Theorem could be improved to give a more general result. Also some motivating examples are provide

Selmer group statistics in the congruent number family of elliptic curves

Prvi deo disertacije se bavi skupovima zbirova hA = {a1 + · · · + ah ∈ Zd : a1, . . . , ah ∈ A}, gde je A konačni skup u Zd. Poznato je da postoji konstanta h0 ∈ N i polinom pA(X) takav da je pA(h) = |hA| za h ⩾ h0. Međutim, malo se zna o polinomu, kao i o konstanti h0. Konus CA nad skupom A sadrži informacije o hA, za svako h ∈ N. Kada skup A ima d + 2 elementa, mogu se eksplicitno opisati polinom pA i konstanta h0. Kada A ima d + 3 elementa, nalazi se gore ograniqee za broj elemenata skupa hA. Drugi deo disertacije se bavi Selmerovim grupama u familiji eliptičkih krivih pridruenih kongruentnim brojevima. Beskvadratan prirodan broj n je kongruentan ako i samo ako postoji pravougli trougao sa celobrojnim duinama stranica qija povrxina je n. Poznato je da je prirodan broj n kongruentan ako i samo ako je rang eliptiqke krive En : y2 = x3 − n2x kao algebarske grupe razliqit od nule. Selmerove grupe pridruene izogenijama eliptiqkih krivih En su zanim ive, jer ihov rang nije mai od ranga krive En, pa kada je rang Selmerovih grupa nula, tada je i rang krive En jednak nuli. Elementi Selmerovih grupa se mogu predstaviti kao particije odreenog grafa, pa se na taj naqin moe na'i distribucija ranga Selmerovih grupa.irst part of dissertation examines sumsets hA = {a1 + · · · + ah ∈ Zd : a1, . . . , ah ∈ A}, where A is a finite set in Zd. It is known that there exists a constant h0 ∈ N and a polynomial pA(X) such that pA(h) = |hA| for h ⩾ h0. However, little is known of polynomial pA and constant h0. Cone CA over the set A contains information about hA, for all h ∈ N. When A has d + 2 elements, polynomial pA and constant h0 can be explicitly described. When A has d + 3 elements, an upper bound is found for the number of elements of hA. Second part of dissertation examines Selmer groups of elliptic curves in the con- gruent number family. A squarefree natural number is congruent if and only if there exists a right triangle with area n whose sides all have integer lengths. It is known that n is a congruent number if and only if elliptic curve En : y2 = x3 − n2x has nonzero rank as an algebraic group. Selmer groups of isogenies on En are interesting, because their rank is not smaller than the rank of En, so when the Selmer groups have rank zero, then the elliptic curve En also has rank zero. Elements of these Selmer groups can be represented as partitions of a particular graph, from which one may find the distribution of ranks of Selmer groups