WORKSHOP FOR YOUNG RESEARCHERS IN MATHEMATICS May 22-23, 2014 ABSTRACTS

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1 WORKSHOP FOR YOUNG RESEARCHERS IN MATHEMATICS May 22-23, Organized by: the Faculty of Mathematics and Computer Science, Ovidius University Constanta, the Mathematical Institute of the Romanian Academy Bucharest and University of Missouri - St. Louis, USA ABSTRACTS Horrocks theory and applications Cristian ANGHEL Horrocks theory was along many years a useful tool in the study of vector bundles on projective spaces. I intend to review it briefly and to present some applications obtained by Coanda, Coanda-Trautmann and Malaspina-Rao. The last part will be devoted to the role of Horrocks ideas in a joint paper with Coanda and Manolache on globally generated vector bundles. Some applications of the Newton Polygon Method Nicolae Ciprian BONCIOCAT We present some applications of the Newton Polygon Method in the study of the factorization of polynomials over discrete valuation domains. We first show how to use the Newton Polygon Method to find information on the irreducible factors of a given polynomial. We then provide several irreducibility criteria of Eisenstein-Schönemann- Dumas-type for polynomials with integer coefficients, criteria that are given by some divisibility conditions for their coefficients with respect to two or more prime numbers. 1

2 TBA Adrian DIACONU University of Minnesota USA & TBA On the resolution of binomial edge ideals Ahmet DOKUYUCU Lumina University, Romania & Ovidius University of Constanta, Romania In this talk we discuss about binomial ideals arising from graphs. Given a simple graph G on the vertex set [n], one may associate ( with it a binomial ) ideal J G in the x1 x polynomial ring K[X] over a field K, where X = 2... x n. J y 1 y 2... y G is generated n by maximal minors of X, f ij = x i y j x j y i with {i, j} edge of G, and is called the binomial edge ideal of G. The interest in studying binomial edge ideals partially comes from the fact that they turned out to have applications in statistics. In our talk, we are going to discuss homological properties of some classes of binomial edge ideals. We mainly focus on some results obtained in two recent papers. A generalization of Ramanujan s congruence Radu GABA One famous congruence due to Ramanujan states that the Fourier coefficients of the normalized cusp form of weight 12 for the full modular group are congruent with the Fourier coefficients of an Eisenstein series of the same weight, modulo the numerator 691 of the Bernoulli number B 1 2. Generalizations to modular forms of higher weight for the modular group are also known. We use the theory of period polynomials to prove a generalization of Ramanujan s congruence for modular forms of even weight k for the congruence subgroup Gamma0(p) with p prime. Inside this space of modular forms, we show that there exists a cuspidal Hecke eigenform congruent to a specific Eisenstein series, modulo any large prime dividing p k 1. This is joint work in progress with Alex Popa. 2

3 Recent results on two conjectures of Stanley Lukas KATTHÄN Universitat Osnabrück, Germany In this talk I will present the conjecture of Stanley that every Cohen-Macaulay simplicial complex is partitionable, and explain its connection to the famous Stanley conjecture. By a recent result, the former is in fact equivalent to a special case of the later. If time permits, I will also explain the connection between the Stanley depth and the lcm lattice of a monomial ideal. L 2 -Betti numbers of hypersurface complements Laurentiu MAXIM University of Wisconsin-Madison, USA I will present vanishing results for the L 2 -cohomology of complements to complex affine hypersurfaces in general position at infinity. Curve singularities and Poincaré series Julio José MOYANO-FERNÁNDEZ Universitat Osnabrück, Germany The aim of the talk is a short introduction to some invariants of curve singularities, namely the value semigroup and the Poincar series. The trace formula for Hecke operators on congruence subgroups Alexandru POPA We give a simple formula for the trace of Hecke operators on modular forms for congruence subgroups. The proof is also very simple and avoids the technicalities usually associated with proving the trace formula. It is based on an approach for the full modular group sketched by Don Zagier 20 years ago, by computing the trace of Hecke operators on the space of period polynomials associated with modular forms. This approach has been recently finalized and sharpened in a joint work with Don Zagier, and we apply it to congruence subgroups using the theory of period polynomials developed together with Vicentiu Pasol. 3

4 McMullen inversion formula and related computations Vicenţiu PAŞOL We try to expose how can one understand the Deligne-Mumford compactifications of moduli stacks using generating functions (which encode different strata). For genus zero curves with marked points, this is exactly McMullen computation (as a generalization of Getzler s identity) Geometry of most Alexandrov surfaces Joël ROUYER An Alexandrov surface (with curvature bounded below) is a closed topological surface endowed with an intrinsic metric for which the Topogonov s Theorem holds (here, considered as an axiom). The space of all compact Alexandrov surfaces, endowed with the Gromov-Hausdorff metric, is a Baire space. Classical examples of Alexandrov surfaces are convex surfaces and closed Riemannian surfaces. Typical properties of convex surfaces are for long studied, but the investigation of typical properties of Alexandrov surfaces is very recent. After introducing Alexandrov surfaces and their space, I will present some of their typical properties, concerning endpoints, conical points, Gaussian curvature, and simple closed geodesics. On the defining equations of tangent cones of numerical semigroup rings Dumitru STAMATE University of Bucharest & Let H be a numerical semigroup minimally generated by a 1 < dots < a r. We show that if we bound the width wd(h) = a r a 1, then the Betti numbers of the tangent cone gr m K[H] are bounded as well. We conjecture what these bounds are in terms of the width and we present evidence to support this. This is joint work with Juergen Herzog. 4

5 Farthest points on most Alexandrov surfaces Costin VÎLCU In this talk I shall present some properties of critical points for distance functions on most Alexandrov surfaces with curvature bounded below, where most means all, except those in a first category set. The accent will be on farthest points (i.e., global maxima for distance functions). 5

6 WORKSHOP FOR YOUNG RESEARCHERS IN MATHEMATICS May 22-23, Organized by: the Faculty of Mathematics and Computer Science, Ovidius University Constanta, the Mathematical Institute of the Romanian Academy Bucharest and University of Missouri - St. Louis, USA ABSTRACTS Constrained Facility Location Problems. Numeric methods and algorithms Diana ALEXANDRESCU University of Craiova, Romania This paper studies the facility location problem in linear normed spaces, provided that the distances are replaced by the sum of increasing convex functions of norm. This problem can be seen as a generalization of the celebrate Weber problem: Given n distinct points in R 3, find another point x such that the weighted sum of distances from x to these n points is minimum, where the weights w 1,..., w n are positive real numbers. In case when w i = 1, i = 1,..., n, we obtain the problem given first by Fermat in Let E be a linear normed space and the functions ϕ i : R + R, i = 1, 2,..., n. We consider the distinct points a 1,..., a n E and f : E R given by n f (x) = ϕ i ( x a i ). i=1 Our purpose is to find conditions on the space E and on the functions ϕ 1,..., ϕ n such that the point of global minimum for the function f exists. We also give conditions in which this point is unique. In the second part of this paper, we study the facility location problem subject to constraints in the plane and on the sphere. We also propose algorithms that approximate the solution of the facility location problem subject to constraints in plane and also on the sphere. Several numeric examples are also presented. 1

7 A note on the bilinear versions of Swartz s Theorem Gabriela BADEA Ovidius University of Constanta, Romania In this talk we give some characterizations of the nuclear and multiple 1 summing operators on c 0 (X) c 0 (Y ), which actually are the bilinear versions of Swartz s theorem for these types of operators. Thus, although on c 0 c 0 these two classes of operators coincide, we will show that on c 0 (X) c 0 (Y ) their behavior is different.some examples of such operators are also discussed. Bochner-Kolmogorov theorem Iulian CIMPEAN We prove the Bochner-Kolmogorov theorem on the existence of the limit of projective systems of second countable Hausdorff (non-metrizable) spaces with tight probabilities, such that the projection mappings are merely measurable functions. The motivation of the revisit of this classical result is an application to the construction of the continuous time fragmentation processes and related branching processes, based on a measurable identification between the space of all fragmentation sizes considered by J. Bertoin and the limit of a projective system of spaces of finite configurations. New estimates for the best constant of a Hardy type inequality with multi-singular potential Cristian CAZACU In this talk we are concerned with the Hardy constant µ Ω (Ω) := inf u 2 dx u H0 Ω 1(Ω) V u2 dx, in a smooth domain Ω R N, N 2,, where V denotes the unbounded potential V := with the singular poles a 1,..., a n Ω. 1 i<j n a i a j 2 x a i 2 x a j 2, 2

8 We discuss both situations when all singularities are located either in the interior or on the boundary of Ω. We particularly determine µ (Ω) for some geometrical configurations of Ω. Otherwise, regardless of geometry, we show uniform upper and lower bounds for µ (Ω). On the existence of solutions for inequality problems of hemivariational type Nicusor COSTEA In this talk we present some existence results for nonlinear hemivariational inequalities. The connection with the variational formulation of some problems in contact mechanics is also discussed. On the spectrum of some eigenvalue problems Maria FARCASEANU University of Craiova, Romania The goal of this presentation is to emphasize different situations regarding the nature of the spectrum of some eigenvalue problems involving elliptic differential operators. More precisely, we will show that the spectrum of such an eigenvalue problem can be either discrete, or continuous or a combination of the above two cases involving a continuous part plus an isolated point Long-time behavior for nonlocal problems Liviu IGNAT, Tatiana IGNAT, Denisa STANCU-DUMITRU In this talk we will present some nonlocal evolution problems that involve operators of the type: Lu(x) = J(x y)(u(y) u(x)) dy R d We analyze the asymptotic behavior of the solutions of the following nonlocal convectiondiffusion equation u t = J u u + G u 2 u 2. 3

9 The results are mainly obtained by scaling arguments and a new compactness argument that is adapted to nonlocal evolution problems. The compactness tool is the following one: Let Ω R d be an open set. Let ρ : R d R be a nonnegative smooth radial function with compact support, non identically zero, and ρ n (x) = n d ρ(nx). Let {f n } n 1 be a sequence of functions in L p ((0, T ) Ω) such that and T n p 0 Ω Ω T 0 Ω f n p M ρ n (x y) f n (t, x) f n (t, y) p dxdydt M. If {f n } n 1 is weakly convergent in L p ((0, T ) Ω) to f then f L p ((0, T ), W 1,p (Ω)) for p > 1 and f L 1 ((0, T ), BV (Ω)) for p = 1. Let p > 1. Assuming that Ω is a smooth bounded domain in R d, ρ(x) ρ(y) if x y and that t f n L p ((0,T ),W 1,p (Ω)) M then {f n } n 1 is relatively compact in L p ((0, T ) Ω). Superposition operators between higher-order Sobolev spaces with applications to non-existence results of higher-order regular solutions Florin ISAIA Transilvania University of Brasov, Romania In this talk, a well-known non-existence result of Pohozaev for a Dirichlet problem with Laplacian is generalized to some non-existence results of higher-order regular solutions for some divergence-type equations and these results are subjected to the following natural principle: the stronger (respectively weaker) are the assumptions on the given data, the larger (respectively smaller) is the Sobolev space in which nontrivial solutions can be found. To do this, we use some recent developments of superposition operators between higher-order Sobolev spaces. Eigenvalue problems in Orlicz-Sobolev spaces for rapidly growing operators in divergence form Mihai MIHAILESCU University of Craiova & 4

10 Eigenvalue problems involving the p-laplacian and rapidly growing operators in divergence form are studied in an Orlicz-Sobolev setting. An asymptotic analysis of these problems leads to a full characterization of the spectrum of an exponential type perturbation of the Laplace operator. This is a joint work with Marian Bocea. On AT(n) property Radu MUNTEANU University of Bucharest, Romania T. Giordano and D. Handelman reformulated matrix valued random walks and their associated group actions in terms of dimension spaces, in order to deal with measure theoretic classification of ergodic actions of discrete groups. Their approach lead to a notion of rank called AT(n), n 1 This new concept generalizes approximate transitivity (shortly AT), a property of ergodic actions introduced by A. Connes and E. J. Woods in the theory of von Neumann algebras, which occurs for n = 1. It was proved by A. Connes and E. J. Woods that any measure preserving automorphism which is AT has zero entropy. We give a necessary condition for shift maps to be AT(n). We use this condition in order to prove that Bernoulli shifts are not AT(n) for any n 1. As a consequence, we obtain that any finite measure preserving transformation which is AT(n) has zero entropy. Some connections with the theory of von Neumann factors of type III 0 will be discussed. The asymptotic behavior of solutions to non-homogeneous eigenvalue problems in Orlicz-Sobolev spaces Denisa STANCU-DUMITRU The main interest of this talk is the limiting behavior of the sequence of positive first eigenfunctions for a class of nonhomogeneous eigenvalue problems in the setting of Orlicz- Sobolev spaces. After possibly extracting a subsequence, the sequence of positive first eigenfunctions converges uniformly to a viscosity solution of a nonlinear PDE involving the -Laplacian. This is a joint work with Marian Bocea (Loyola University Chicago) and Mihai Mihăilescu (University of Craiova) 5