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A: Báo Cáo M I Toàn Th A: B¡o c¡o mời toàn thº A1VMC741: Tannakian duality and applications Phung Ho Hai Institute of Mathematics, Hanoi Tannakian duality aims to answer the question: how much can the representations of a group tell us about the group itself. The starting point was the celebrated duality for locally compact abelian groups due to Pontryagin (1930s). For com- pact groups, an affirmative answer was given by the works of Tannaka and Krein (1930s): a compact group is uniquely determined by its finite dimensional complex representations. The case of affine algebraic groups was studied by Grothendieck and his students Saavedra and Deligne (1970-1980). There are many applications of Tannakian duality ranging from representations theory, number theory, algebraic geometry to mathematical physics, most notably the invention of quantum groups due to Drinfeld. The aim of this talk is to review recent results on Tannakian duality and explain some applications to the study of the fundamental group of algebraic schemes. A2VMC402: Bài to¡n ngược trong c¡c qu¡ tr¼nh truy·n nhi»t và khuy¸ch t¡n Đinh Nho Hào Vi»n To¡n học, Hà Nëi Nhi·u hi»n tượng cõa tự nhi¶n, khoa học, kỹ thuªt, y học, địa vªt lý . được mô t£ b¬ng c¡c mô h¼nh to¡n học, nhưng mët sè tham sè, điều ki»n trong c¡c mô h¼nh này không đo đạc được trực ti¸p mà ph£i x¡c định qua c¡c đo đạc gi¡n ti¸p. Đó ch½nh là c¡c bài to¡n ngược. C¡c bài to¡n ngược thường đặt không ch¿nh, nghĩa là không ph£i với dú ki»n đo đạc nào bài to¡n cũng có nghi»m và n¸u như nghi»m tồn t¤i, th¼ chúng chưa h¯n là duy nh§t; ngoài ra nghi»m không phụ thuëc li¶n tục vào dú ki»n cõa bài to¡n. V¼ nghi»m không phụ thuëc li¶n tục vào dú ki»n cõa bài to¡n, n¶n mët sai sè nhỏ trong dú ki»n cũng có thº d¨n đến nhúng sai sè lớn tùy ý trong lời gi£i, do đó vi»c gi£i sè c¡c bài to¡n đặt không ch¿nh là không đơn gi£n. Trong b¡o c¡o này, chúng tôi đề cªp đến mët sè bài to¡n ngược trong qu¡ tr¼nh truy·n nhi»t và khu¸ch t¡n được mô t£ b¬ng phương tr¼nh đạo hàm ri¶ng parabolic và elliptic. Chúng tôi tr¼nh bày k¸t qu£ v· t½nh gi£i được, đánh gi¡ ên định, phương ph¡p ch¿nh hóa và phương ph¡p sè húu hi»u để gi£i c¡c bài này. A3VMC346: Variational convergence of unifunctions and bifunctions with applications in optimization 1 Phan Quoc Khanh International University, VNU-HCMC One calls a convergence notion a variational convergence if it preserves extremal properties in nature, i.e., if elements of a convergent sequence possess such proper- ties then so does the limit. In this talk we present first epi convergence of univariate functions and its variational properties. Then, we focus on the two extensions to bivariate functions: lopsided convergence and epi/hypo convergence. These three notions are the main kinds of variational convergence. We discuss also conditions additional to the convergence to ensure higher-level and almost complete preserva- tions of variational properties. Among these conditions tightness assumptions are the most important. We provide also various characterizations of the mentioned no- tions of convergence. Applications in both deterministic and stochastic optimization problems are included in our presentation as well. A4VMC723: Some recent results on pseudoconvex domains of infinite type Tran Vu Khanh Tan Tao University, Long An Pseudoconvexity is one of the most central concepts in modern complex analysis. There are two kinds of pseudoconvexity: finite type and infinite type. A lots of work has been done about pseudoconvexity of finite type. We are interested in the infinite type case. In this report we present some recent results on pseudoconvex domains of infinite type. The content of the report consists of: 1. Lp-estimates for the Henkin solution to the @¯-problem; 2. Regularity of the @¯-Neumann problem; 3. Lower bounds of the Bergman metric and Kobayashi metric; 4. Estimates for the “type" of the boundary. A5VMC753: Representations of finite groups and applications Pham Huu Tiep University of Arizona, USA In the first part of the talk we will survey recent results in the representation theory of finite groups. In the second part we will describe some applications of these results in group theory, number theory, and algebraic geometry 2 D: Đại sè - H¼nh học - Tôpô D1VMC742: Subgroups of the multiplicative group of a division ring Bui Xuan Hai (joint work with Nguyen Anh Tu and Nguyen Kim Ngoc) University of Science, VNU-HCMC Let D be a division ring and D∗ be the multiplicative group of D. There is a closed relation between subgroup structure of D∗ and algebraic properties of D. The first result reflecting this relation is the famous theorem of Wedderburn in 1905 asserting that every finite division ring is commutative. This remarkable result have stimulated many further investigations of mathematicians (among them we recall Jacobson, Faith, Kaplansky, Herstein,...) to generalise it. At the present, this problem of research remains still one of subjects which attracts the attention of many authors around the world. A particular interest is that to study maximal subgroups of D∗. On the other hand, it is well-known from old results that subnormal subgroups in D∗ behave as D∗ in several manners. For instance, from the ealier result of Scott in 1957, we see that it does not exist any abelian non-central normal subgroup in a non-commutative division ring. This is a particular case of the most important result concerning subnormal subgroup structure obtained later in 1964 by Stuth asserting that every soluble subnormal subgroup of D∗ is central. We notice also Herstein's result in 1956 asserting that xD∗ is infinite for every non-central element x of D∗. This result was extended by Scott in 1957, when he proved that jxD∗ j = jDj. Moreover, Scott also proved that if G is a non-central subnormal subgroup then for every non-central element x of D, the division subring generated by xG is D. There are also other results of several authors showing that subnormal subgroups of D∗ are, roughly speaking, “ big". Our purpose in this report is to present subgroup structure of an arbitrary subnormal subgroup G in a division ring D and its ifluence to the structure of D . In particular, we study maximal subgroups of G. For the last time, several authors study maximal subgroups of D∗ and many nice properties of such subgroups were obtained. Studying maximal subgroups of G, we get in many cases the similar results for these subgroups as the results obtained previously for maximal subgroups of D∗. Another problem we study is the question of the existence of non-cyclic free sub- groups in subnormal subgroups and maximal subgroups of a division ring. This problem plays an important role for the understanding of the structure of division rings. D2VMC754: Homological realization of a combinatorial identity of G. Andrews 3 Nguyen Dang Ho Hai College of Science, Hue University In studying the generating function of Minc's partitions, G. Andrews established a combinatorial identity which suggests the existence of a minimal injective resolution of an interesting object in the category of unstable modules over the modulo two Steenrod algebra. This talk gives a report on the joint work with Lionel Schwartz and Tr¦n. N. Nam, where we construct this injective resolution and derive some of its homotopical consequences. D3VMC327: On the k-Buchsbaum property of powers of Stanley-Reisner ideals Nguy¶n Công Minh Hanoi National University of Education This is a joint work with Yukio Nakamura. Let S = K[x1; x2; :::; xn] be a polynomial ring over a field K. Let ∆ be a simplicial complex whose vertex set is contained in f1; 2; :::; ng. For an integer k ≥ 0, we investigate the k-Buchsbaum property (t) t of residue class rings S=I and S=I for the Stanley-Reisner ideal I = I∆. We characterize the k-Buchsbaumness of such rings in terms of the simplicial complex ∆ and the power t. D4VMC396: Some results on local cohomology modules Le Thanh Nhan Thai Nguyen College of Sciences Let (R; m) be a Noetherian local ring, I an ideal of R and M a finitely generated d i R-module with dim M = d: It is well known that HI (M) and Hm(M) are Artinian d i R-modules for all i. When A = HI (M) or A = Hm(M); in general A does not satisfy the following property AnnR(0 :A p) = p for all p 2 Var(AnnRA): (∗) The first aim of this talk is to characterize the property (*) for these Artinian local cohomology modules in the relations with the catenarity, the universal catenarity, the Cohen-Macaulayness of the formal fibers of the base ring R as well as the closedness of pseudo supports of M. Then we describe the attached primes, the co-supports d i and the associativity formular for multiplicity of HI (M) and Hm(M), cf. [2,5,6]. Secondly, consider the case where R is a quotient of a Gorenstein local ring. Denote by K(M) the canonical module of M.
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