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Topics in Algebraic Geometry’ International Conference of The Indian Mathematics Consortium in cooperation with American Mathematical Society Symposium on `Topics in Algebraic Geometry' Organisers: Ramesh Sreekantan (Indian Statistical Institute, Bangalore, India) and Patrick Brosnan (University of Maryland, USA ) A1-Connectedness in Reductive Groups Chetan Balwe IISER Mohali, India We examine the notion of A1-connectedness, in the sense of the A1-homotopy theory of Morel-Voevodsky, on schemes, focusing specifically on reductive groups. We do so by compar- ing this notion with the equivalence relation on points induced by \naive" A1-homotopies. This allows us to employ geometric methods and, in some cases, relate the sheaf of A1- connected components with the notion of R-equivalence. We obtain a complete character- ization of A1-connected reductive groups. This talk is based on joint work with Anand Sawant. Hessenberg Varieties and the Shareshian-Wachs Conjecture Patrick Brosnan University of Maryland, USA I will explain joint work with Tim Chow, which proves a conjecture of Shareshian and Wachs relating Hessenberg varieties to chromatic symmetric polynomials. Hodge Theory on Matroids Eric Katz Ohio State University, USA The chromatic polynomial of a graph counts its proper colourings. This polynomial's coefficients were conjectured to form a unimodal sequence by Read in 1968. This conjecture was extended by Rota in his 1970 address to assert the log-concavity of the characteristic polynomial of matroids which are the common generalizations of graphs and linear subspaces. We discuss the resolution of this conjecture which is joint work with Karim Adiprasito and June Huh. The solution draws on ideas from the theory of algebraic varieties, specically Hodge theory, showing how a question about graph theory leads to a solution involving Grothendieck's standard conjectures. ICTIMCAMS 14-17 December 2016 International Conference of The Indian Mathematics Consortium in cooperation with American Mathematical Society Bloch-Quillen Formula for Singular Schemes Amalendu Krishna Tata Institute of Fundamental Research, Mumbai, India The Bloch-Quillen formula in the theory of algebraic cycles provides a formula for the Chow groups of 0-cycles on a smooth quasi-projective scheme in terms of the Zariski co- homology of the Quillen K-theory sheaves. Such a formula for singular schemes is not yet known, except in some very special cases. In this talk, I will present such a formula of certain class of singular schemes. Modular Forms and Higher Chow Cycles Ramesh Sreekantan Indian Statistical Institute, Bangalore, India A consequence of some conjectures in Arithmetic Algebraic Geometry is a link between higher Chow cycles on the self-product of the universal abelian surface over a Shimura curve and modular forms on the curve. We will explain this precisely and prove the conjecture in the special case when the family is the self-product of the universal elliptic curve over a modular curve. As a result of this we obtain some results on the algebraicity of higher Greens functions predicted by Gross and Zagier. Motivic Intersection Complex of an Arbitrary Threefold Vaibhav Vaish Indian Statistical Institute, Bangalore, India We formalize a notion of \punctual gluing" of t-structures. This allows us to construct analogue of certain S.Morel's weight truncations in the motivic setting. As an application we can construct the analogue of intersection complex for an arbitrary threefold in Voevodsky's triangulated category of mixed motives. Symposium on Algebraic Groups Organisers: K.N. Raghavan (IMSc, Chennai, India) and V. Lakshmibai (Northeastern University, Boston, USA) Elliptic Quantum Groups Sachin Gautam Perimeter Institute for Theoretical Physics, Canada ICTIMCAMS 14-17 December 2016 International Conference of The Indian Mathematics Consortium in cooperation with American Mathematical Society In 1995 G. Felder introduced an elliptic R-matrix, which quantizes the classical dynamical r-matrix arising from the study of conformal blocks on elliptic curves. The elliptic R-matrix satisfies a dynamical analog of the Yang-Baxter equation and can be used to define the elliptic quantum group of sln in the same vein as the usual R-matrices gives rise to quantum groups via the RT T formalism of Faddeev, Reshetikhin and Takhtajan. In this talk I will explain Felder's definition and present its generalization to the case of arbitrary Kac-Moody Lie algebras analogous to the Drinfeld's new presentation of Yangians and quantum loop algebras. I will also present a method of constructing representations of the elliptic quantum group using q-difference equations. Our construction gives rise to a classification of irreducible representations of the elliptic quantum group, which is reminiscent of the Drinfeld's classification of irreducible representations of Yangians. This talk is based on a joint work with V. Toledano Laredo. Representation Ring of Levi Subgroups versus Cohomology Ring of Flag Varieties Shrawan Kumar University of North Carolina, Chapel Hill, USA Recall the classical result that the cup product structure constants for the singular cohomology with integral coefficients of the Grassmannian of r-planes coincide with the Littlewood-Richardson tensor product structure constants for GL(r). Specifically, the result ∗ asserts that there is an explicit ring homomorphism φ : Reppoly(GL(r)) to H (Gr(r; n)), n where Gr(r; n) denotes the Grassmannian of r-planes in C and Reppoly(L( r)) denotes the polynomial representation ring of GL(r). This talk seeks to achieve one possible generalization of this classical result for GL(r) and the Grassmannian Gr(r; n) to the Levi subgroups of any reductive group G and the corresponding flag varieties. On Complete Reducibility in Characteristic p A. J. Parameswaran Tata Institute of Fundamental Research, Mumbai, India P Serre showed that if semi-simple representations Vi of a group Γ are such that (dimVi − 1) < p, then their tensor product is semisimple. Later he proved that semisimple represen- tations restrict to semisimple representations of a subgroup Γ if it is a Gcr subgroup. Recently, Deligne generalized the first result to doubly saturated subgroup schemes of a reductive group. In this talk, we consider the case when Γ is a doubly saturated Gcr subgroup scheme of a reductive group G and generalize Serre's results to low height representations. Further we show an analogue of Luna's ´etaleslice theorem for low height representations of reductive groups. ICTIMCAMS 14-17 December 2016 International Conference of The Indian Mathematics Consortium in cooperation with American Mathematical Society This is a joint work with V. Balaji and P. Deligne. Odd Dimensional Representations in Bratelli Diagrams of Weyl Groups and Bijective McKay Correspodences Amritanshu Prasad Institute of Mathematical Sciences, Chennai, India The induced subgraphs formed by odd-dimensional representations in the Bratelli di- agrams of Weyl groups of type An, Bn and Dn turn out to have surprisingly nice struc- ture. This structure suggests explicit bijections between the odd dimensional representations of these groups and one-dimensional representations of their 2-Sylow subgroups (bijective McKay correspondences). This talk is based on joint work with Arvind Ayyer, Karimilla Bi, and Steven Spallone. A Polynomial Time Algorithm to Determine Membership in the Null Cone of Matrix Semi-invariants Venkata Subrahmanyam Chennai Mathematical Institute, Chennai, India In 1967, J. Edmonds introduced the problem of computing the rank over the rational function field of an n × n matrix T with integral homogeneous linear polynomials. We consider the non-commutative version of Edmonds'problem: compute the rank of T over the free skew field. This problem has been proposed from several different perspectives in the study of, for example, the free skew field itself (Cohn 1973), matrix spaces of low rank (Fortin-Reutenauer,2004), Edmonds' original problem (Gurvits, 2004), and more recently, non-commutative arithmetic circuits with divisions (Hrubeˇsand Wigderson, 2014). It is known that this problem relates to the following invariant ring of matrix semi- invariants denoted R(n; m). For a field F it is the ring of invariant polynomials for the action of SL(n; F ) × SL(n; F ) on tuples of matrices - (A; C) 2 SL(n; F ) × SL(n; F ) sends ⊕m T T (B1; :::; Bm) 2 M(n; F ) to (AB1C ; :::; ABmC ). Then those T with non-commutative rank < n correspond to points in the nullcone of R(n; m). Recently Garg et al. (FOCS 2016) showed that the non-commutative rank problem is in P over Q, building on an algorithm of Gurvits (JCSS 2004). Our main result is a deterministic algorithm for the non-commutative rank problem us- ing polynomially many arithmetic operations over any field. Over Q the bit complexity of the algorithm is also polynomial. It is constructive in that it produces witnesses certi- fying the non-commutative rank. We give another proof that the nullcone of R(n; m) is defined by invariants of polynomial degrees, a result recently proved by Derksen and Makam (arXiv:1512.03393). This is joint work with G´labor Ivanyos and Yoming Qiao. ICTIMCAMS 14-17 December 2016 International Conference of The Indian Mathematics Consortium in cooperation with American Mathematical Society Equivariant K-Theory of Flag Varieties Revisited and Related Results V. Uma Indian Institute of Technology, Madras, India In this talk we shall obtain many results on the multiplicative structure constants of the T -equivariant Grothendieck ring KT
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