Daniel S. Sage

Exact relations for effective tensors of composites: Necessary conditions and sufficient conditions

Typically the elastic and electrical properties of composite materials are strongly microstructure dependent. So it comes as a nice surprise to come across exact formulae for effective moduli that are universally valid no matter what the microstructure. Such exact formulae provide useful benchmarks for testing numerical and actual experimental data and for evaluating the merit of various approximation schemes. They can also be regarded as fundamental invariances existing in a given physical context. Classic examples include Hills formulae for the effective bulk modulus of a two-phase mixture when the phases have equal shear moduli, Levins formulae linking the effective thermal expansion coefficient and effective bulk modulus of two-phase mixtures, and Dykhnes result for the effective conductivity of an isotropic two-dimensional polycrystalline material. Here we present a systematic theory of exact relations embracing the known exact relations and establishing new ones. The search for exact relations is reduced to a search for matrix subspaces having a structure of special Jordan algebras. One of many new exact relations is for the effective shear modulus of a class of three-dimensional polycrystalline materials. We present complete lists of exact relations for three-dimensional thermoelectricity and for three-dimensional thermopiezoelectric composites that include all exact relations for elasticity, thermoelasticity, and piezoelectricity as particular cases.

Moduli Spaces of Irregular Singular Connections

In the geometric version of the Langlands correspondence, irregu- lar singular point connections play the role of Galois representations with wild ramification. In this paper, we develop a geometric theory of fundamental strata to study irregular singular connections on the projective line. Funda- mental strata were originally used to classify cuspidal representations of the general linear group over a local field. In the geometric setting, fundamen- tal strata play the role of the leading term of a connection. We introduce the concept of a regular stratum, which allows us to generalize the condition that a connection has regular semisimple leading term to connections with non-integer slope. Finally, we construct a symplectic moduli space of mero- morphic connections on the projective line that contain a regular stratum at each singular point.

Perverse coherent sheaves and the geometry of special pieces in the unipotent variety

Let X be a scheme of finite type over a Noetherian base scheme S admitting a dualizing complex, and let UX be an open set whose complement has codimension at least 2. We extend the Deligne�Bezrukavnikov theory of perverse coherent sheaves by showing that a coherent intermediate extension (or intersection cohomology) functor from perverse sheaves on U to perverse sheaves on X may be defined for a much broader class of perversities than has previously been known. We also introduce a derived category version of the coherent intermediate extension functor. Under suitable hypotheses, we introduce a construction (called �S2-extension�) in terms of perverse coherent sheaves of algebras on X that takes a finite morphism to U and extends it in a canonical way to a finite morphism to X. In particular, this construction gives a canonical �S2-ification� of appropriate X. The construction also has applications to the �Macaulayfication� problem, and it is particularly well-behaved when X is Gorenstein. Our main goal, however, is to address a conjecture of Lusztig on the geometry of special pieces (certain subvarieties of the unipotent variety of a reductive algebraic group). The conjecture asserts in part that each special piece is the quotient of some variety (previously unknown for the exceptional groups and in positive characteristic) by the action of a certain finite group. We use S2-extension to give a uniform construction of the desired variety.

THE GEOMETRY OF FIXED POINT VARIETIES ON AFFINE FLAG MANIFOLDS

Let G be a semisimple, simply connected, algebraic group over an algebraically closed field k with Lie algebra g. We study the spaces of parahoric subalgebras of a given type containing a fixed nil-elliptic element of gk((�)), i.e. fixed point varieties on affine flag manifolds. We definea natural class of k�-actions on affine flag manifolds, generalizing actions introduced by Lusztig and Smelt. We formulate a condition on a pair (N,f) consisting of N 2 g k((�)) and a k�-action f of the specified type which guarantees that f induces an action on the variety of parahoric subalgebras containing N. For the special linear and symplectic groups, we characterize all regular semisimple and nil-elliptic conjugacy classes containing a representative whose fixed point variety admits such an action. We then use these actions to find simple formulas for the Euler characteristics of those varieties for which the k � -fixed points are finite. We also obtain a combinatorial description of the Euler characteristics of the spaces of parabolic subalgebras containing a given element of certain nilpotent conjugacy classes of g.

TWISTED FROBENIUS-SCHUR INDICATORS FOR HOPF ALGEBRAS

Abstract The classical Frobenius–Schur indicators for finite groups are character sums defined for any representation and any integer m ⩾ 2 . In the familiar case m = 2 , the Frobenius–Schur indicator partitions the irreducible representations over the complex numbers into real, complex, and quaternionic representations. In recent years, several generalizations of these invariants have been introduced. Bump and Ginzburg, building on earlier work of Mackey, have defined versions of these indicators which are twisted by an automorphism of the group. In another direction, Linchenko and Montgomery have defined Frobenius–Schur indicators for semisimple Hopf algebras. In this paper, the authors construct twisted Frobenius–Schur indicators for semisimple Hopf algebras; these include all of the above indicators as special cases and have similar properties.

Isomonodromic Deformations of Connections with Singularities of Parahoric Formal Type

In previous work, the authors have developed a geometric theory of fundamental strata to study connections on the projective line with irregular singularities of parahoric formal type. In this paper, the moduli space of connections that contain regular fundamental strata with fixed combinatorics at each singular point is constructed as a smooth Poisson reduction. The authors then explicitly compute the isomonodromy equations as an integrable system. This result generalizes work of Jimbo, Miwa, and Ueno to connections whose singularities have parahoric formal type.

A Theory of Minimal

The theory of minimal K-types for p-adic reductive groups was developed in part to classify irreducible admissible representations with wild ramification. An important observation was that minimal K-types associated to such representations correspond to fundamental strata. These latter objects are triples (x, r, beta), where x is a point in the Bruhat-Tits building of the reductive group G, r is a nonnegative real number, and beta is a semistable functional on the degree r associated graded piece of the Moy-Prasad filtration corresponding to x. Recent work on the wild ramification case of the geometric Langlands conjectures suggests that fundamental strata also play a role in the geometric setting. In this paper, we develop a theory of minimal K-types for formal flat G-bundles. We show that any formal flat G-bundle contains a fundamental stratum; moreover, all such strata have the same rational depth. We thus obtain a new invariant of a flat G-bundle called the slope, generalizing the classical definition for flat connections. The slope can also be realized as the minimum depth of a stratum contained in the flat G-bundle, and in the case of positive slope, all such minimal depth strata are fundamental. Finally, we show that a flat G-bundle is irregular singular if and only if it has positive slope.

K

Let V be a complex representation of the compact group G. The subrepresentation semiring associated to V is the set of subrepresentations of the algebra of linear endomorphisms of V with operations induced by the matrix operations. The study of these semirings has been motivated by recent advances in materials science, in which the search for microstructure-independent exact relations for physical properties of composites has been reduced to the study of these semirings for the rotation group SO(3). In this case, the structure constants for subrepresentation semirings can be described explicitly in terms of the 6j-symbols familiar from the quantum theory of angular momentum. In this paper, we investigate subrepresentation semirings for the class of quasisimply reducible groups defined by Mackey [“Multiplicity free representations of finite groups,” Pac. J. Math. 8, 503 (1958)]. We introduce a new class of symbols called twisted 6j-symbols for these groups, and we explicitly calculate the structure constan...

-Types for Flat

In joint work with C. Bremer, the author has developed a geometric theory of fundamental strata which provides a new approach to the study of meromorphic G-connections on curves (for complex reductive G). In this theory, a fundamental stratum associated to a connection at a singular point plays the role of the local leading term of the connection. In this paper, we illustrate this theory for \(G = \mathfrak{g}\mathfrak{l}_{2}(\mathbb{C})\) (i.e. for connections on rank two vector bundles). In particular, we show how this approach can be used to construct explicit moduli spaces of irregular singular connections on the projective line with specified singularities and formal types.

G

AbstractLet G be a complex, semisimple, simply connected algebraic group withLie algebra