## Uladzimir Shtukar, NC Central University, “Canonical bases, subalgebras, reductive pairs of Lie algebras, and possible applications”

Subalgebras of Lie algebra of Lorentz group will be discussed as the basic examples at the beginning of the report. The corresponding analysis is performed by canonical bases for subspaces of a vector space. All canonical bases for 5-dimensional and…

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## Jamie Pommersheim, Reed College, “Euler-Maclaurin summation formulas for polytopes”

Discovered in the 1730s, the classical Euler-Maclaurin formula may be viewed as a formula for summing the values of a function over the lattice points in a one-dimensional polytope. Several years ago, Berline and Vergne generalized this formula to polytopes…

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## Rosa Orellana, Dartmouth College, “The partition algebra, symmetric functions and Kronecker coefficients”

The Schur-Weyl duality between the symmetric group and the general linear group allows us to connect the representation theory of these two groups. A consequence of this duality is the Frobenius formula which connects the irreducible characters of the general…

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## Max Glick, University of Connecticut, “The Berenstein-Kirillov group and cactus groups”

Berenstein and Kirillov have studied the action of Bender-Knuth moves on semistandard tableaux. Losev has studied a cactus group action in Kazhdan-Lusztig theory; in type A this action can also be identified in the work of Henriques and Kamnitzer. We…

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## Mark Shimozono, Virginia Tech, “Quiver Hall-Littlewood symmetric functions and Kostka-Shoji polynomials”

We associate to any quiver a family of symmetric functions, defined by creation operators which are generalizations of Jing's creation operators. For the cyclic quiver the coefficient polynomials were studied by Finkelberg and Ionov. Shoji has recently shown that the…

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## Ben Cox, College of Charleston, “On the universal central extension of certain Krichever-Novikov algebras”

We will describe results on the center of the universal central extension of certain Krichever-Novikov algebras. In particular we will describe how various families of classical and non-classical orthogonal polynomials appear. We will also provide certain new identities of elliptic…

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## Rekha Biswal, Université Laval, “Demazure flags, Chebyshev polynomials and mock theta functions”

The g-stable Demazure modules are a lot of interest because of their connections to representation theory of quantum affine algebras. These modules are indexed by a pair (ell, lambda) where ell is a positive integer and lambda is a dominant…

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## Richard Rimanyi, UNC Chapel Hill, Counting partitions and quantum dilogarithm identities

In the theory of Donaldson-Thomas invariants for quivers one finds identities for quantum dilogarithm series. The combinatorial interpretation of the simplest of these identities is equivalent to a clever way of counting partitions. The combinatorial interpretation of more involved dilogarithm…

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## Leonardo Mihalcea, Chern-Schwartz-MacPherson classes for Schubert cells: geometry and representation theory

A compact manifold has a tangent bundle, and a natural question is to find a replacement for the Chern classes of the tangent bundle, in the case when the space is singular. The Chern-Schwartz-MacPherson (CSM) classes are homology classes which…

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## Heekyoung Hahn, Duke University, Langlands’ beyond endoscopy proposal and related questions on algebraic groups and combinatorics

Langlands' beyond endoscopy proposal for establishing functoriality motivates the study of irreducible subgroups of $\mathrm{GL}_n$ that stabilize a line in a given representation of $\mathrm{GL}_n$. Such subgroups are said to be detected by the representation. In this talk we present…

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## Joshua Hallam, Wake Forest University, Whitney duals of graded partially ordered sets

To each graded poset one can associate two sequences of numbers; the Whitney numbers of the first kind and the Whitney numbers of the second kind. One sequence keeps track of the Möbius function at each rank level and other…

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## Shira Viel, NC State,Folding and dominance: relationships among mutation fans for surfaces and orbifolds

The $n$-associahedron is a well-known $n$-dimensional polytope whose vertices are labeled by triangulations of an $(n+3)$-gon with edges given by diagonal flips. The $n$-cyclohedron is defined analogously using centrally-symmetric triangulations of a $(2n+2)$-gon, or, modding out by the symmetry, triangulations…

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## Tomoyuki Arakawa, RIMS and MIT, Vertex algebras and symplectic varieties

In my talk I will discuss some remarkable correspondence between symplectic varieties and vertex algebras, which has been discovered by physicists in the study of the four-dimensional N = 2 superconformal field theories. In the special class of the four-dimensional N = 2 superconformal…

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## Michael Singer, NC State, Walks, Difference Equations and Elliptic Curves

Many questions in combinatorics, probability and thermodynamics can be reduced to counting lattice paths (walks) in regions of the plane. A standard approach to counting problems is to consider properties of the associated generating function.  These functions have long been well understood for walks…

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