## Talks (titles and abstracts)

Anton Alekseev: The Horn problem and planar networks

The Horn equalities (proved by Klyachko and Knutson-Tao) describe eigenvalues of a sum of two Hermitian matrices in terms of eigenvalues of the summands. Surprizingly, the same set of inequalities comes up in the theory of planar networks equipped with Boltzmann weights taking values in the tropical semi-field. The link between two setups is provided by the theory of Poisson-Lie groups.

The talk is based on a joint work with M. Podkopaeva and A. Szenes.

Martina Balagovic (short talk): Rational Cherednik algebras in positive characteristic

Vladimir Bavula (short talk): An analogue of the Dixmier conjecture is true for the algebra of polynomial integro-differential operators

In 1968, Dixmier posed six problems for the algebra of polynomial differential operators, i.e. the Weyl algebra. In 1975, Joseph solved the third and sixth problems and, in 2005, I solved the fifth problem and gave a positive solution to the fourth problem but only for homogeneous differential operators. The remaining three problems are still open. The first problem/conjecture of Dixmier (which is equivalent to the Jacobian Conjecture as was shown in 2005-07 by Tsuchimito, Belov and Kontsevich) claims that the Weyl algebra 'behaves' as a finite field extension. In more detail, the first problem/conjecture of Dixmier asks: is true that an algebra endomorphism of the Weyl algebra is an automorphism? In 2010, I proved that this question has an affirmative answer for the algebra of polynomial integro-differential operators. In my talk I will explain the main ideas, the structure of the proof and recent progress on the first problem/conjecture of Dixmier.

Yuri Berest: Derived representation schemes and cyclic homology

The affine representation scheme $$Rep_n(A)$$ parametrizing the n-dimensional representations of an associative algebra A defines a functor on the category of algebras. In this talk, I will describe the derived functor of $$Rep_n$$ in the category of $$DG$$ schemes and construct higher trace maps relating the (stable) homology of $$DRep_n$$ to cyclic homology. Time permitting, I will discuss variations and some applications of this construction in low-dimensional topology.

The talk is based on joint work with Ajay Ramadoss.

Alexander Braverman: Semi-infinite Schubert varieties and difference equations

William Crawley-Boevey: Multiplicative preprojective algebras as a receptacle for monodromy

Multiplicative preprojective algebras associated to quivers were introduced by Crawley-Boevey and Shaw in order to study (in case the underlying quiver is star-shaped) problems concerning the monodromy of differential equations on the Riemann sphere. I will show how monodromy for suitable systems of differential equations on a union of Riemann surfaces leads to representations of multiplicative preprojective algebras for an arbitrary quiver.

Will Donovan (short talk): Window shifts, flop equivalences and Grassmannian twists

Laszlo Feher: The Ruijsenaars self-duality map as a mapping class symplectomorphism

We explain that the self-duality symplectomorphism of the completely integrable compactified trigonometric Ruijsenaars-Schneider system arises from the natural action of the mapping class group on the moduli space of flat SU(n) connections on the one-holed torus. The talk is based on joint work with C. Klimcik reviewed in arXiv:1203.3300.

Victor Ginzburg: D-modules, W-algebras, and the affine Grassmannian

Iain Gordon: Macdonald positivity via the Harish-Chandra D-module

David Jordan: Quantized multiplicative quiver varieties

We introduce a new class of algebras $$D_q(Mat_d(Q))$$ associated to a quiver $$Q$$ and dimension vector $$d$$, which yield a flat (PBW) $$q$$-deformation of the algebra of differential operators on the space of matrices associated to $$Q$$. This algebra admits a $$q$$-deformed moment map from the quantum group $$U_q(gl_d)$$, acting by base change at each vertex. The quantum Hamiltonian reduction, $$A^\xi_d(Q)$$, of $$D_q$$ by $$\mu_q$$ at the character $$\xi$$, is simultaneously a quantization of the Crawley Boevey and Shaw's multiplicative quiver variety, and a $$q$$-deformation of Gan and Ginzburg's quantized quiver variety.

Specific instances of the data $$(Q,d,\xi)$$ yield $$q$$-deformations of familiar algebras in representation theory: for example, the spherical DAHA's of type $$A$$ arise from Calogero-Moser quivers, quantizations of parabolic character varieties (Deligne Simpson moduli spaces) arise from comet-shaped quivers, and algebras of difference operators on Kleinian singularities arise from affine Dynkin quivers.

Mikhail Kapranov: Higher Segal spaces

Segal spaces form a class of simplicial topological spaces which are used to model higher categories. The talk will discuss a "2-dimensional" generalization of this class, which relates to the usual Segal spaces in roughly the same way as triangulations of a plane polygon relate to subdivisions of an interval. These "2-Segal spaces" provide a general framework for constructing Hall algebras (the classical Hall algebras correspond to the Waldhausen space of algebraic K-theory). In a very special case, 2-Segal spaces reduce to set-theoretic solutions of the pentagon equation.

Joint work with T. Dyckerhoff.

Alastair King: Grassmannian cluster algebras

I will describe a categorification of the homogeneous coordinate ring of a Grassmannian, regarded as a cluster algebra

Sophie Morier-Genoud: From Hurwitz problem on sums-of-squares identities to wireless communication

At the end of the 19th century Hurwitz formulated a problem on the existence of sums-of-squares identities. This problem remains widely open. It turns out that the question can be rephrased in various ways and arises in many different contexts, algebraic, geometric, topologic... Furthermore, Hurwitz's work on sums-of-squares is extensively used by engineers in information theory to construct certain codes for wireless communication. We will recall the problem of Hurwitz and discuss some of the related topics. We will describe a method to construct solutions to the problem explicitly. The method is based on nonassociative algebras generalizing the octonions.

Eric Opdam: Dirac induction for graded affine Hecke algebras

The Dirac operator for graded affine Hecke algebras was introduced by Barbasch, Ciubotaru and Trapa (2010). Based on this construction we define Dirac induction for graded affine Hecke algebras, both algebraically and analytically. This has applications to harmonic analysis and points at a remarkable link between the representation theory of the Pin-cover of a Weyl group and the elliptic and discrete series characters of the associated graded affine Hecke algebras. In particular the discrete series representations can be constructed in this way.

(Based on joint work with Dan Ciubotaru and Peter Trapa).

Jeremy Pecharich (short talk): Derived symplectic reduction

We will discuss the notion of a derived symplectic form on a derived scheme due to recent work of Pantev, Toen, Vaquie, and Vezzosi. We will then show that the natural derived enhancement of classical symplectic reduction has a derived symplectic form.

Markus Reineke: GW/QM correspondence

The tropical vertex, a group of formal automorphisms of a torus, relates two seemingly unrelated geometries: the Gromow-Witten theory of certain toric surfaces (by work of Gross-Pandharipande-Siebert), and moduli spaces of representations of certain quivers. We will review these geometries and discuss a general factorization formula in the tropical vertex relating Gromov-Witten invariants and Euler characteristics of quiver moduli. Examples and special classes admitting explicit formulas will be discussed.

Olivier Schiffmann: Hall algebras of curves

Eric Vasserot: Affine W-algebras and quiver varieties

Andrei Zelevinsky: Triangular bases in quantum cluster algebras

In a joint work in progress with Arkady Berenstein, we develop a new approach to the problem of constructing a "natural" basis in an acyclic quantum cluster algebra. This approach is based on a suitable modification of Lusztig's lemma. Thus it is close in spirit to the well known-constructions of the Kazhdan-Lusztig basis in a Hecke algebra and of Lusztig's canonical basis for quantum groups.

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