The RTGC Research Seminar is held every Friday from 4:10 - 5:30 in 939 Evans (unless otherwise specified).
 Date Title Speaker 01/31 Quiver varieties, stability conditions, and Mirkovic-Vilonen polytopes (Friday, 4:10 - 5:30, 939 Evans) Peter Tingley , Loyola University, Chicago 02/07 Sigma models, elliptic cohomology and the Witten genus (Friday, 4:10 - 5:30, 939 Evans) Daniel Berwick Evans , Stanford 02/14 Classical Lie superalgebras, translation functors, parabolic induction and categorification of Fock space. (Friday, 4:10 - 5:30, 939 Evans) Vera Serganova , UC Berkeley 02/28 The Amplituhedron (Friday, 3:00 - 5:30, 939 Evans) Jaroslav Trnka, Caltech 03/07 Isomonodromic tau-functions, conformal blocks and supersymmetric gauge theory. (Friday, 4:10 - 5:30, 939 Evans) Joerg Teschner , DESY, University of Hamburg 03/13 Algebraic Geometry and Computer Vision (Thursday, 2:10 pm, 891 Evans) Luke Oeding , Auburn University 03/21 Poisson sigma models, Lie third theorem and Lagrangian correspondences (Friday, 4:10 - 5:30, 939 Evans) Ivan Contreras, UC Berkeley 04/04 Soergel bimodules and the Decomposition theorem (Friday, 4:10 - 5:30, 939 Evans) Ben Elias, MIT 04/11 Khovanov homology from categorified quantum groups (Friday, 4:10 - 5:30, 939 Evans) Hoel Queffelec , Institut de mathématiques de Jussieu 04/18 Affine permutations and rational slope parking functions (Friday, 4:10 - 5:30, 939 Evans) Monica Vazirani , UC Davis 04/25 Double total ramifications for curves of genus 2 (Friday, 4:10 - 5:30, 939 Evans) Nicola Tarasca, University of Utah 04/25 Non semi-simple TQFTs and Reidemeister torsion (Friday, 3:10 - 4:00, 939 Evans) Christian Blanchet , Institut de Mathématiques de Jussieu 05/02 Heisenberg-picture TQFTs (Friday, 4:10 - 5:30, 939 Evans) Theo Johnson-Freyd , Northwestern

Abstracts:

Quiver varieties, stability conditions, and Mirkovic-Vilonen polytopes, Peter Tingley :

Lusztig's nilpotent quiver varieties can be used to index the crystal basis for U−(g), where g is a simply laced complex simple Lie algebra. Various combinatorial objects also can be used to index this crystal, and we are interested in how they relate to the geometry. In particular we consider Mirkovic-Vilonen (MV) polytopes. To relate these to quiver varieties one uses Harder Narasimhan filtrations for various stability conditions on representations of the preprojective algebra. Perhaps the most interesting, we use this to extend the theory of MV polytopes beyond finite type. This is joint work with Joel Kamnitzer and Pierre Baumann. I will also mention recent work with John Claxton involving explicit type A combinatorics.

Sigma models, elliptic cohomology and the Witten genus, Daniel Berwick Evans :

I will describe a geometric model for elliptic cohomology with complex coefficients motivated by the physics of supersymmetric sigma models. One upshot of this language is a straightforward construction of the Witten genus of a smooth manifold equipped with a rational string structure. Time permitting, I will explain a relationship between (delocalized) equivariant elliptic cohomology and gauged sigma models.

Classical Lie superalgebras, translation functors, parabolic induction and categorification of Fock space., Vera Serganova :

After brief review of representation theory of Lie superalgebras, we will discuss two categorification constructions. The first one categorifies gl(infinity) by translation functors. The second construction categorifies free fermion action on Fock space using parabolic induction.

The Amplituhedron, Jaroslav Trnka:

In this talk we show that scattering amplitudes in N=4 SYM correspond to “The Amplituhedron”, a natural generalization of convex polygons into the Grassmannian. The amplitude itself is represented by the form with logarithmic singularities on the boundaries of this space. All physical concepts like Locality or Unitarity are just derived properties from Amplituhedron geometry.

Isomonodromic tau-functions, conformal blocks and supersymmetric gauge theory., Joerg Teschner :

We'll present a simple explanation for the recently discovered relations between the isomonodromic tau-functions and Virasoro conformal blocks at c=1. We will point out possible implications for the correspondence between Liouville theory and supersymmetric gauge theories discovered by Alday, Gaiotto and Tachikawa.

Algebraic Geometry and Computer Vision, Luke Oeding :

In Computer Vision and multi-view geometry one considers several cameras in general position as a collection of projection maps. One would like to understand how to reconstruct the 3-dimensional image from the 2-dimensional projections. [Hartley-Zisserman] (and others such as Alzati-Tortora and Papadopoulo-Faugeras) described several natural multi-linear (or tensorial) constraints which record certain relations between the cameras such as the epipolar, trifocal, and quadrifocal tensors. (Don't worry, I will stop at quadrifocal tensors!) A greater understanding of these tensors is needed for Computer Vision, and Algebraic Geometry and Representation Theory provide some answers. I will describe a uniform construction of the epipolar, trifocal and quadrifocal tensors via equivariant projections of a Grassmannian. Then I will use the beautiful Algebraic Geometry and Representation Theory, which naturally arrises in the construction, to recover some known information (such as symmetry and dimensions) and some new information (such as defining equations). Part of this work is joint with Chris Aholt (Microsoft).

Poisson sigma models, Lie third theorem and Lagrangian correspondences, Ivan Contreras:

In this talk we will present an alternative solution of the problem of integration of Poisson structures, through the phase space construction of the Poisson sigma model with boundary. We will describe this integration in terms of immersed canonical relations and we will mention some possible extensions of this construction for different categories, in a way to understand the quantization of Poisson manifolds via symplectic groupoids.

Soergel bimodules and the Decomposition theorem, Ben Elias:

In a 1979 paper, Kazhdan and Lusztig introduced the so-called Kazhdan-Lusztig basis of the Hecke algebra associated to a Coxeter group. They posited a number of deep algebraic conjectures about this basis, which inspired the blossoming field of geometric representation theory. Not long thereafter, these conjectures were proven for Weyl groups using the intersection cohomology of Schubert varieties, the key tool being the Decomposition Theorem of Beilinson-Bernstein-Deligne. However, this proof will not generalize to arbitrary Coxeter groups, because of the lack of any corresponding geometric setup. Soergel proposed an approach to proving these conjectures algebraically. He constructed a category of bimodules, known asSoergel bimodules, which agree with the equivariant intersection cohomology of Schubert varieties in Weyl group type, but are defined in an algebraic and combinatorial way for any Coxeter group. The Soergel conjecture states that the "characters" of the indecomposable Soergel bimodules are precisely the Kazhdan-Lusztig basis. This is an algebraic analog of the decomposition theorem, and it implies Kazhdan and Lusztig's conjectures. Inspired by de Cataldo and Migliorini's Hodge-theoretic proof of the Decomposition Theorem, Geordie Williamson and I recently gave an algebraic proof of the Soergel conjecture for a general Coxeter group. The key idea was to show that Soergel bimodules have the Hodge-theoretic properties expected of an equivariant intersection cohomology space.

Khovanov homology from categorified quantum groups, Hoel Queffelec :

About 15 years ago, Khovanov introduced an homological invariant of knots categoryfying the combinatorial skein version of the Jones polynomial. He later developed, with Lauda, a categorified analog of quantum groups, in which the definitions of the Jones polynomial in terms of representation theory have their roots. I will review both Khovanov homology and diagrammatic categorifications of Uq(slm), before presenting the map relating both categorifications. This allows for a presentation of Khovanov homology entirely in terms of higher representation theory, and paves the way for an extension of cobordism categories to general sln invariants.

Affine permutations and rational slope parking functions, Monica Vazirani :

The Catalan numbers 1, 2, 5, 14 ... comprise one of the most well-known sequences in combinatorics. This sequence enumerates over 100 families of combinatorial objects. Some of these families include the set of non-decreasing parking functions on $[n]$, the points in the root lattice $Q$ intersect a dilation $D_n^{n+1}$ of the fundamental alcove, the $n+1$-restricted" $w \in \widehat S_n / S_n$, and the dimension of the finite-dimensional representation $e L_{\frac{n+1}{n}}$ of the spherical Cherednik algebra $e H_n e$. The above families and the bijections between them all generalize from $(n+1,n)$ to $(m,n)$ when $\gcd(m,n)=1$. Further, we can move from Catalan numbers to $m^{n-1}$ by considering all parking function $PF_{\frac mn}$, all alcoves in the dilation $D_n^{m}$, the $m$-restricted" $w \in \widehat S_n$, and the dimension of the finite-dimensional representation $L_{\frac{m}{n}}$ of $H_n$. Parking functions carry interesting combinatorial statistics. I'll discuss some of these statistics, how we hope they interact with $L_{\frac{m}{n}}$, and how they do arise in the space of diagonal harmonics, the geometry of certain affine Springer fibres. This is joint work with Eugene Gorsky and Mikhail Mazin.

Double total ramifications for curves of genus 2, Nicola Tarasca:

Inside the moduli space of curves of genus 2 with 2 marked points, the loci of curves admitting a map to P1 of degree d totally ramified at the two marked points have codimension two. In this talk I will show how to compute the classes of the compactifications of such loci in the moduli space of stable curves. This is a special case of what is generally known as the Eliashberg's problem. I will also discuss the relation with the related work of Hain, Grushevsky-Zakharov, and Chen-Coskun.

Non semi-simple TQFTs and Reidemeister torsion, Christian Blanchet :

This is joined work with Francois Costantino, Nathan Geer and Bertrand Patureau. Costantino, Geer and Patureau have defined a series of 3-manifolds invariants based on nilpotent representations of quantum sl(2) at a root of unity. The corresponding category is non semi-simple, and the usual modular category framework does not apply. They get a series of invariants, indexed by an integer r>=2, defined for 3-manifolds with cohomology class over C/2Z, or equivalently C∗ flat connection. We show that these CGP invariants extend to TQFTs with interesting features. For the smallest root of unity i.e. r=2, the construction produces a TQFT for a canonical normalisation of Reidemeister torsion.

Heisenberg-picture TQFTs, Theo Johnson-Freyd :

The Atiyah--Segal axioms for quantum field theory generalize the Schr\"odinger picture'' of quantum mechanics (Hilbert spaces of states, partition functions, etc.). I will describe a small modification that corresponds instead to the Heisenberg picture'' (algebras of observables). As examples, I will describe some versions of "fully-extended" quantum Chern--Simons Theory: one built from the category of comodules of a quantized function algebra, and another built from the Temperley--Lieb category. The latter is defined over Z and fully extended, and (perhaps most importantly) at generic level.''

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