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for events the day of Tuesday, September 24, 2013.

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     August 2013           September 2013          October 2013    
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Tuesday, September 24, 2013

11:00 am in 243 Altgeld Hall,Tuesday, September 24, 2013

A differential graded approach to derived manifolds

David Carchedi (Bonn)

Abstract: Given two smooth maps of manifolds $f:M \to L$ and $g:N \to L,$ if they are not transverse, the fibered product $M \times_L N$ may not exist, or may not have the expected dimension. In the world of derived manifolds, such a fibered product always exists as a smooth object, regardless of transversality. In fact, every (quasi-smooth) derived manifold is locally of this form. In this talk, we briefly explain what derived manifolds ought to be, why one should care about them, and how one can describe them. We end by explaining a bit of our joint work with Dmitry Roytenberg in which we make rigorous some ideas of Kontsevich to give a model for derived intersections as certain differential graded manifolds.

1:00 pm in 243 Altgeld Hall,Tuesday, September 24, 2013

Divergence of Weil-Petrsson geodesic rays.

Babak Modami ((UIUC Math))

Abstract: The Weil-Petrsson (WP) geodesic flow is a non-uniformly hyperbolic flow on the moduli space of Riemann surface. We review some results about a kind of symbolic coding of the flow using laminations and subsurface coefficients. Then we apply some estimates on WP metric and its derivatives in the thin part of moduli space to show that the strong asymptotics of a class of WP geodesic rays is determined by the associated laminations. As a result we give a symbolic condition for divergence of WP geodesic rays in the moduli space.

2:00 pm in 345 Altgeld Hall,Tuesday, September 24, 2013

Polynomials for $GL_p \times GL_q$-orbit closures on the flag variety

Benjamin Wyser (UIUC)

Abstract: Whenever a subgroup of $GL_n$ acts on the flag variety $GL_n/B$ with finitely many orbits, it is a general fact that a polynomial representative of the cohomology class of a given orbit closure $Y$ can be found by first giving a representative of a closed orbit $Q$ below $Y$ in weak order, then applying to that representative some sequence of divided difference operators. However, there are choices of that initial representative to be made, and some choices may be more geometrically natural than others. In the case of the $B$-action on $GL_n/B$, the orbit closures are the Schubert varieties, and it is more or less a settled matter that the Schubert polynomials give the most natural choices of representatives of the classes, both from a combinatorial standpoint and from a geometric one, the latter having been elucidated by A. Knutson and E. Miller via Gröbner degenerations of matrix Schubert varieties. In this talk, I will consider the $K$-action on $GL_n/B$, where $K$ is the symmetric subgroup $GL_p \times GL_q$. Here, the question of which representatives are most natural is not as well-studied. I will describe a particular choice of representatives which seems particularly natural, and detail some of their known properties, including their self-consistency, their monomial positivity, and their known interpretation as multidegrees of Gröbner degenerations in certain cases, making contact with the aforementioned work of Knutson-Miller, and also related work of Knutson-Miller-Yong. I will also describe a number of conjectures regarding the cases which are not currently as well-understood from the standpoint of Gröbner geometry. Finally, I will indicate how the methods used to investigate these cohomological questions can be adapted to the problem of studying the local structure of $K$-orbit closures on $GL_n/B$. This is joint work with Alexander Yong.

2:00 pm in Altgeld Hall 347,Tuesday, September 24, 2013

Maximum independent sets in random d-regular graphs

Jian Ding (U Chicago Stat)

Abstract: Satisfaction and optimization problems subject to random constraints are a well-studied area in the theory of computation. These problems also arise naturally in combinatorics, in the study of sparse random graphs. While the values of limiting thresholds have been conjectured for many such models, few have been rigorously established. In this context we study the size of maximum independent sets in random d-regular graphs. We show that for d exceeding a constant d(0), there exist explicit constants A, C depending on d such that the maximum size has constant fluctuations around A*n-C*(log n) establishing the one-step replica symmetry breaking heuristics developed by statistical physicists. As an application of our method we also prove an explicit satisfiability threshold in random regular k-NAE-SAT. This is joint work with Allan Sly and Nike Sun.

3:00 pm in 347 Altgeld Hall,Tuesday, September 24, 2013

Topology and the Eigenvalues of the Laplacian on a Network

Jared Bronski   [email] (UIUC Math)

Abstract: The Laplacian on a graph or network arises in many appled problems including problems of synchronization and convergence to consensus. In the case where the edge weights are all positive the spectral theory of the Laplacian is well understood: the eigenvalues are all non-positive, with the number of zero eigenvalues equal to the number of components of the network. In many applications, however, the edge weights may be of either sign, and the spectral theory is much less well developed. In this talk we present a theory for such signed Laplacians which gives best possible information on the signs of e eigenvalues of the matrix in terms of the topology of the graph. We close with some applications to social networks with positive and negative edges (likes and dislikes).

3:00 pm in 241 Altgeld Hall,Tuesday, September 24, 2013

Extending 3-coloring of a face in triangle-free planar graphs

Bernard Lidický   [email] (UIUC Math)

Abstract: A graph $G$ is $k$-colorable if there exists a mapping $c: V(G) \rightarrow [1,\ldots,k]$ such that $c(u) \neq c(v)$ for every edge $uv$. A graph $G$ is $(k+1)$-critical if it is not $k$-colorable but every proper subgraph of $G$ is $k$-colorable. When enumerating critical graphs, it is often handy to know if a partial coloring of a graph $G$ extends to some subgraph of $G$. In particular, in the study of critical graphs embedded on surfaces, it is useful to know when a coloring of the outer face of a plane graph extends to the whole graph. Such result is known for 3-coloring and plane graphs of girth 5. For girth 4, there is a description for graphs with outer face of length up to 7. We describe the case for outer face of length 8. Unlike the previous for length 7, ours does not use the discharging method but nowhere-zero flows. This is a joint work with Zdeněk Dvořák.

3:00 pm in 243 Altgeld Hall,Tuesday, September 24, 2013

On S-duality and T-duality and algebro-geometric proof of modularity conjectures in BPS counting theories

Artan Sheshmani (Ohio State)

Abstract: We construct an algebraic-geometric framework to calculate the partition functions of "massive black holes" enumerating invariants of supersymmetric D4-D2-D0 BPS states in type IIA string theory. Using S-duality, the entropy of such black holes can be related to a certain N=2, d=4 Super Yang-Mills theory on a divisor in a threefold. Physicists: Gaiotto, Strominger, Yin, Denef, Moore, via careful study of such S-duality, have conjectured that these partition functions have modular properties. We give a rigorous mathematical proof of their conjectures in different geometric setups. This is a report of joint project with Amin Gholampour and Richard Thomas. We also use an algebro-geometric analogue of the string theoretic D4/D2 T-duality to prove the modularity properties of certain PT stable pair invariants over threefolds given by smooth and Nodal surface fibrations over a curve. Here our strategy is to use combination of degeneration techniques, conifold transitions, and wall crossing of Bridgeland stability conditions. This is a report of joint project with Gholampour and Yukinobu Toda.

4:00 pm in 243 Altgeld Hall,Tuesday, September 24, 2013

An Introduction to Tropical Geometry: Part II

Nathan Fieldsteel (UIUC Math)

Abstract: A continuation of last week's seminar, we will begin by tying up some loose ends from last time. We will then present more of the general theory of tropical geometry, and discuss connections to polyhedral geometry, hyperplane arrangements, and grassmanians, time permitting. Professors are welcome to attend.