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Current contacts: Vasily Dolgushev, Ed Letzter or Martin Lorenz.
The Seminar usually takes place on Mondays at 1:30 PM in Room 617 on the sixth floor of Wachman Hall. Click on title for abstract.
Cris Negron, UNC Chapel Hill
I will discuss the notion of a noncommutative complete intersection algebra (ncci), and relations with support theory. I will provide a number of examples of such ncci’s, and explain how they appear naturally in several Hopf algebraic contexts. For a number of Hopfy examples, I will explain how this notion can be leveraged to provide a classification thick ideals in the corresponding derived category of representations. More specifically, I will explain how a certain geometry which one associates to a ncci can be used to provide the proposed classification. This is joint work with Julia Pevtsova.
Sam Taylor, Temple University
In this mostly expository talk, we'll recall the theory of Perron polynomials of directed graphs and the connection to computing the growth rate of directed cycles. As the lengths of edges of the graph are varied, so are the corresponding growth rates and these values are packaged as 'roots' of a single 'polynomial'.
We'll then apply this theory to counting closed orbits of certain flows on 3-manifolds. Although some aspects of this theory are old (they are part of the Thurston—Fried—McMullen fibered-face theory), I'll show how the theory can be reconstructed (and generalized!) using a certain combinatorial structure on the manifold. The basic idea is to find a (canonical) directed graph in the manifold to which the theory of the first paragraph applies.
Niranjan Ramachandran, University of Maryland
I will report on some recent joint work with S. Lichtenbaum on the special value of zeta functions of elliptic curves over Q. We will begin with number fields and their zeta functions and build up from there.
Samuel Taylor, Temple University
POSTPONED
Martin Lorenz, Temple University
I will wrap up this mini-course with a discussion of modular forms. The goal is to explain how (commutative) algebras of Hecke operators arise from a construction analogous to the one that yielded the (noncommutative) Iwahori-Hecke algebras considered in the first two talks.
Martin Lorenz, Temple University
To start with, I will specialize the "corner" of the group algebra that was discussed in my first talk to the case where the group $G$ is the general linear group of degree $n$ over the field with $q$ elements and $B$ is the subgroup consisting of the upper triangular matrices in $G$. The resulting corner is then isomorphic to the (one-parameter) Hecke algebra $H_n(q)$ as defined in the first talk. After that, I plan to move on to modular forms and explain how algebras of Hecke operators also arise from an analogous construction.
Martin Lorenz, Temple University
Hecke algebras and their representations have been much studied in algebra and they have also played an important role in the construction of the Jones-Conway polynomial in knot theory. I will start this short series of talks by defining Iwahori-Hecke algebras (for the symmetric groups, as in my current graduate course on representations of braid groups). Then I will discuss a more general group-theoretical approach leading to the same algebras when suitable specialized. Finally, I shall endeavor to explain how the latter approach also yields the familiar Hecke operators in the theory of modular forms. The talks should be widely accessible, not only to students in my aforementioned course.
Apoorva Khare, Indian Institute of Science, Bangalore
In 1971, Graham and Pollak showed that if $D_T$ is the distance matrix of a tree $T$ on $n$ nodes, then $\det(D_T)$ depends only on $n$, not $T$. This independence from the tree structure has been verified for many different variants of weighted bi-directed trees. In my talk:
1. I will present a general setting which strictly subsumes every known variant, and where we show that $\det(D_T)$ - as well as another graph invariant, the cofactor-sum - depends only on the edge-data, not the tree-structure.
2. More generally - even in the original unweighted setting - we strengthen the state-of-the-art, by computing the minors of $D_T$ where one removes rows and columns indexed by equal-sized sets of pendant nodes. (In fact we go beyond pendant nodes.)
3. We explain why our setting is the "most general possible", in that allowing greater freedom in the parameters leads to dependence on the tree-structure.
4. Our results hold over an arbitrary unital commutative ring. This uses Zariski density, which seems to be new in the field, yet is richly rewarding.
We then discuss related results for arbitrary strongly connected graphs, including a third, novel invariant. If time permits, a formula for $D_T^{-1}$ will be presented for trees $T$, whose special case answers an open problem of Bapat-Lal-Pati (Linear Alg. Appl. 2006), and which extends to our general setting a result of Graham-Lovasz (Advances in Math. 1978). (Joint with Projesh Nath Choudhury.)
Daniil Rudenko, University of Chicago
I will start with explaining the classical relation between scissor congruence theory, volumes of hyperbolic polytopes and mixed Hodge structures. Next I will explain how to construct a rational elliptic surface out of every non-Euclidean tetrahedra. This surface "remembers" the trigonometry of the tetrahedron: the length of edges, dihedral angles and the volume can be naturally computed in terms of the surface. The main property of this construction is self-duality: the surfaces obtained from the tetrahedron and its dual coincide. This leads to some unexpected relations between angles and edges of the tetrahedron. For instance, the cross-ratio of the exponents of the spherical angles coincides with the cross-ratio of the exponents of the perimeters of its faces. The construction is based on relating mixed Hodge structures, associated to the tetrahedron and the corresponding surface.
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