# Algebra Seminar 2020

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.

• Monday January 27, 2020 at 13:30, Wachman 617
Non-Euclidean tetrahedra, mixed Hodge structures and rational elliptic surfaces

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.

• Monday February 3, 2020 at 13:30, Wachman 617
Distance matrices of trees: invariants, old and new

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.)

• Monday February 17, 2020 at 13:30, Wachman 617
Hecke algebras and Hecke operators

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.

• Monday February 24, 2020 at 13:30, Wachman 617
Hecke algebras and Hecke operators, II

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.

• Monday March 16, 2020 at 13:30, Wachman 617
TBA

Niranjan Ramachandran, University of Maryland

TBA

• Monday March 23, 2020 at 13:30, Wachman 617
TBA

Samuel Taylor, Temple University

TBA

• Monday March 30, 2020 at 13:30, Wachman 617
TBA

Samuel Taylor, Temple University

TBA