Colloquium

Speaker: Christian Maire (University of Franche-Comte)

"The Theorem of Golod-Shafarevich in number theory"

Time: 15:30

Room: MC 107

For a finite group group of prime power order which can be defined minimally by d generators and r relations, the Golod-Shafarevich theorem asserts that $r> d ^ 2/4$. This has important consequences in number theory. After recalling the whole context, I will present a generalization of the GS theorem, and some new consequences of this generalized version. We will take the time to present the arithmetic objects, so that the lecture is intended for a general audience.

Geometry and Combinatorics

Speaker: Ahmed Umer Ashraf (Western)

"Finding 4T relations for matroids"

Time: 14:30

Room: MC 108

4T relations of chord diagrams are very useful in computing Vassiliev invariant of a knot. These relations can be purely put in the language of graph theory. Stanley’s chromatic symmetric function of the graph satisfies a variant of these relations. We will describe these ideas and see the relevant notions in the matroid theoretic setting.

Geometry and Topology

Speaker: Paul Goerss (Northwestern University)

"Dualizing modules in stable homotopy theory"

Time: 15:30

Room: MC 108

Let $G$ be a very nice $p$-adic analytic group; I have in mind examples such as $\mathsf{Gl}_n(\mathbb{Z}_p)$. The category of continuous $G$-modules has a very elegant theory of duality reflecting Poincare duality for $G$. We would very much like to extend this to stable homotopy theory where, in various contexts, it would help explain some deep structure we have seen so far only through computations. It is easy enough to define the dualizing objects, but then we are left with understanding them. It turns out that if we are only interested in finite subgroups of $G$ (which would be a serious start) we can get away with classical computations with Stiefel-Whitney classes. This is an on-going project with Agnes Beaudry, Mike Hopkins, and Vesna Stojanoska.

Geometry and Combinatorics

Speaker: Udit Mavinkurve (Western)

"Bimonoids for hyperplane arrangements II"

Time: 14:30

Room: MC 108

In the first talk, we introduced Tits' projection map for real hyperplane arrangements, and gave a purely combinatorial description of this map in the special case of braid arrangements.

In this talk, we will see how this projection map appears naturally in the Hopf theory for Joyal's category of combinatorial species, motivating us to generalize the definition of species from braid arrangements to Coxeter arrangements and further to hyperplane arrangements. This talk is based on the work of Aguiar and Mahajan.

Geometry and Combinatorics

Speaker: Brian Hepler (University of Wisconsin)

"The Weight Filtration on the Constant Sheaf on a Parameterized Surface"

Time: 15:30

Room: MC 108

$\def\Q{\mathbb Q}$ On an $n$-dimensional locally reduced complex analytic space $X$ on which the shifted constant sheaf $\Q_X[n]$ is perverse, it is well-known that, locally, $\Q_X[n]$ underlies a mixed Hodge module of weight $<= n$ on $X$, with weight $n$ graded piece isomorphic to the intersection cohomology complex $IC_X$ with constant $\Q$ coefficients. In this paper, we identify the weight $(n-1)$ graded piece $Gr_{n-1}^W \Q_X[n]$ in the case where X is a “parameterized space", using the comparison complex, a perverse sheaf naturally defined on any space for which the shifted constant sheaf $\Q_X[n]$ is perverse. 

Geometry and Topology

Speaker: Nathan Grieve (Michigan State)

"Distance to divisors and concepts that surround stability"

Time: 15:30

Room: MC 108

I will report on recent results which deal with the manner in which diophantine arithmetic measures of distance to divisors relate to concepts of stability for polarized projective varieties. These results build on many previous insights including those of Boucksom-Chen, Evertse-Ferretti, K. Fujita, A. Levin, C. Li, McKinnon-Roth, Ru-Vojta, Ru-Wang and J. Silverman. Some emphasis will be placed on the case of K-(in)stability for Fano varieties. At the same time, I will present motivational examples which arise within the context of toric varieties.

Colloquium

Speaker: Douglas Park (Waterloo)

"Geography of simply connected symplectic 4-manifolds"

Time: 15:30

Room: MC 107

I hope to give an elementary survey of recent works on the geography problem of simply connected smooth 4-dimensional manifolds. I will focus mainly on the existence and uniqueness of symplectic 4-manifolds that satisfy certain topological conditions. One such condition that I wish to explore in detail is the signature being nonnegative.