Transformation Groups Seminar

Speaker: Xin Fu (Ajou University)

"The integral cohomology ring of four-dimensional toric orbifolds"

Time: 09:30

Room: https://westernuniversity.zoom.us/s/93798234275, Passcode: 520011

Toric orbifolds introduced by Davis and Januszkiewicz are topological analogs of projective toric varieties. When a toric orbifold is smooth, its integral cohomology ring is isomorphic to a quotient ring of the Stanley-Reisner ring. Such a formula holds for the singular case over rational coefficients, but integrally it becomes more complicated. For instance, the cohomology of a weighted projective space is additively isomorphic to the cohomology of a complex projective space, but the ring structure differs. In this talk, we focus on toric orbifolds $X$ in four dimensions. If $X$ has a smooth fixed point, we construct a basis for its integral cohomology and present their cup products in a matrix whose entries are explicitly determined by the characteristic function. This is joint work with Tseleung So and Jongbaek Song.

Department Meeting

Speaker: (Western)

"Department Meeting"

Time: 15:30

Room:

Transformation Groups Seminar

Speaker: Stephen Theriault (University of Southampton)

"Polyhedral products for connected sums of simplicial complexes"

Time: 09:30

Room: https://westernuniversity.zoom.us/s/93798234275, Passcode: 520011

We investigate how the homotopy type of a polyhedral product changes under the operation of taking the connected sum of two simplicial complexes. This is obtained as a consequence of a more general result that considers how the homotopy type of a polyhedral products changes under the operation of gluing two simplicial complexes together along a common full subcomplex.

Colloquium

Speaker: Muxin Han (Florida Atlantic University)

" Four-dimensional Spinfoam Quantum Gravity with Cosmological Constant"

Time: 15:30

Room: MC 107

We present a formulation of 4-dimensional Lorentzian spinfoam quantum gravity with cosmological constant. The construction of spinfoam amplitudes uses the state-integral model of PSL(2,C) Chern-Simons theory and the implementation of simplicity constraint. The formulation has 2 key features: (1) spinfoam amplitudes are all finite, and (2) With suitable boundary data, the semiclassical asymptotics of the vertex amplitude has two oscillatory terms, with phase plus or minus the 4-dimensional Lorentzian Regge action with cosmological constant for the constant curvature 4-simplex.

Geometry and Combinatorics

Speaker: Mohabat Tarkeshian (Western)

"The geometry of Markov random graphs"

Time: 15:30

Room: MC 108

Random graphs are at the intersection of probability and graph theory: it is the study of the stochastic process by which graphs form and evolve. In 1959, Erdős and Rényi defined the foundational model of random graphs on n vertices. Subsequently, Frank and Strauss (1986) added a Markov twist to this story by describing a topological structure on random graphs that encodes dependencies between local pairs of vertices. The general model that describes this framework is called the exponential random graph model (ERGM). It is used in social network analysis and appears in statistical physics as in the ferromagnetic Ising model. It can also be thought of as a generalization of a p-spin infinite-range spin glass model. We characterize the parameters that determine when an ERGM has desirable properties (e.g., stable, Lorentzian) using a well-developed dictionary between probability distributions and their corresponding generating polynomials.

Transformation Groups Seminar

Speaker: Sergio Chaves (University of Rochester)

"Free and flat extension pairs in equivariant cohomology"

Time: 09:30

Room: https://westernuniversity.zoom.us/s/93798234275, Passcode: 520011

The equivariant cohomology of a space with an action of a group $G$ inherits a canonical module structure over the cohomology ring of $BG$. In this talk, we study pair of groups $K \subseteq G$ such that some algebraic properties of the $G$-equivariant cohomology are captured by the action of the subgroup $K$.

Motivated by compact connected Lie group, torus and cyclic group actions, we generalize these reductions into the notions of free and flat extension pairs in equivariant cohomology.

If time permits, we will discuss related results into the equivariant cohomology of canonical cyclic group actions on surfaces arising as real moment-angle complexes.

Colloquium

Speaker: Michael Yampolsky (University of Toronto Mississauga)

"How to lose at Monte Carlo"

Time: 15:30

Room: MC107

I will talk about the theoretical challenges to the numerical study of dynamical systems. I will broadly discuss what practitioners attempt to compute, and whether such computations are always possible. Such questions lead to interesting mathematics with surprising practical implications. As an instructive example of the limitations on our ability to compute things, I will describe a "nice" one-dimensional dynamical system for which a numerical approximation of the long-term statistical behavior of the orbits is not possible. In particular, the Monte Carlo simulation provably fails for it.

Geometry and matrix analysis

Speaker: Benjamin Lovitz (Northeastern University)

"Nullstellensatz-inspired algorithms for certifying entanglement of subspaces"

Time: 09:30

Room: zoom

In this talk, I will discuss the computational primitive of determining whether a given linear subspace S of pure states contains any product states. If this is not the case, then we say that S is entangled. Certifying that a subspace S is entangled has applications, for example, in certifying entanglement of mixed states (via the range criterion), and constructing entanglement witnesses. One way to certify entanglement in S is via a Nullstellensatz certificate. While a very high degree certificate may be necessary to certify all entangled subspaces, we prove that already the degree-2 certificate (computable in polynomial time) certifies entanglement of generic subspaces of dimension up to a constant multiple of the maximum possible. This is surprising, given that the best-known algorithm for certifying entanglement of a subspace in the worst case scales exponentially in the system size. A robust variant of this primitive, which asks how far S is from product in Hausdorff distance, has similar applications. We develop a robust version of the Nullstellensatz certificate for computing this variant. Specifically, we construct a hierarchy of eigenvalue computations that compute this distance exactly in the limit. Another related problem is to find the product elements contained in S, if any exist. We develop an algorithm, again inspired by the Nullstellensatz certificate, for solving this problem under certain genericity conditions. As a consequence, we obtain new algorithms for tensor rank decompositions that work for generic tensors of bounded rank. The set of product states forms a variety (called the Segre variety), and it is natural to ask whether these techniques generalize to arbitrary varieties. We prove that they do under certain conditions on the variety. We also extend these results to varieties over the real numbers. This talk is based on joint work with Nathaniel Johnston and Aravindan Vijayaraghavan.

Transformation Groups Seminar

Speaker: Matthew Staniforth (University of Southampton)

"Higher Whitehead maps in polyhedral products"

Time: 09:30

Room: (cancelled)

We define generalised higher Whitehead maps in polyhedral products and study their properties and the relations among them. By investigating the interplay between the homotopy theoretic properties of polyhedral products and the combinatorial properties of simplicial complexes, we describe new families of relations among these maps, while recovering and generalising known identities among Whitehead products.