Transformation Groups Seminar

Speaker: Kumar Shukla (Western)

"Piecewise polynomial ring as equivariant cohomology of toric varieties"

Time: 09:30 - 11:30

Room: MC 106

We will define piecewise polynomial ring over a fan and discuss some properties of this ring. Then we will show that the equivariant cohomology of a toric variety over a fan Σ (with vanishing odd cohomology) is given by piecewise polynomials over Σ.

Colloquium

Speaker: Cihan Okay (Bilkent University)

"Simplicial distributions and contextuality"

Time: 12:30 - 13:30

Room: MC 108

In modern homotopy theory, spaces are represented by combinatorial models called simplicial sets. Their elegant formulation gives them great expressive power to capture spaces up to homotopy. Simplicial distributions are basic mathematical objects that mix simplicial sets with probabilities. That is, they model probability distributions on spaces. In my talk, I will show how simplicial distributions provide a framework for studying a central quantum feature associated with probabilities, known as contextuality. A typical measurement scenario consists of a set of measurements and outcomes, whereas simplicial distributions can be defined for spaces of measurements and outcomes. Our approach unifies and goes beyond two earlier approaches: the sheaf-theoretic (Abramsky-Brandenburger) and group cohomological (Okay-Roberts-Bartlett-Raussendorf).

Transformation Groups Seminar

Speaker: Kumar Shukla (Western)

"Piecewise polynomial ring as equivariant cohomology of toric varieties II"

Time: 09:30 - 11:30

Room: MC 108

We will define piecewise polynomial ring over a fan and discuss some properties of this ring. Then we will show that the equivariant cohomology of a toric variety over a fan $\Sigma$ (with vanishing odd cohomology) is given by piecewise polynomials over $\Sigma$.

This is the second part.

Geometry and Topology

Speaker: Georg Wille (Philipps-Universitaet Marburg)

"Discrete homotopy theory of Cayley graphs of Coxeter groups"

Time: 15:30 - 16:30

Room: MC 107

Discrete homotopy theory is a way of analyzing graphs with topological ideas. Among other discrete analogs of familiar concepts, it is possible to define discrete homotopy groups and discrete homology groups of a graph which give insight into the connectivity of the graph. In the talk, an introduction to the emerging field will be given. Additionally, a notion of Cat(0) graphs will be introduced, whose theory resembles the theory of Cat(0) cubical complexes. The Cartan-Hadamard theorem for cubical complexes implies that Cat(0) cubical complexes are contractible; a combinatorial analog of this theorem for graphs will be presented. As an application, the discrete homotopy types of Cayley graphs of Coxeter groups will be determined.

Colloquium

Speaker: Anibal Medina-Mardones (Western)

"Framed polytopes and higher structures"

Time: 15:30 - 16:30

Room: MC 108

A framed polytope is the convex closure of a finite set of points in Euclidean space together with an ordered linear basis. An n-category is a category that is enriched in the category of (n-1)-categories. Although these concepts may initially appear to be distant peaks in the mathematical landscape, there exists a trail connecting them, blazed in the 90's by Kapranov and Voevodsky. We will traverse this path, widening and improving it as we address some of their conjectures along the way. If time permits, using a special embedding of the combinatorial simplex, we will connect this trail to the one ascending Mount Steenrod. This connection will enable us to express the combinatorics of cup-i products in convex geometric terms, dual to those introduced earlier in the talk to define the nerve of higher categories.

Transformation Groups Seminar

Speaker: Tao Gong (Western)

"On contractibility of quotients of real toric varieties from Weyl groups"

Time: 09:30 - 11:30

Room: MC 108

Given a reduced crystallographic root system with a fixed simple system, it is associated to a Weyl group $W$ and a polytope $P$ which is the convex hull of the $W$-orbit of a dominant weight. The polytope $P$ is associated to a real toric varieties $X_P^{\mathbb{R}}$. We will see that the underlying topological space $X_P^{\mathbb{R}}/W$ is contractible when the rank is below 7.

Ph.D. Candidacy Exam Lecture

Speaker: Anif Shikder (Western)

"An Operator Approach to All-Pairs Shortest Path for Random Graphs"

Time: 15:00 - 16:00

Room: MC 108

We introduce an analytical model for Erdos–Renyi graphs called GRE and derive expressions approximating its adjacency matrix. We then find expressions approximating its l-path matrix. After formulating key graph measures such as the clustering coefficient, the distance matrix, and the average shortest distance in terms of the l-path matrix, we deduce analytical expressions approximating them. Our expressions do not contain any free parameters and show fast convergence valid non-asymptomatically over the entire probability space. The proposed approach is generalizable to other random graphs and can be used to approximate algebraically well-defined graph measures to investigate finite-sized real-world networks.

Geometry and Topology

Speaker: Diego Manco Berrio (Western)

"Pseudosymmetric multifunctors: coherence and applications to K-theory"

Time: 15:30 - 16:30

Room: MC 107

In this talk I will introduce multicategories and symmetric and pseudo symmetric multifunctors between them. By work of Elmendorf, Mandell and Yau, K-theory is a symmetric multifunctor and inverse K-theory is a pseudo symmetric multifunctor. Via a coherence result for pseudo symmetric multifunctors we prove that inverse K-theory preserves certain multiplicative structures. In this way we generalize an old result by Thomason which establishes that any connective spectrum can be realized as the K-theory of some symmetric monoidal category. We also give new examples of pseudo symmetric multifunctors.

PhD Thesis Defence

Speaker: Alan Flatres (Western)

"Group living features can challenge predictions for the evolution of altruistic behaviors"

Time: 13:00 - 14:00

Room: MC 204

Altruistic behaviors occur when an individual decreases its personal fitness to help another individual. Such behaviors occur across a range of species and environments, and they take different forms. The diversity of altruistic behaviors is also characterized by various group living features, including group structure and social interactions. In this thesis, I develop models to study how the specificities of group living can influence the evolution of altruistic behaviors. I use inclusive fitness models to understand how the social environment the group creates, the ecological factors, and the benefits of altruistic behaviors impact the evolution of social behaviors. In the first model, I study the evolution of delayed dispersal with group size benefits. Dispersal tends to be delayed when breeding opportunities are scarce, i.e., when the habitat is saturated. I find that habitat saturation is not always associated with a high level of dispersal. This finding challenges previous results and highlights the need to model environmental feedback explicitly. In a second model, I measure how redirected help can emerge when individuals disperse near their relatives. Redirected help happens when an individual whose entire brood fails reallocates the effort it would have expended on parental care to help a related neighbor. The adaptive significance of this strategy may look straightforward, but if the population is viscous, the helper also competes with its relatives. This population viscosity creates additional costs and benefits that can restrain the evolution of altruism. To investigate the evolution of redirected help in a viscous population, I use an infinite-island model where redirected help can provide survival or fecundity benefits to the recipients. I find that the survival benefits associated with redirected help sometimes promote the emergence of help better than fecundity benefits, which contradicts previous findings. In a third model, I delve more deeply into the evolution of redirected help by explicitly accounting for spatial structuring within the population. I find that switching to a spatially explicit model has repercussions for the evolution of redirected help. For instance, the influence of offspring dispersal on the evolution of redirected help is reversed between the two models. My findings highlight the impact of spatial structure on the evolution of social behaviors. Overall, my thesis shows that different group living features can challenge predictions on the evolution of social behaviors.

Colloquium

Speaker: Tatyana Barron (Western)

"Multisymplectic and polysymplectic forms on manifolds"

Time: 15:30 - 16:30

Room: MC 107

Let $M$ be a smooth manifold. Let $k$, $r$ be positive integers. A $k$-plectic (multisymplectic) form on $M$ is a closed non-degenerate $(k+1)$-form on $M$. Let's define a polysymplectic form on $M$ as an ${\mathbb{R}}^r$-valued closed non-degenerate 2-form on $M$. I will discuss the Darboux theorem and the Poisson bracket.