Geometry and Topology

Speaker: Yael Karshon (Tel Aviv/Toronto)

"Symplectic excision"

Time: 15:30

Room: MC 107

A central question in symplectic geometry is to determine which symplectic manifolds are symplectomorphic. We provide novel tools to answer this question in a new context: We use time-independent incomplete Hamiltonian flows to excise interesting closed subsets of positive codimension from symplectic manifolds. In this talk I will focus on our reduction of this symplectic question to a differential-topological question. In tomorrow's colloquium I will discuss this differential-topological question, but the two talks will be independent of each other. This is joint work with Xiudi Tang, available on the arXiv.

Colloquium

Speaker: Yael Karshon (Tel Aviv/Toronto)

"Harnessing incomplete vector fields"

Time: 15:30

Room: MC 107

Let $M$ be an open subset of $\mathbb{R}^n$, or, more generally, a manifold. A vector field on $M$ integrates to a flow, which -- unless the vector field is complete -- is not everywhere defined for all times. Its time-one map is a diffeomorphism between open subsets of $M$. We say that this diffeomorphism excises a closed subset $Z$ of $M$ if its domain is the complement of $Z$ in $M$ and its image is all of $M$.

If a closed subset $Z$ of $M$ is diffeomorphic to the epigraph of a lower-semicontinuous function, we build a vector field on $M$ whose time-one flow excises $Z$ from $M$. Examples of such $Z$ include the ray $[0,\infty)$, what we call a "Cantor brush", and a "box with a tail". We do not know if this sufficient condition for excisability is necessary.

This is part of a joint project with Xiudi Tang about excisability in symplectic geometry. In the Geometry and Topology seminar on Wednesday I will discuss other parts of this work, but the two talks will be independent of each other.

Colloquium

Speaker: Tristan Collins (Toronto)

"From Calabi-Yau manifolds to optimal transport and free-boundaries"

Time: 15:30

Room: MC 107

Calabi-Yau manifolds are fundamental objects in geometric analysis and theoretical physics. I will discuss some recent progress concerning the construction of new, non-compact Calabi-Yau manifolds. In particular, I will highlight connections to questions in nonlinear partial differential equations including optimal transportation and free boundary problems. Based on joint works with Y. Li, and F. Tong and S.-T. Yau.

Transformation Groups Seminar

Speaker: Tao Gong (Western)

"Weyl group action on the coweight lattice modulo 2"

Time: 09:30

Room: MC 108

To understand the Weyl group action on the corresponding real toric variety, it is necessary to understand that action on the coweight lattice modulo 2. We will generalize the construction of alcoves to get the orbit space and the size of each orbit.

Graduate Seminar

Speaker: Meagan James (Western)

"Mapping Class Groups, Homeomorphisms, Curve Graphs, and Ivanov’s Metaconjecture"

Time: 15:30

Room: MC 107

The curve graph of a surface encodes the intersection patterns of the isotopy classes of essential simple closed curves in the surface. In 1997, Ivanov showed that the curve graph can be used as a combinatorial model to study the extended mapping class group of a surface. Since then, with the objective of finding results analogous to that of Ivanov, several alternative versions of the curve graph have been defined and investigated in relation to certain algebraic invariants of a surface. For instance, Long–Margalit–Pham–Verberne–Yao successfully showed that the fine curve graph can be used as a combinatorial model for the group of homeomorphisms of a surface. In this expository talk, we will begin by introducing mapping class groups and several curve graphs associated to a surface. From here, we will investigate the relationship between curve graphs and homeomorphisms of a surface, stating several powerful theorems and even some preliminary results regarding separating curves. Finally, we will briefly discuss adjacent areas of research and the search for evidence supporting Ivanov’s infamous metaconjecture.

Geometry and Combinatorics

Speaker: Mieke Fink (Western)

"Schubert matroids and valuative invariants"

Time: 15:30

Room: MC 108

In this talk I will discuss an algorithm that decomposes an arbitrary matroid polytope into some elementary pieces, called Schubert matroids. The algebraic structure in which the decomposition takes place is the valuative group of matroids, which received attention recently as the homology of permutohedral toric varieties.

Geometry and Topology

Speaker: Yufeng Li (University of Cambridge)

"Extensional concepts in intensional type theory, revisited"

Time: 15:30

Room: MC 107

We revisit Martin Hofmann's now-classical result on the relationship between extensional and intensional type theories by building on insights from homotopy type theory and leveraging the structure of left semi-model category on the category of models of type theory. Specifically, we demonstrate that extensional type theory (ETT) and intensional type theory (ITT) extended by the axiom of uniqueness of identity proofs (UIP) are equivalent in both a logical and a homotopy-theoretic sense. Whereas Hofmann's original proof was focused on initial, or syntactic, models, our approach generalizes to all cofibrant extensions of the base theories, encompassing types, terms, and propositional equalities. In doing so, this result unifies the verification of the analogue of Hofmann's result for all possible new extensions of intensional type theory at once.

Department Meeting

Speaker: (Western)

"N/A"

Time: 15:30

Room:

Graduate Seminar

Speaker: Achraf Ben Said (Universidad Complutense de Madrid)

"The Norm of Hardy-type Oscillation Operators in the Continuous Settings"

Time: 15:30

Room: MC 108

In this presentation we intend to illustrate in a brief way the study of optimal constants in some inequalities in $L^p$ norm of operators involving the Hardy operator defined as $Hf (x) = (1/x) \int_{0}^x f(t) dt$. In 1925, Hardy proves that $H$ is bounded from $L^p$ to $L^p$, if $1 < p ≤ ∞$, with norm exactly equal to $p′$. This result has given rise to a new field of study of great interest in Mathematical Analysis and Operator Theory. The aim of this talk is to go into this line of research by exposing the results that have been obtained recently, the open problems in this field, such as the problem formulated in 1996 by G. Bennett and, finally, to present some of our contributions giving solutions to several questions within this subject.