This conference aims to expose graduate students in algebra, geometry, and topology to current research, and provide them with an opportunity to present and discuss their own research.
It also intends to provide a forum for graduate students to engage with each other as well as expert faculty members in their areas of research.
Most of the talks at the conference will be given by graduate students, with four given by distinguished keynote speakers.
The organizers of the GTA Philly Conference share the values and commitment to promoting diversity, equity, and inclusion as expressed by the American Mathematical Society.
"The American Mathematical Society recognizes the breadth of people, thought, and experience that contribute to mathematics.
We value the contributions of all members of our mathematics community to improve mathematics research, education, and the standing of the mathematical sciences.
We welcome everyone interested in mathematics as we work to build a community that is diverse, respectful, accessible, and inclusive.
We are committed to ensuring equitable access to mathematics opportunities and resources for people regardless of gender, gender identity or expression, race, color, national or ethnic origin, religion or religious belief, age, marital status, sexual orientation, disabilities, veteran status, immigration status, or any other social or physical component of their identity."
There is no registration fee. Members of gender, racial, and ethnic groups underrepresented in mathematics are encouraged to register.
Abstract: In the eighties, Goldman discovered two Lie algebra structures on two vector spaces generated by free homotopy classes of closed curves on a surface. In one case, the basis is given by the classes of oriented curves, and in the other, by the classes of unoriented curves.
In this talk, we will explain the definition of these Lie brackets, which combines transversal intersection structure with reconnection of curves.
We will describe how the algebraic structure then captures minimal intersection structure of curves on surfaces, in particular counting minimal intersections of a general curve with simple curves and showing the central elements are parallel to the boundary. (The proof uses both hyperbolic geodesic geometry and the effect of Thurston earthquakes on angles at intersection points.)
In the case of surfaces with boundary, the set of free homotopy classes of closed, oriented curves is in one-to-one correspondence with the set of cyclic, reduced words (in a certain alphabet). We will discuss an algorithm to compute the bracket and how implementation of the algorithm lead us to conjetures (some of which, later on, became theorems)
Time permits, we will discuss how the study of these Lie algebras lead to the discovery of String Topology (jointly with Dennis Sullivan)
Title: When are Links Weakly Concordant to Boundary Links
Abstract: Knots are circles embedded into Euclidean space. Links are
knots with multiple components. The classification of links is
essential for understandingn the fundamental objects in
low-dimensional topology: 3- and 4-dimensional manifolds since every
3- and 4-manifold can be represented by a weighted link. When
studying 3-manifolds, one considers isotopy as the relevant
equivalence relation whereas when studying 4-manifolds, the relevant
condition becomes knot and link concordance. The nicest of links are
called boundary links since they are closest to a knot: they bound
disjointly embedded surface in Euclidean space, called a multi-Seifert
surface. The strategy to understand link concordance, starting with
Levine in the 60s, was to first understand link concordance for
boundary links and then to try to relate other links to boundary
links. However, this point of view was foiled in the 90's when Tim
Cochran and Kent Orr showed that there were links (with all known
obstructions vanishing i.e. Miilnor's invariants) that were not
concordant to any boundary link. In this work, Chris Davis, Jung Hwan
Park, and I consider weaker equivalence relations on links filtering
the notion of concordance, called n-solvable equivalence. We will
show that most links are 0- and 0.5-solvably equivalent but that for
larger n, that there are links not n-solvably equivalent to any
boundary link (thus cannot be concordant to a boundary link). This
is joint work with C. Davis and J.H. Park.
Abstract: Tropical geometry is a combinatorial shadow of algebraic
geometry that replaces varieties with objects from polyhedral geometry
and related combinatorics. In this talk I will introduce this area,
and describe some of the applications. I will then explain how
commutative algebra over the tropical semiring (addition is minimum,
multiplication is addition) enters this world to develop a tropical
scheme theory. No background in algebraic geometry will be assumed.
Title: The Waist Inequality and Positive Scalar Curvature
Abstract: The topology of three-manifolds with positive scalar curvature has been (mostly) known since the solution of the Poincare conjecture by Perelman.
Indeed, they consist of connected sums of spherical space forms and S^2 x S^1's.
In spite of this, their "shape" remains unknown and mysterious.
Since a lower bound of scalar curvature can be preserved by a codimension two surgery, one may wonder about a description of the shape of such manifolds based on a codimension two data (in this case, 1-dimensional manifolds).
In this talk, I will show results from a recent collaboration with Y. Liokumovich elucidating this question for closed three-manifolds.