Date	Speaker	Title
May 29	Vassily Manturov Moscow Institute of Physics and Technology and BIMSA	The photography method: The state of the art and open problems
May 22	Friedrich Bauermeister Dartmouth College	Topological consequences of null-geodesic refocusing and applications to $Z^x$ manifolds
Nov 14	Rustam Sadykov KSU	Fusing critical points of smooth functions (extended version)
Oct 17	Jaewon Chang Harvard	Distinguished Basis of Floer Cohomology
Oct 3	Corey Bregman Tufts	Moduli spaces of 3-manifolds with boundary
Sept 26	Patricia Sorya UQAM	Non-integral Dehn surgeries characterize composite knots

May 29, 2025; Vassily Mantiurov "The photography method: The state of the art and open problems." Zoom Recorded Talk 870 912 2782, please ask Vladimir for the password

Abstract: In 2023, the speaker formulated the photography method which allows one to solve various equations and construct invariants of various topological object. We start from some object (say, pentagon) and its state (say, triangulation) with some data (for example, edge lengths of triangles) together with a transformation rule (say, a triangulation flip). After that, by using some geometrical consideration, one can "automatically" prove that such transformation lead to a solution of some equation (for example, the Ptolemy transormation gives a solution to the pentagon equation) which leads to invariants of various objects (for example, braids). After that, one can "freely" (without any calculations) get a formula coming from some geometrical data (say, lengths in the hyperbolic space). Having such a formula, one automatically gets its algebraic proof which allows one to pass from geometric objects (lengths, angles) to variables. This method is very vast. In particular, Dijkgraaf-Witten invariants can be interpreted in terms of the photography method. We mention just some relations of further research: invariants of knots, braids, manifolds of arbitrary dimensions, solutions to the pentagon, Yang-Baxter and other equations, relations to cluster algebras, tropical geometry and other parts of mathematics

May 22, 2025; Friedrich Bauermeister "Topological consequences of null-geodesic refocusing and applications to $Z^x$ manifolds " Zoom Recorded Talk 870 912 2782, please ask Vladimir for the password

Abstract: Let $(M,h)$ be a connected, complete Riemannian manifold, let $x\in M$ and $l>0$. Then $M$ is called a $Z^x$ manifold if all geodesics starting at $x$ return to $x$ and it is called a $Y^x_l$ manifold if every unit-speed geodesic starting at $x$ returns to $x$ at time $l$. It is unknown whether there are $Z^x$ manifolds that are not $Y^x_l$-manifolds for some $l>0$. By the Bérard-Bergery theorem, any $Y^x_l$ manifold of dimension at least $2$ is compact with finite fundamental group. We prove the same result for $Z^x$ manifolds $M$ for which all unit-speed geodesics starting at $x$ return to $x$ in uniformly bounded time. We also prove that any $Z^x$ manifold $(M,h)$ with $h$ analytic is a $Y^x_l$ manifold for some $l>0$. We start by defining a class of globally hyperbolic spacetimes (called observer-refocusing) such that any $Z^x$ manifold is the Cauchy surface of some observer-refocusing spacetime. We then prove that under suitable conditions the Cauchy surfaces of observer-refocusing spacetimes are compact with finite fundamental group and show that analytic observer-refocusing spacetimes of dimension at least $3$ are strongly refocusing. We end by stating a contact-theoretic conjecture analogous to our results in Riemannian and Lorentzian geometry.

November 14, 2024: Rustam Sadykov "Fusing critical points of smooth functions (extended version)"

Abstract: In general Morse functions on manifolds do not minimize the number of critical points as critical points of Morse functions sometimes can be fused to produce fewer but more complicated critical points. I will present obstructions preventing fusibility of critical points of smooth functions in certain cases, and present non-trivial examples where critical points can be fused.

October 17, 2024: Jaewon Chang "Distinguished Basis of Floer Cohomology"

Abstract: Beginning from the physicists' observation that Calabi-Yau manifolds come in pairs of "mirror dual" families, the mirror symmetry conjectures were proposed to formulate the intricate connection between symplectic geometry on a space and algebraic geometry on its dual. One version of this is the homological mirror symmetry conjecture, which implies a correspondence between (wrapped) Lagrangian Floer chain complexes and corresponding Hom spaces in the category of coherent sheaves. One notable aspect of studying chain complexes and their homology groups is that the methods for calculation and their properties can vary significantly. For example, while there are vast studies of canonical bases of the cohomology rings on the algebraic side, the descriptions of Lagrangian Floer cohomology rings often fail to give a distinguished basis. In this talk, I will discuss the case of certain Lagrangians in (C*)^2, and describe distinguished bases of them. Under mirror symmetry, this will also give distinguished bases of the ring of functions on P^1 minus 4 points.

October 3, 2024: Corey Bregman "Moduli spaces of 3-manifolds with boundary"

Abstract: Let M be a connected, orientable 3-manifold with nonempty boundary. In this talk, we study the classifying space for the diffeomorphism group of M fixing the boundary pointwise and show that it has the homotopy type of a finite CW complex. This confirms a conjecture of Kontsevich for orientable 3-manifolds. After explaining the connection to moduli spaces, the proof will take us on a crash course through 3-manifold topology over the past 70 years, and will feature a combination of results geometrization of 3-manifolds with a topological poset parametrizing embedded spheres in M. This is joint work with Rachael Boyd and Jan Steinebrunner.

September 26, 2024: Patricia Sorya "Non-integral Dehn surgeries characterize composite knots"

Abstract: A Dehn surgery slope p/q is said to be characterizing for a knot K if the homeomorphism type of the p/q-Dehn surgery along K determines the knot up to isotopy. I show that if K is a composite knot, then every non-integral slope is characterizing for K. This yields the first and only examples to date of a complete and non-empty list of non-characterizing slopes for a knot.