9:30 AM — 10:30 AM
Speaker: John Baldwin, Boston College
Location: McDonnell A10

L-spaces and knot traces

There has been a great deal of interest in understanding which knots are characterized by which of their Dehn surgeries. We study a 4-dimensional version of this question: which knots are determined by which of their traces? We prove several results that are in stark contrast with what is known about characterizing surgeries, most notably that the 0-trace detects every L-space knot. Our proof combines tools in Heegaard Floer homology with results about surface homeomorphisms and their dynamics.

1:45 PM — 2:45 PM
Speaker: Juan Muñoz-Echániz, Stony Brook University
Location: McDonnell A01

Boundary Dehn twists on 4-manifolds and Milnor fibrations of surface singularities

A 4-manifold with boundary on a Seifert-fibered space admits a `boundary Dehn twist’ diffeomorphism, obtained by a fibered version of the classical 2-dimensional Dehn twist. This diffeomorphism arises naturally as (a power of) the monodromy of Milnor fibrations of surface singularities. In this talk I will discuss non-triviality results for boundary Dehn twists on symplectic fillings of Seifert-fibered spaces, using tools from Seiberg—Witten theory. Some applications include:

- The ADE singularities are the only weighted-homogeneous isolated hypersurface singularities in complex dimension 2 whose monodromy has finite order in the smooth mapping class group.

- There are exotic R^4’s which admit exotic (compactly-supported) diffeomorphisms.

Joint with Hokuto Konno, Jianfeng Lin and Anubhav Mukherjee.

3:00 PM — 4:00 PM
Speaker: Jørgen Ellegaard Andersen, University of Southern Denmark
Location: McDonnell A01

Gaussian Boson Sampling, Gaussian weighted Integrals and their generalization to curved phase spaces

First the notion of Gaussian Boson Sampling in the context of general symplectic manifolds which admits Kähler structures will be introduced together with the corresponding Gaussian weighted Integral problem. The simplest possible, yet very interesting case, is the case of flat Euclidean space with its standard symplectic structure. This example describes the mathematics behind the Gaussian Boson Sampling Quantum computing platform which has been realized in a number of quantum physical labs and quantum computing companies around the world using quantum optics. We will demonstrate how these quantum computing platforms can be used to solve Gaussian weighted Integral problems, in fact some of them exponentially faster than using the traditionally Monte Carlo simulation techniques. We will then turn to the general setting and explore these techniques and problems in the context of 2+1 dimensional TQFT in terms of geometric quantization of moduli spaces and their relation to more standard quantum computing platforms. Finally, we will consider these techniques and problems in the context of general curved phases space, e.g. general symplectic manifolds admitting Kähler structures.

4:15 PM — 5:15 PM
Speaker: Marco Marengon, Alfréd Rényi Institute of Mathematics
Location: McDonnell A01

Sliceness problems in some 4-manifolds

A knot in S^3 is called (smoothly) slice in a 4-manifold X if it bounds a smooth disc in X - B^4. An approach proposed by Freedman-Gompf-Morrison-Walker to disprove the smooth 4-dimensional Poincaré conjecture amounts to showing that there exists a knot which is slice in an exotic copy of S^4, but not in the standard one. More generally, one can use sliceness and similar notions to study exotic manifolds. In the talk I will mostly focus on two results:

1) Every knot with unknotting number at most 21 is slice in the K3 surface (joint work with Stefan Mihajlovic);

2) There exists a 2-component link that is not smoothly slice in a blown-up CP^2 (joint work with Clayton McDonald).

The second point is connected to the search for exotic copies of the blown-up CP^2.