Mathematics

Molecular interactions lie at the core of biochemistry and biology, and their understanding is crucial to the advancement of biotechnology, therapeutics, and diagnostics. Most existing tools make “ensemble” measurements and report a single result, typically averaged over millions of molecules or more. These measurements can miss rare events, averaging out the natural variations or sub-populations within biological samples, and consequently obscure insights into multi-step and multi-state reactions. The ability to make and connect robust and quantitative measurements on multiple scales - single molecules, cellular complexes, cells, tissues - is a critical unmet need. In this talk, I will introduce a general method called “CLiC” imaging to image molecular interactions one molecule at time with precision and control, and under cell-like conditions. CLiC works by mechanically confining molecules to the field of view in an optical microscope, isolating them in nanofabricated features, and eliminates the complexity and potential biases inherent to tethering molecules. By imaging the trajectories of many single molecules simultaneously and in a dynamic manner, CLiC allows us to investigate and discover the design rules and mechanisms which govern how therapeutic molecules or molecular probes interact with target sites on nucleic acids; and how molecular cargo is released inside cells from lipid nanoparticles. In this talk, I will discuss applications of our imaging platform to better understand DNA, RNA, protein interactions, as well as emerging classes of genetic medicines, gene editing and drug delivery systems. I will highlight current and potential future applications to connect our observations from the level of single molecule to single cells, and opportunities for collaboration as we set up our labs at UBC.

I will introduce a skeleton obtained directly from monomial relations in a finite quiver without cycles, and relate the construction to some classical examples in mirror symmetry. This is work in progress with David Favero.

The Jones polynomial of a knot, discovered in 1983, is a very
subtle invariant that is related to a great deal of mathematics and
physics. This talk will be an overview of quantum field theories in
dimensions 2, 3, 4 and 5 that are intimately related to the Jones
polynomial of a knot and a more contemporary refinement of it that is known
as Khovanov homology.

This is a guest lecture in the PIMS Network Wide Graduate Course in Differential Equations in Algebraic Geometry.

Gauge-theoretic moduli spaces are often noncompact, and various techniques have been introduced to study their asymptotic features. Seminal work by Taubes shows that in many situations where the failure of compactness for sequences of solutions is due to the noncompactness of the gauge group, diverging sequences of solutions lead to what he called Z_2 harmonic spinors. These are multivalued solutions of a twisted Dirac equation which are branched along a codimension two subset. This leads to a number of new problems related to these Z_2 harmonic spinors as interesting geometric objects in their own right. I will survey this subject and talk about some recent work in progress with Haydys and Takahashi to compute the index of the associated deformation problem.

Speaker Biography

Rafe Mazzeo is an expert in PDEs and Microlocal analysis. He did his PhD at MIT, and was then appointed as Szegő Assistant Professor at Stanford University, where he is now Professor and Chair of the Department of Mathematics. He has served the mathematical community in many important ways, including as Director of the Park City Mathematics Institute.