Talks

The quantum difference equation (qde) is the $q$-difference equation which is proposed by Okounkov and Smirnov to encode the $K$-theoretic twisted quasimap counting for the Nakajima quiver varieties. In this talk, we will give a direct quantum toroidal algebra $U_{q,t}(\hat{\hat{\mf{sl}}}_{n})$ construction for the qde of the affine type $A$ quiver varieties. We will show that there is a really explicit and concise formula for the quantum difference operators. Moreover we will show that the degeneration limit of the quantum difference equation is equivalent to the Dubrovin connection for the quantum cohomology of the affine type A quiver varieties, which will give the description of the monodromy representation of the Dubrovin connection via the monodromy operators in the quantum difference equation.

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4 months after it opened, the Tacoma Narrows bridge, one of the world's
longest suspension bridges at the time, collapsed spectacularly in the first
storm that hit it. Though it was built to exceed all of the standrds at the
time, something  clearly went wrong. The failure was filmed from almost the
beginning to the end (about 1 hour), and that film has been shown to almost
all first year physics or engineering classes as an example of resonance, that
explanation is clearly nonsense. What happened? Why did it collapse. The
explanation is closely linked to, for example, the reason that clarinets or
flutes, or even violins, make their music. With Daniel Green (whom you may
remember from his time at Toronto) we were able to show in detail what
happened.

Populations often adapt to spatially heterogeneous environments via substitutions or allele frequency changes at many loci spread across the genome. Subsequent contact between diverged populations can produce unfit hybrids, which inhibits genetic exchange across the genome, thus setting the stage for further divergence and speciation. How such polygenic divergence builds up or is maintained depends on the coupled dynamics of multiple loci under selection, making it challenging to predict. In this talk, I will outline simple theoretical approximations based on effective migration rates that capture how coupling or linkage disequilibria (LD) between loci influences the maintenance of polygenic adaptive divergence. This analysis allow us to clarify how genetic architecture (i.e., the numbers, selective and dominance effects and linkage relationships of selected variants) influences divergence in the face of gene flow and genetic drift. I will conclude my talk with a broader perspective on...