Any finitely presented group can be the fundamental of a smooth 4-manifold. So, generally, the study of 4-manifolds has been restricted to simply connected cases or, thanks to Freedman and Teichner's profound results, to groups with subexponential growth (where surgery theoretical results still hold). In this talk, we'll explore this a bit further.
My talk concerns bordism rings of compact smooth manifolds equipped with a smooth action by a finite group. I will start by recalling classical results on the subject from the 60's and 70's, mostly due to Conner-Floyd, Boardman, Stong and Alexander. Afterwards I will discuss joint work with Stefan Schwede in which we prove an algebraic universal property for the collection of all bordism rings of manifolds with commuting involutions.
After a brief motivation for mirror symmetry, we will discuss how to sistematically produce the mirror symmetry functor for the product, following the paper of Abouzaid and Smith. The technology presented will help us relate Wehrheim-Woodward functors to Fourier-Mukai ones. For the most part of the talk we will handle symplectic 2-tori, but we will ask ourselves whether what we performed holds in greater generality. Does this point towards a 2-categorical framework for mirror symmetry?
This talk will provide an overview of recent developments in Fourier restriction theory, which one could describe as the study of exponential sums over restricted frequency sets with geometric structure, typically arising in PDE or number theory. Decoupling inequalities measure the square root cancellation behavior of these exponential sums.
I will give an overview of how oscillatory integrals arise in analytic number theory, especially the theory of L-functions and automorphic forms. Usually the integrals we are faced with are multiple-yet-low dimensional, so that they are approachable by repeated one-dimensional stationary phase estimates.
We will begin with an introduction of the key notions in Hodge theory.We will then explain how Hodge theory is used to prove results (called Ax-Schanuel) about transcendental degrees of function fields.
After a brief motivation for mirror symmetry, we will discuss how to sistematically produce the mirror symmetry functor for the product, following the paper of Abouzaid and Smith. The technology presented will help us relate Wehrheim-Woodward functors to Fourier-Mukai ones. For the most part of the talk we will handle symplectic 2-tori, but we will ask ourselves whether what we performed holds in greater generality. Does this point towards a 2-categorical framework for mirror symmetry?
Based on the analogy between number and function rings, for a fixed prime $p$, delta geometry is developed based on the notion of a $p$-derivation $\delta$ which plays the role of 'differentiation' for number rings. Such a $p$-derivation $\delta$ comes from the $p$-typical Witt vectors and as a result, delta geometry naturally encodes valuable arithmetic information, especially those that pertain to lifts of Frobenius.
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