Abstracts
Week of March 26, 2023
| Geometry, Symmetry and Physics | Emergent Flattening of Moment-Curve-Like Geometries |
4:30pm -
LOM 214
|
In this talk, I discuss how an infinite dimensional convex geometry of interest to physicists exhibits "flattening," which manifests as emergent equalities among naively independent coordinates. This flattening behavior is intrinsically tied to the infinite dimensional nature of the convex geometry, as these emergent equalities only appear in the infinite dimensional limit. In more detail, the space of causal and unitary theories, called the EFT-Hedron, is identified as the intersection of a convex region given by the Minkowski sum of two moment curves and a hyperplane in an infinite dimensional projective space. I use linear programming to provide strong numeric evidence that the EFT-hedron "flattens out." For example, restricting a finite fraction of the coordinates to be even-zeta values, the remaining coordinates are (conjecturally) fixed to take odd-zeta values. I will conclude by briefly sketching how this conjecture relates to Type-I superstring theory, which corresponds to a particular point in the EFThedron. |
| Colloquium | The arithmetic of power series and applications |
4:15pm -
LOM 214
|
Borel and Dwork gave conditions on when a nice power series with rational number coefficients comes from a rational function in terms of meromorphic convergence radii at all places. Such a criterion was used in Dworkâs proof of the rationality of zeta functions of varieties over finite fields. Later, the work of Andre, Bost and many others generalized the rationality criterion of Borel–Dwork and deduced many applications in the arithmetic of differential equations and elliptic curves. In this talk, we will discuss some further refinements and generalizations of the criteria of Andre and Bost and their applications to the unbounded denominators conjecture for modular forms, and irrationality of 2-adic zeta value at 5 and some other linear independence problems. This is joint work with Frank Calegari and Vesselin Dimitrov. |
| Analysis | Degeneration of hyperbolic surfaces and spectral gaps for large genus |
4:00pm -
LOM 205
|
The study of "small" eigenvalues of the Laplacian on hyperbolic surfaces has a long history and has recently seen many developments. In this talk I will focus on the recent work (joint with Yunhui Wu and Haohao Zhang) on the higher spectral gaps, where we study the differences of consecutive eigenvalues up to $\lambda_{2g-2}$ for genus $g$ hyperbolic surfaces. We show that the supremum of such spectral gaps over the moduli space has infimum limit at least 1/4 as genus goes to infinity. The analysis relies on previous joint works with Richard Melrose on degenerating hyperbolic surfaces |
| Friday Morning Seminar | Friday Morning Seminar |
9:30am -
LOM 215
|
A relaxed-pace seminar on impromptu subjects related to the interests of the audience. Everyone is welcome. The subjects are geometry, probability, combinatorics, dynamics, and more! |