Published on Department of Mathematics Calendar (https://calendar.math.yale.edu)

Week of February 5, 2023

February 6, 2023
Group Actions, Geometry and Dynamics [3]	Short closed geodesics in higher rank arithmetic locally symmetric spaces	Lam Pham - Brandeis University	4:00pm -
Geometry, Symmetry and Physics [4]	Mirror symmetry for Q-Fano 3-folds	Paul Hacking - UMass Amherst	4:30pm - LOM 214

February 8, 2023
Colloquium [5]	The critical exponent: old and new.	Beibei Liu - MIT	4:15pm - LOM 214

February 10, 2023
Friday Morning Seminar [6]	Friday Morning Seminar		9:30am - LOM 215

Abstracts

Week of February 5, 2023

February 6, 2023
Group Actions, Geometry and Dynamics [3]	Short closed geodesics in higher rank arithmetic locally symmetric spaces	4:00pm -	A well-known conjecture of Margulis predicts the existence of a uniform lower bound on the systole of any irreducible arithmetic locally symmetric space. In joint work with F. Thilmany, we proved that this conjecture is equivalent to a weak version of the Lehmer conjecture, a well-known problem from Diophantine geometry. In joint work with M. Fraczyk, we recently established a uniform lower bound for simple Lie groups of higher rank conditional on a uniform lower bound on Salem numbers, a much weaker – but still open – problem. I will discuss these results and some tools used in the proofs and present additional results which highlight the structure of the bottom of the length spectrum.
Geometry, Symmetry and Physics [4]	Mirror symmetry for Q-Fano 3-folds	4:30pm - LOM 214	This is a report on work of my graduate student Cristian Rodriguez. A Q-Fano 3-fold is a complex projective variety with mild singularities such that its 1st Chern class is positive. Q-Fano 3-folds with b_2=1 arise as end products of Mori's minimal model program. Thousands of families are expected, whereas there are only 17 in the smooth case. We will describe mirror symmetry for Q-Fano 3-folds in terms of the Strominger-Yau-Zaslow conjecture and Kontsevich's homological mirror symmetry conjecture, building on work of Auroux. The mirror of a Q-Fano 3-fold is a K3 fibration over the affine line such that the total space is log Calabi--Yau and some power of the monodromy at infinity is maximally unipotent. In 95 cases the Q-Fano is realized as a hypersurface in weighted projective space and we describe the mirror K3 fibration explicitly.

February 8, 2023
Colloquium [5]	The critical exponent: old and new.	4:15pm - LOM 214	The critical exponent is an important numerical invariant of discrete groups acting on negatively curved Hadamard manifolds, Gromov hyperbolic spaces, and higher-rank symmetric spaces. In this talk, I will focus on discrete groups acting on hyperbolic spaces (i.e., Kleinian groups), which is a family of important examples of these three types of spaces. In particular, I will review the classical result relating the critical exponent to the Hausdorff dimension using the Patterson-Sullivan theory and introduce new results about Kleinian groups with small or large critical exponents.

February 10, 2023
Friday Morning Seminar [6]	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!

Links
[1] https://calendar.math.yale.edu/list/calendar/grid/week/2023-W05 [2] https://calendar.math.yale.edu/list/calendar/grid/week/2023-W07 [3] https://calendar.math.yale.edu/seminars/group-actions-geometry-and-dynamics [4] https://calendar.math.yale.edu/seminars/geometry-symmetry-and-physics [5] https://calendar.math.yale.edu/seminars/colloquium [6] https://calendar.math.yale.edu/seminars/friday-morning-seminar [7] https://calendar.math.yale.edu/list/calendar/grid/week/abstract/2023-W05 [8] https://calendar.math.yale.edu/list/calendar/grid/week/abstract/2023-W07