Kan simplicial manifolds, also known as "Lie infinity-groupoids", are simplicial Banach manifolds which satisfy conditions similar to the horn lling conditions for Kan simplicial sets. Group-like Lie infinity-groupoids (a.k.a "Lie infinity-groups") have been used to construct geometric models for the higher stages of the Whitehead tower of the orthogonal group. With this goal in mind, Andre Henriques developed a smooth analog of Sullivan's realization functor from rational homotopy theory which produces a Lie infinity-group from certain commutative dg-algebras (i.e. L_infinity-algebras).

In this talk, I will present a homotopy theory for both these commutative dg-algebras and for Lie infinity-groups, and discuss some examples that demonstrate the compatibility between the two. Conceptually, this work can be interpreted either as a C^\infty-analog of classical results of Bouseld and Gugenheim in rational homotopy theory, or as a homotopy-theoretic analog of classical theorems from
Lie theory. This is based on joint work with A. Ozbek (UNR grad student) and C. Zhu (Gottingen).

Convex surface theory and bypasses are extremely powerful tools
for analyzing contact 3-manifolds.  In particular they have been
successfully applied to many classification problems.  After reviewing
convex surface theory in dimension three,  we explain how to generalize many
of their properties to higher dimensions.   This is joint work with Yang
Huang.

In this talk we will focus on constructing "super-isolated abelian varieties". These are abelian varieties that have isogeny class which contains a single isomorphism class. Their motivation comes from security concerns in elliptic and hyperelliptic curve cryptography. Using a theorem of Honda and Tate, we transfer the problem of finding such varieties to a problem in algebraic number theory. Finding these varieties turns out to be related to finding primes of the form n2 + 1 and to solving Pell's equation.

The Hull-Strominger system is a system of nonlinear PDEs describing the geometry of compactification of heterotic strings with torsion to 4d Minkowski spacetime, which can be regarded as a generalization of Ricci-flat Kähler metrics coupled with Hermitian Yang-Mills equation on non-Kähler Calabi-Yau 3-folds. The Anomaly flow is a parabolic approach to understand the Hull-Strominger system initiated by Phong-Picard-Zhang. We show that in the setting of generalized Calabi-Gray manifolds, the Hull-Strominger system and the Anomaly flow reduce to interesting elliptic and parabolic equations on Riemann surfaces. By solving these equations, we obtain solutions to the Hull-Strominger system on a class of compact non-Kähler Calabi-Yau 3-folds with infinitely many topological types and sets of Hodge numbers. This talk is based on joint work with Zhijie Huang and Sebastien Picard.

Conjugation invariant ensembles of random matrices have long formed one of the basic paradigms in Random Matrix Theory. Apart from the Gaussian case, the matrix elements of a conjugation invariant random matrix are highly correlated, and this fact has traditionally been viewed as prohibiting the use of moment methods in the spectral analysis of invariant ensembles. However, it turns out that there is a very natural and appealing version of the moment method available for these ensembles which seems to have been overlooked. I will describe the rudiments of this method, and some of its applications. Based on joint work with Sho Matsumoto.