Deciding nonnegativity of real polynomials is a key question in real algebraic geometry with crucial importance in polynomial optimization. It is well-known that in general this problem is NP-hard, therefore one is interested in finding sufficient conditions (certificates) for nonnegativity, which are easier to check. Since the 19th century, sums of squares (SOS) are a standard certificate for nonnegativity, which can be recognized using semidefinite programming (SDP). This approach, however, has some issues, especially in practice if the optimization problem has many variables or high degree.
In this talk I will introduce sums of nonnegative circuit polynomials (SONC). SONC polynomials are certain sparse polynomials having a special structure in terms of their Newton polytopes and supports and serve as a nonnegativity certificate for real polynomials, which is independent of sums of squares.
Moreover, I will provide an overview about polynomial optimization via SONC polynomials. Similar as SOS correspond to SDP, the new SONC certificates correspond to geometric programming and relative entropy programming. Based on a Positivstellensatz for SONC polynomials we establish a converging hierarchy of efficiently computable lower bounds for constrained optimization problems.
The talk is based on joint work with Sadik Iliman, Adam Kurpisz, and Timo de Wolff.

In this talk, we prove global well-posedness of a system
describing behavior of dilute flexible polymeric fluids. This model is
based on kinetic theory, and a main difficulty for this system is its
multi-scale nature. A new function space, based on moments, is
introduced to address this issue, and this function space allows us to
deal with larger initial data.

This is an introduction to the Birch and Swinnerton-Dyer
Conjecture on L-functions of elliptic curves. The talk is aimed
at graduate and undergraduate students who are strongly encouraged
to attend.

Most of the basic results on martingale problems extend to the setting in which the generator depends on a control. The ``control'' could represent a random environment, or the generator could specify a classical stochastic control problem. The equivalence between the martingale problem and forward equation (obtained by taking expectations of the martingales) provides the tool for extending linear programming methods introduced by Manne in the context of controlled finite Markov chains to general Markov stochastic control problems. The controlled martingale problem can also be applied to the study of constrained Markov processes (e.g., reflecting diffusions), the boundary process being treated as a control. Time permitting: the relationship between the control formulation and viscosity solutions of the corresponding resolvent equation will be discussed. Talk includes joint work with Richard Stockbridge and with Cristina Costantini.

In this talk, we will discuss a particular Hermitian flow on compact or complete non-compact complex manifolds. By using the flow, we will discuss the existence of Kahler-Einstein metric on Hermitian manifolds with quasi-negative bisectional curvature.