Finite element methods are numerical methods that approximate solutions to PDEs using functions on a mesh representing the problem domain. Discontinuous-Petrov Galerkin Methods are a class of finite element methods that are aimed at achieving stability of the Petrov-Galerkin finite element approximation through a careful selection of the associated trial and test spaces. In this talk, I will present DPG theorems as they apply to linear problems, and then approaches for those theorems in the case of semi-linear problems. In particular, I will explore a particular case of semilinear problems, that allows for results in the linear case to hold.

A central theme in the theory of von Neumann algebras is to determine all possible tensor product decompositions of a given factor. I will present a recent joint work with Yusuke Isono where we use the rigidity of full factors and a new flip automorphism approach in order to study this problem. Among other things, we show that a separable full factor admits at most countably many tensor product decompositions (up to stable unitary conjugacy). We also establish new primeness and Unique Prime Factorization results for crossed products coming from compact actions of irreducible higher rank lattices (e.g. $SL_n(\mathbb{Z})$ for $n>2$) as well as noncommutative Bernoulli shifts with arbitrary base (not necessarily amenable).

We will discuss some important classical results from Glimm and Thoma about the existence of ``big'' irreducible representations. No prerequisites required beyond curiosity.

In this talk, we explore explicit cross sections to the horocycle and geodesic flows on $\operatorname{SL}(2, \mathbb{R})/G_q$, with $q \geq 3$. Our approach relies on extending properties of the primitive integers $\mathbb{Z}_\text{prim}^2 := \{(a, b) \in \mathbb{Z}^2 \mid \gcd(a, b) = 1\}$ to the discrete orbits $\Lambda_q := G_q (1, 0)^T$ of the linear action of $G_q$ on the plane $\mathbb{R}^2$. We present an algorithm for generating the elements of $\Lambda_q$ that extends the classical Stern-Brocot process, and from that derive another algorithm for generating the elements of $\Lambda_q$ in planar strips in increasing order of slope. We parametrize those two algorithm using what we refer to as the \emph{symmetric $G_q$-Farey map}, and \emph{$G_q$-BCZ map}, and demonstrate that they are the first return maps of the geodesic and horocycle flows resp. on $\operatorname{SL}(2, \mathbb{R})/G_q$ to particular cross sections. Using homogeneous dynamics, we then show how to extend several classical results on the statistics of the Farey fractions, and the symbolic dynamics of the geodesic flow on the modular surface to our setting using the $G_q$-BCZ and symmetric $G_q$-Farey maps. This talk is self-contained, and does not assume any prior knowledge of Hecke triangle groups or homogeneous dynamics.

An important result of X.-J. Wang states that a convex ancient solution to mean curvature flow either sweeps out all of space or lies in a stationary slab (the region between two fixed parallel hyperplanes). We will describe recent results on the construction and classification of convex ancient solutions and convex translating solutions to mean curvature flow which lie in slab regions, highlighting the connection between the two. Work is joint with Theodora Bourni and Giuseppe Tinaglia.

Tverberg's Theorem says that a set with sufficiently many points in $\mathbb{R}^d$ can always be partitioned into $m$ parts so that the $(m-1)$-simplex is the (nerve) intersection pattern of the convex hulls of the parts. In this talk we will talk about intersection patterns and how Tverberg's Theorem is but a special case of a more general situation where other simplicial complexes arise as nerves.

In this talk, we shall discuss recent progress in the global
uniqueness issues for inverse boundary problems for second order elliptic
equations, such as the conductivity and magnetic Schrodinger equations,
with low regularity coefficients. Generally speaking, in an inverse
boundary problem, one wishes to determine the coefficients of a PDE inside
a domain from the knowledge of its solutions along the boundary of the
domain. While ubiquitous in practice, the mathematical analysis of such
problems is quite challenging, and the consideration of the low regularity
setting, motivated by applications, brings additional substantial
difficulties. In this talk, we shall discuss the case of full, as well as
partial, measurements, both for domains in the Euclidean space, as well as
in the more general setting of transversally anisotropic compact
Riemannian manifolds with boundary. Some of the important ingredients in
our approach are semiclassical Carleman estimates with limiting Carleman
weights with an optimal gain of derivatives, precise smoothing estimates,
as well as a construction of Gaussian beam quasimodes in a low regularity
setting. This is joint work with Gunther Uhlmann.

In this talk, I'll give a brief introduction on the study of the Nevanlinna theory (theory of holomorphic curves) using the stochastic calculus, following the works fo B. Davis, T. K. Carne and A. Atsuji etc.. In particular, I will outline the ideas and compare its method with the classical approach, and outline its advantage on studying the maps on the (general) Kahler manifolds (or even the Rieamnnian manifolds).

Biomolecules are inherently flexible and their dynamics affects ligand binding. To investigate the internal dynamics of proteins and nucleic acids, we apply all-atom molecular dynamics
simulations. However, atomistic level of detail makes simulations too computationally demanding to describe folding and global motions so reduced representations of molecules are often
applied. These so-called coarse-grained models are sufficient to capture global collective motions on biologically relevant spatial and temporal scales. However, due to reduction of the
degrees of freedom, coarse-grained models require parameterizations of the potential energy function (force field). Moreover, coarse-grained force field parameters are typically not transferable between different molecules and problems.

I will present our efforts to design an automatic parameterization procedure to obtain force fields for reduced models of biomolecules. The procedure for the optimization of potential
energy parameters is based on metaheuristic methods. I will also show examples of applications to dynamics of proteins and nucleic acids.

This is a joint work with Wanke Yin. We discuss the connection between the type defined by the Lie-Bracket of vector fields, the finite type condition in terms of the trace of Levi form and the order of contact with smooth complex submanifolds. We discuss a partial solution to an old conjecture of Bloom asked about 40 years ago.

Formulated in 1984, Green's Conjecture predicts that one can recognize
the intrinsic complexity of an algebraic curve from the syzygies of its
canonical embedding. Green's Conjecture for a general curve has been
resolved using geometric methods in two landmark papers by Voisin in the
early 00s. I will explain how the theory of Koszul modules provides an
alternative solution to this problem, by relating it via Hermite
reciprocity to the study of the syzygies of the tangent developable
surface to a rational normal curve. Joint work with M. Aprodu, G.
Farkas, S. Papadima, and J. Weyman.