For a quiver with potential, we can associate a vanishing cycle to each representation space. If there is a nice torus action on the potential, the vanishing cycles can be expressed in terms of truncated Jacobian algebras. We study how these vanishing cycles change under the mutation of Derksen-Weyman-Zelevinsky. We use Ringel-Hall algebras as the main organizing tools. The wall-crossing formula leads to a categorification of quantum cluster algebras under the assumption of existence of certain potential. This is a special case of A. Efimov's result, but our approach is more concrete and down-to-earth. We also obtain a formula relating the representation Grassmannians under sink-source reflections. In this talk, I will start with basic definitions of mutations of quivers with potentials, vanishing cycles, and Hall algebras.

We study the problem of finding best rank-1 approximations for both
symmetric and nonsymmetric tensors. For symmetric tensors, this is
equivalent to optimizing homogeneous polynomials over unit spheres; for
nonsymmetric tensors, this is equivalent to optimizing multi-quadratic
forms over multi-spheres. We propose semidefinite relaxations, based on
sum of squares representations, to solve these polynomial optimization
problems. Some numerical experiments are presented to show that this
approach is practical in getting best rank-1 approximations.

\def\RR{{\mathbb R}}

Where do polynomial inequalities come from? We have polynomials $p$ and $q$ acting on tuples $x$ of numbers in $\RR^n$. Suppose $p(x) > 0$ for all $x$ in $\RR^n$ making $q(x) > 0$. Is there some algebraic relationship between $p,q$ equivalent to this? That is a lot to hope for, but an ``algebraic certificate'' equivalent to an inequality often exists and this is the substance of much of real algebraic geometry (RAG). It is a subject which bloomed in the last 50 years.

Now consider $n$ tuples $X:= \{ X_1, \dots, X_n \}$ of symmetric matrices $X_j$ and polynomials $p$ and $q$ acting on such tuples, for example, $n=2$ and $$p(X):= X_1 {X_2}^3 + {X_2}^3 X_1 + {X_1}^5.$$ The polynomial yields a value $p(X)$ that is a symmetric matrix, and we can consider the same issues as in classical RAG. We have polynomials $p$ and $q$. Suppose $p(X)$ is positive definite for all $X$, making $q(X)$ a positive definite matrix. Recall a positive definite matrix is one whose eigenvalues are all $>0$. A theory parallel to RAG for converting such inequalities to algebra formulas has emerged in the last ten years. The motivation came from problems in systems engineering but spread out from this in many directions.

The talk will give a taste of selections from this smorgasboard.

We'll discuss various constraints on rational cuspidal curves in the projective plane, and consider the question of whether such curves are necessarily equivalent to a line, under a birational automorphism of the plane.

This is the first in a series of seven Food for Thought talks on the Millennium Prize Problems. In this talk we give an introduction to the Clay Mathematics Institute and its Millennium Prize Problems program. We then focus our attention on the question of whether \textbf{P}, the class of problems with polynomial-time algorithms, equals \textbf{NP}, the class of problems with polynomial-time nondeterministic algorithms. We will define \textbf{P} and \textbf{NP} in terms of Turing machines. We will then move on to some diverse contemporary approaches to proving either \textbf{P} = \textbf{NP} or \textbf{P} $\neq$ \textbf{NP}, and discuss the implications that such a theorem would have. Along the way, we will see some famous results that have appeared out of the struggle to solve this hallmark riddle of theoretical computer science.

Biological processes are carried out through molecular conformational transitions, ranging from the structural changes within biomolecules to the formation of macromolecular complexes and the associations between the complexes themselves. These transitions cover a vast range of timescales and are governed by a tangled network of molecular interactions. The resulting hierarchy of interactions, in turn, becomes encoded in the experimentally measurable “mechanical fingerprints” of the biomolecules, their force–extension curves. However, how can we decode these fingerprints so that they reveal the kinetic barriers and the associated timescales of a biological process? Here, we show that this can be accomplished with a simple, model-free transformation that is general enough to be applicable to molecular interactions involving an arbitrarily large number of kinetic barriers. Specifically, the transformation converts the mechanical fingerprints of the system directly into a map of force-dependent rate constants. This map reveals the kinetics of the multitude of rate processes in the system beyond what is typically accessible to direct measurements. With the contributions from individual barriers to the interaction network now “untangled”, the map is straightforward to analyze in terms of the prominent barriers and timescales. Practical implementation of the transformation is illustrated with simulated biomolecular interactions that comprise different patterns of complexity—from a cascade of activation barriers to competing dissociation pathways.

Some central limit results on stochastic processes in a compact connected 2-point homogeneous space $E(d)$ of growing dimension $d$ are reformulated within the theory of polynomial convolution structures. This approach stresses the algebraic-topological relationship between those structures and the asymptotic properties of the stochastic processes under consideration, in particular of random walks and Gaussian processes on $E(d)$ with $d\to\infty$.