Our settings are one-dimensional compact symbolic systems. We discuss the flexibility of the pressure function of a continuous potential (observable) with respect to a parameter regarded as the inverse temperature. The points of non-differentiability of this function are of particular interest in statistical physics since they correspond to qualitative changes of the characteristics of a dynamical system referred to as phase transitions. It is well known that the pressure function is convex, Lipschitz, and has an asymptote at infinity. We show that these are the only restrictions. We present a method to explicitly construct a continuous potential whose pressure function coincides with any prescribed convex Lipschitz asymptotically linear function starting at a given positive value of the parameter.
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This is based on joint work with Anthony Quas.

For a non-amenable group $G$, there may be many (exotic) group $C^*$-algebras that lie naturally between the universal and the reduced $C^*$-algebra of $G$. Let $G$ be a simple Lie group or an appropriate locally compact group acting on a tree. I will explain how the $L^p$-integrability properties of different spherical functions on $G$ (relative to a maximal compact subgroup) can be used to distinguish between different (exotic) group $C^*$-algebras. This recovers results of Samei and Wiersma. Additionally, I will explain that under certain natural assumptions, the aforementioned exotic group $C^*$-algebras are the only ones coming from $G$-invariant ideals in the Fourier-Stieltjes algebra of $G$.
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This is based on joint work with Dennis Heinig and Timo Siebenand.

Interior Point methods and Sequential Quadratic Programming (SQP) methods have become two of the most crucial methods for solving large-scale nonlinear optimization problems. The two methods take very different approaches to solving the same problem. SQP methods find approximate solutions to a sequence of linearly constrained quadratic subproblems in which a quadratic model of the Lagrangian is minimized subject to a linear model of the constraints. Typically, the QP subproblems are solved using an active-set method, giving the problem a major-minor iteration pattern in which each iteration of the active-set method solves an indefinite system. In contrast, interior point methods follow a continuous path towards the optimal solution by perturbing the first-order optimality conditions of the problem. In this talk, we discuss a shifted primal dual interior point method and its potential applicability in solving the QP subproblem of an SQP method. We also discuss some of
the potential issues with this approach that we hope to overcome.

A classical question in differential topology is the following: Classify all simply-connected, closed, smooth (2n)-manifolds whose only non-trivial homology groups are $H_0, H_n$ and $H_{2n}$.
In this talk I will survey the history of the high dimensional side of this question and how its resolution requires a surprisingly deep understanding of the Adams spectral sequence computing the stable homotopy groups of spheres. Time permitting, I will then discuss how the situation changes as we relax our topological restrictions on the manifold (for example allowing $H_{n-e}$, $H_{n-e+1}$, ... $H_{n+e}$ to be non-trivial for a small number e).
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This talk represents joint work with Jeremy Hahn and Andy Senger.

Set theory has long been considered the foundation of all mathematical thought. However, people who prove theorems on computers for a living don't use set theory anymore--and now some suggest that mathematicians should do the same. We'll discuss the problems people have with Set Theory and its main alternative, Type Theory.

How many ways can a positive integer be written as the sum of four squares? There is a simple formula for the number of ways, which goes back to Jacobi. I'll introduce modular forms and sketch how they provide an answer to this question.

I present new stochastic multi-particle models for global optimization of nonconvex functions on the sphere. These models belong to the class of Consensus-Based Optimization methods. In fact, particles move over the manifold driven by a drift towards an instantaneous consensus point, computed as a combination of the particle locations weighted by the cost function according to Laplace's principle. The consensus point represents an approximation to a global minimizer. The dynamics is further perturbed by a random vector field to favor exploration, whose variance is a function of the distance of the particles to the consensus point. In particular, as soon as the consensus is reached, then the stochastic component vanishes. In the first part of the talk, I present the well-posedness of the model on the sphere and we derive rigorously its mean-field approximation for large particle limit.
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In the second part I address the proof of convergence of numerical schemes to global minimizers provided conditions of well-preparation of the initial datum. The proof combines the mean-field limit with a novel asymptotic analysis, and classical convergence results of numerical methods for SDE. We present several numerical experiments, which show that the proposed algorithm scales well with the dimension and is extremely versatile. To quantify the performances of the new approach, we show that the algorithm is able to perform essentially as good as ad hoc state of the art methods in challenging problems in signal processing and machine learning, namely the phase retrieval problem and the robust subspace detection.
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Joint work with H. Huang, L. Pareschi, and P. S"{u}nnen

A generalization of Mazur's theorem states that there are 26
possibilities for the torsion subgroup of an elliptic curve over a
quadratic extension of $\mathbb{Q}$. If $G$ is one of these groups, we
count the number of elliptic curves of bounded naive height whose
torsion subgroup is isomorphic to $G$ in the case of imaginary quadratic
fields.

A projective manifold is algebraically hyperbolic if the
degree of any curve is bounded from above by its genus times a
constant, which is independent from the curve. This is a property which
follows from Kobayashi hyperbolicity. We prove that hyperkahler
manifolds are not algebraically hyperbolic when the Picard rank is at
least 3, or if the Picard rank is 2 and the SYZ conjecture on existence
of Lagrangian fibrations is true. We also prove that if the automorphism
group of a hyperkahler manifold is infinite, then it is algebraically
non-hyperbolic.
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These results are a joint work with Misha Verbitsky.