In a couple of recent works with Khare and Larsen we have
developed a method to construct finite Lie groups of type $C_n, B_n$ and $G_2$ (and potentially more) as Galois groups over the field of rational numbers. The main tools are functorial lifts and self-dual automorphic forms on $GL(n)$ as a source of $l$-adic representations (Kottwitz, Clozel, Harris-Taylor). In this talk I will discuss how one can control the image of the $l$-adic representation by picking local components of automorphic representations: deeply embedded tame parameters and Jordan subgroups, as main tools.

The examples will be: [1] Spectral determinants of conformally covariant operators (e.g. conformal Laplace and Dirac operators), and [2] Total $Q$-curvature. Both of these
satisfy conformal invariance. We recently found a striking universality in the variational structure when the base manifold is the round sphere, in any such problem, by using
representation theory of the conformal group. As a corollary it gives us a neat proof of some of the extremal results in Kate Okikiolu's paper from Annals (2001), and of analogous results for many other examples. Lastly, I will mention the proof
of $B\% \ r-$ Schopka's conjecture on dimensional asymptotics of explicit determinants on spheres (i.e. the extremal values in the examples [1] above). Some of the work is joint with Bent Orsted.

We formulate $p$-adaptive finite element methods based on the technique called gradient recovery. The answers to two open questions on the design of transit elements and the existence of approximate formulae for the error will be addressed.

here has been much recent progress on the classification of
discrete series representations of classical groups. In particular, we have the following two partial classifications: One, due to Jiang and Soudry, of generic representations in terms of Shahidi's L-invariants, and second, due to DeBacker and Reeder, depth zero representations can be organized into into $L$-packets, which are characterized in terms of character distributions. In particular, the two results are expressed in terms of two different languages. We show that the two classifications coincide where they overlap,
that is, for generic representations of depth 0. This result is relevant for the work on the Inverse Problem in Galois Theory.

We present a simple method for solving a free boundary value
problem of locating either minimum free energy transition path
ensemble or minimum potential energy transition path between
two configurations on a corresponding energy surface. Our method, called gradient-augmented Harmonic Fourier Beads, employs the global Fourier representation of the path that is a curve interpolation of a discrete set of points on the surface - beads. To optimize the path curve, the method computes energy gradients for each bead from either convex optimization or molecular dynamics or Monte Carlo simulations subject to harmonic restraints. The path optimization is driven by primitive Steepest Descent procedure. Line integration of the Fourier transformed forces along the path curve provides complete and accurate structural and energetic information regarding all the intermediates and saddle points present. The utility of the HFB method is demonstrated by computing potentials of mean force for various transformations of diverse molecular systems.

Justification Logic is a relatively new field that studies provability,
knowledge, and belief via proofs, or justifications, explicitly present in
the language. Many justification logics have been developed that closely
resemble modal epistemic logics of knowledge and belief with one important
difference: instead of modality box with existential epistemic reading
'there exists a proof of $F$,' justification logics operate with constructs $t
:F$, where a justification term $t$ represents a blueprint of a Hilbert-style
proof of $F$. The machinery of explicit justifications can be used to analyze
well-known epistemic paradoxes such as Gettier's examples, to study
self-referential properties of modal logics, and to avoid Logical
Omniscience. This talk will focus on quantitative analysis of justification
logics. We will give an overview of what is known about their decidability
and complexity of the decision procedure. We will also analyze a realization
procedure that provides a bridge from a modal epistemic logic to its
justification counterpart. We will discuss the complexity of one such
realization procedure as well as provide its qualitative analysis that leads
to interesting corollaries about self-referentiality of modal logics.

Given an algebraic automorphic representation of a genus two symplectic group over a totally real number field with an Iwahori-spherical component at a finite place $w$, we shall show - following Sorensen - that there is a congruent automorphic representation with a tamely ramified principal series component
at $w$. Thus after a base change to a finite solvable totally real extension, the level at $w$ can be lowered.

Let $S=\{ (a_1, b_1), (a_2,b_2),\dots, (a_n, b_n)\}$ be a set of lattice points in the first quadrant $\{(x,y): x \geq 0, y \geq 0\}$ and set
\[
\Delta_S (x,y) = \det \lVert x_i^{a_j} y_i^{b_j} \rVert_{i,j=1}^n
\]
Let $\mathbf{M}_S$ denote the linear span of all the partial derivatives of $\Delta (x,y)$. Computer data reveals that these vector spaces of polynomials intersect in the most remarkable ways. Since the eary 90's we have accumulated a variety of conjectures most of which are still open. In this talk we will give a glimpse of this amazing mathematical Kaleidoscope.

Recent developments in the field of Numerical Relativity have not only provided key insights of binary black hole systems but also began influencing its future role. Undoubtedly one of the most important future drivers in the near future of the field will be its role as another element within the study of spectacular astrophysical phenomena involving strongly gravitation scenarios. Connecting (yet to be observed) gravitational waves with observations within the electromagnetic spectra will be one ultimate goal of this enterprise. This talk will summarise some interesting results on the binary black hole problem, connect with the data analysis efforts and illustrate the study of magnetized binary neutron star systems as a stepping stone towards more ambitious goals.

Let $\pi$ be an algebraic automorphic representation of a reductive group $G$ over a totally real number field $F$. Assume $G$ is anisotropic at infinity, and $\pi$ is not congruent to an automorphic character. Suppose $w$ is a finite place of $F$ where the component of $\pi$ is unramified and congruent to the
trivial representation. Then there is an automorphic representation $\pi'$ of $G$ congruent to $\pi$, with the same central character and type at infinity, whose component at w is more ramified than that of $\pi$. Applications in rank one and two include showing that Saito-Kurokawa forms are congruent to generic ones, for the genus two symplectic group.

Finite state automata are Turing machines with fixed finite bounds on resource use. Automata lend themselves well to real-time computations and efficient algorithms. Therefore, there is a long tradition of studying that part of mathematics which can be represented by automata. This talk will give a survey of this research. In particular, we discuss three major themes: how complicated can automatic structures be? can they be naturally described? how efficient are the associated algorithms? Examples include Thurston's automatic groups associated with 3-manifolds and automatic structures associated to model checking and program verification in computer science.

Haagerup and I introduced some 10 years ago a very simple
mechanism to construct classes of irreducible subfactors with exotic standard invariants. We will review this construction and describe the planar algebras arising from it (joint work with Das and Ghosh). Voiculescu's notion of freeness appears naturally in this context.