We present a characterization of the Anosov condition adapted for linear representations which preserve a flag. Key to this characterization is the consistency of large eigenvalue configurations, which we will motivate and explore in detail. We'll see that this line of reasoning gives rise to a convex deformation space of reducible Anosov representations, and conclude that for many word hyperbolic groups, any connected component of the character variety consisting of Anosov representations cannot contain reducible representations. Time permitting, we'll discuss work in progress to further characterize the convex domains of Anosov representations.

We'll briefly sketch an argument that every completely reducible Anosov representation in SL(d, R) can be deformed to a non-Anosov one.* The key fact is that completely reducible Anosov representations have consistent configurations of large eigenvalues coming from the block decomposition. We conclude that every component of the corresponding character variety consisting of (equivalence classes of) Anosov representations consists of (equivalence classes of) irreducible representations.

*I note that this fact only holds when the commutator subgroup of the domain group has infinite index.

We will introduce and study the dynamical properties of reducible suspensions of linear representations of non-elementary hyperbolic groups, which are discrete, (almost) faithful, and quasi-isometrically embedded linear representations preserving and acting weakly unipotently on a proper non-zero subspace.  After characterizing when these representations are Anosov, we note that the derived conditions correspond to points in bounded convex domains in a finite-dimensional real vector space. Finally, we will use stronger characterizations of such domains for symmetric Anosov representations to find deformations of Borel Anosov representations which retain some but not all of the Anosov conditions and to compute examples of non-Anosov limits of Anosov representations.

*This seminar talk was canceled due to anonymous threats of disruption and violence against the speaker.

We will introduce a class of reducible representations of word hyperbolic groups analogous to the classical construction of Hitchin representations as deformations of irreducibly embedded Fuchsian representations. After characterizing which of these representations are Anosov, we will make conclusions about the failure modes for the Anosov conditions under deformations.

Though "decolonizing" seems to be a popular buzzword in academic spaces, it's not altogether clear what it means. Ironically, it seldom refers to addressing the core problems associated with colonization: theft of land and culture, loss of (human/nonhuman) life, and destruction of the planet. Our guiding question will be: can mathematicians engage in decolonization, insofar as what we mean by decolonization is struggling against these core issues?

How many simple curves can be drawn on a torus so that any pair of them intersect some specified number of times? We will introduce combinatorial questions of this nature, explore some known cases in an interactive activity, and point to what is known/conjectured for more complicated surfaces. We'll also reframe questions of this type in terms of graph-theoretic questions related to cousins of the curve complex. No prior knowledge of surface topology is required, but curiosity, active participation, and an appreciation of balloons are recommended and encouraged.

It is well-established that all deformations of Fuchsian representations of closed surface groups irreducibly embedded in higher rank special linear groups (that is, Hitchin representations) are Anosov. The same cannot be said for reducibly embedded Fuchsian representations. We will investigate this phenomenon by computing how far a generalization of these representations can be deformed (in specific ways) until they are no longer Anosov.

While a marked hyperbolic structure on a surface is determined up to isotopy by its marked length spectrum, the same cannot be said of the unmarked length spectrum; there are isospectral hyperbolic surfaces which aren't isometric! However, recent results of Baik-Choi-Kim and ongoing work of Aougab-L-Loving-Miller give some hope of rectifying this failure of length spectral rigidity by restricting the spectra to simple curves. We present applications of trace relations in surface groups toward this goal and raise relevant unanswered questions along the way.

The mapping class group is a central player in almost every area of surface topology. After computing some basic examples, we will take a whirlwind tour through various fundamental results concerning the algebra (the Lickorish twist theorem, lantern relations, the Nielsen realization problem, the Dehn-Nielsen-Baer theorem) of the mapping class group and the dynamics of its action on Teichmüller space (Fricke's theorem, the Nielsen-Thurston classification). This talk is meant to be accessible to graduate students with some previous exposure to the fundamental group and Teichmüller space.

A marked hyperbolic metric on a closed surface of genus g > 1 is determined by the lengths of finitely closed curves. However, a natural converse of this result can fail spectacularly: there are arbitrarily large finite collections of curves, all of whose lengths agree in every hyperbolic metric. Leininger gives an equivalence between curves whose lengths agree in every hyperbolic metric and elements of the fundamental group whose characters agree in any SL(2,C) representation. We first review various spectral rigidity (or lack-of-rigidity) results for linear representations of closed surface groups, and then focus on techniques relevant to applications to unmarked simple length spectral rigidity.

How many simple curves can be drawn on a torus so that any pair of them intersect some specified number of times? We will introduce combinatorial questions of this nature, explore some known cases in an interactive activity, and point to what is known and what is conjectured for surfaces of higher genus. We'll also reframe questions of this type in terms of graph-theoretic questions related to cousins of the curve complex. No prior knowledge of surface topology is required, but curiosity and an appreciation of balloons are recommended and encouraged.

How many simple closed curves can be drawn on a surface of finite type so that any pair of them intersect some specified number of times? We will introduce questions of this nature, explore some known cases in an interactive activity, and point to what is known and what is conjectured.  We'll also reframe questions of this type in terms of graph-theoretic questions related to cousins of the curve complex. No prior knowledge of surface topology or graph theory is required, but curiosity and an appreciation of balloons are recommended.