Abstract: 

In the study of circle dynamics, the regularity of a map famously dictates its dynamical behavior—a theme dating back to Denjoy's celebrated C^1 vs. C^2 dichotomy. When considering monotone one-parameter families of circle diffeomorphisms, a natural question arises: how does the regularity of the family govern the regularity of the rotation number as a function of the family parameter? It is known that C^2 families yield a Hölder continuous rotation number with exponent at least 1/2 (Graczyk) and C^1 families guarantee at least log-Hölder continuity (Gorodetski & Kleptsyn). But is this the best we can do for generic C^2 families? And for a generic C^1 family, could Hölder continuity somehow survive? What happens with C^{1+\alpha} families? In this talk, we give partial answers to these questions by establishing a modern analogue to Denjoy's dichotomy: we show that for C2 families, the 1/2-Hölder continuity for the rotation number established by Graczyk is generically optimal, whereas for C1 families, Hölder continuity of the rotation number is generically lost entirely. We also comment on the behavior of families coming from a certain dense subset of the C^{1+\alpha} families. 

The moduli space of cubic surfaces and its compactifications are classical and date back to the mid-nineteenth century. While recent progress has been made in describing compactifications of moduli spaces for fully marked cubic surfaces using Kollár-Shepherd-Barron-Alexeev (KSBA) stable pairs with uniform weights, this talk explores an extension into asymmetric weights.

In particular, we investigate the KSBA compactification of moduli spaces of cubic surfaces with a single marked line by considering an asymmetric weight on one line and a uniform weight on the remaining 26 lines. We provide an explicit wall-and-chamber decomposition of the weight domain, giving 20 distinct chambers. These chambers yield new KSBA coarse moduli spaces that reveal how interactions between the marked line and cubic surface singularities give new wall crossings. We also give explicit descriptions of these weighted stable pairs parameterized by the moduli spaces in each chamber.

Holomorphic motions were introduced by Mane, Sad and Sullivan in the 1980's motivated by applications in holomorphic dynamics. They have since then been applied in various other areas such as Kleinian groups, Teichm\"{u}ller theory and most notably quasiconformal mappings. In this talk, I will discuss recent results on the variation of various notions of dimension (Minkowski, Packing, Hausdorff) under holomorphic motions. The results involve a new class of functions called Inf-harmonic.

The classical representation theory of finite and compact groups is built around character theory and the classification of irreducible representations. A fundamental feature of this theory is complete reducibility: every representation decomposes into a direct sum of irreducible representations.

The representation theory of finite groups over fields of positive characteristic p, however, is far more subtle, since complete reducibility generally fails. In this setting, Sylow p-subgroups, together with homological and geometric methods such as support varieties, play a central role.

Supergroups and superalgebras arise naturally in the mathematical foundations of supersymmetry in theoretical physics, where ordinary commuting variables are combined with anticommuting variables (fermions). In this talk, we will explore how ideas from modular representation theory reappear in the study of representations of supergroups. We will discuss “super” analogues of the Sylow theorems and several related classical results in this new setting.