# Graduate Colloquium

The graduate students of the Mathematics Department coordinate a quarterly colloquium. The talks are generally every Thursday from 4-5 PM, with an informal tea (get-together) beforehand.

**Fall 2021 Graduate Colloquium**

**MONDAYS @ 4pm California Time**

**"tea" @ ~3:30pm, talk @ 4pm**

**Schedule ***(click dates for abstract)*

**Alejandro Bravo Doddoli, UCSC**

*Sub-Riemannian Geometry and Metric Lines*

A sub-Riemannian geometry is given by a Manifold \(M\), non-integrable distribution \(D\) and inner product \(g\) for vector tangent to \(D\). Given this triplet, we will define the concepts of sub-Riemannian distance and geodesic flow. Similar to Riemannian Geometry, geodesics may lose their optimality, after passing their cut time due to Jacobi points and discrete symmetries. In the opposite direction, a geodesic which globally minimizes on \(M\) is called a metric line; in other words, a metric line is an isometric embedding of the real line on \(M\). In the past year the problem of classifying metric lines in Carnot groups, groups whose Lie algebra is nilpotent, has been an active area of research. The main classical theories to approach this problem are the weak KAM theory and optimal synthesis. Richard Montgomery and I developed a mixed technique to solve this problem. We will review the basic concepts of sub-Riemannian and introduce the main ideas from weak KAM theory and optimal synthesis in order to explain our new technique.

**Deewang Bhamidipati, UCSC**

**Fundamental Groups: Topological, Étale, Tannakian**

One cannot escape the similarities between the Galois theory of field extensions and the topological theory of covering spaces and fundamental groups. In both theories, there are objects that sit at the top whose automorphisms form a group: the absolute Galois group in the former case and the fundamental group in the latter. Taking subgroups we get intermediate objects and in some nice cases, the automorphism groups of these intermediate objects arise as quotients of the larger automorphism group.

Grothendieck established an algebraic theory of the fundamental group that not only realises the absolute Galois group as the fundamental group of a geometric object but also encompasses the classification of finite covers of complex algebraic varieties of any dimension (Riemann surfaces, for example) and even notions coming from arithmetic such as specialization modulo a prime.

A third kind of a fundamental group also exists, the Tannakian fundamental group associated to a Tannakian category, which can be seen as a linearisation of the ideas above. The origin is a classical theorem from the theory of topological groups due to Tannaka and Krein.

In this talk, we'll discuss these three notions and see some examples.

**David Rubinstein, UCSC**

*Algebras in Tensor Triangular Categories: Separability, Descent, and Finite Étale extensions*

In this talk, we will discuss some naturally occurring examples and properties of separable ring objects in various tt categories. Given any Monoidal Category \((\mathcal{C}, \otimes, \mathbf{1})\) one can define ring objects \(A\) and then attempt to do Homological Algebra over the ring by studying its associated Module Category \(\mathit{A\mbox{-} Mod}_{\mathcal{C}}\). It is often not the case however that the category of \(A\)-Modules has the same "structure" as \(\mathcal{C}\). For example, given a ring object \(A\) in a tensor-triangulated category, the category of \(A\)-modules is in general not even triangulated. If our ring is "separable" then the category of \(A\)-modules is canonically triangulated, and if \(A\) is commutative it is furthermore a tensor-triangulated category such that the extension of scalars functor is a tensor-triangular functor. The study of these separable ring objects includes smashing localizations as a special case, and also includes what we call finite étale extensions. As the name suggests, extending scalars \(D(R) \xrightarrow{-\otimes_R S} D(S)\) by a commutative étale \(R\)-algebra \(S\) is an example of a finite étale extension; much more surprising is that restriction to a finite index subgroup \(\mathit{Res}_H^G: \mathit{Stab}(kG) \to \mathit{Stab}(kH)\) is as well.

**François Monard, UCSC**

*TeX typesetting, bibliography management and other TeX hacks*

**vim-latex**over

**TeXworks**(keyboard shortcuts are better than clicking around)

*minimal*effort using

**JabRef**;

**ipe**

**todonotes**,

**showkeys**

**Yufei Zhang, UCSC**

*Tensor-Triangular Classification, Support Theory and the Application on Commutative Noetherian Rings*

In most known areas where triangulated category plays a role, say algebraic geometry, stable homotopy theory, modular representation theory, etc., the categories we are interested in are too wild to classify all the objects, except for the trivial examples. In this talk, we will discuss a kind of classification up to the tensor-triangular structure in a tensor-triangulated category, via the Balmer spectrum. To be more precise, we want to understand the lattice of thick tensor-ideals, a special kind of triangulated subcategories. Then we will talk about the support theory for compactly generated tensor triangulated categories. We will see the application to algebraic geometry: the derived category of modules over a commutative noetherian ring and the famous result that the Balmer spectrum of the derived category of perfect complexes is homeomorphic to the spectrum of the ring. Finally, we will discuss the derived category of pseudo-coherent complexes over a commutative noetherian ring.

**Tzu-Mo Kuo, UCSC**

*Geometry of Conformal Manifolds: Preserving Geodesics*

A Riemannian conformal manifold is a manifold with an equivalent class of Riemannian metrics: two metric tensors are equivalent if one is equal to the other one by multiplying a smooth positive function. In this talk, I'll give the definition of Cartan geometry, and overview the construction of normal Cartan bundles and tractor bundles of conformal manifolds with dimensions larger than or equal to 3. With the concept of Cartan geometry, I'll introduce conformal geodesics. The remaining part of the talk is to focus on the problem: if a diffeomorphism preserves geodesics, is it a conformal isometry? I'll review some facts of this problem in Riemannian sense and CR sense. Since the Riemannian case (done by Shoshichi Kobayashi) needs the holonomy group concept, I'll take a quick look at holonomy groups, de Rham decomposition and conformal de Rham decomposition (done by Stuart Armstrong).

**Monday, November 15, 2021, 4pm**

**Mita Banik, UCSC**

*Hamiltonian Dynamics and Pseudo-rotations*

The study of Hamiltonian systems on a symplectic manifold plays a fundamental role both in geometry and classical mechanics. One distinguishing feature of these systems is that, with some exceptions, they tend to have infinitely many periodic orbits. This fact is usually referred to as the Conley Conjecture and has been established for a broad class of manifolds. However the conjecture fails for very simple manifolds such as the two-sphere. These manifolds admit Hamiltonian diffeomorphisms with few periodic orbits, the so-called pseudo-rotations, which are of particular interest in Hamiltonian dynamics. The existence of pseudo-rotations has topological implications on the symplectic manifold such as the deformed quantum-product and constraints on minimal Chern number. We introduce symplectic topological tools such as Hamiltonian Floer homology and Gromov-Witten invariants essential to study these problems and briefly survey recent results in these directions.

**Monday, November 22, 2021, 4pm**

**Xu Gao, UCSC**

*Stable Simplexes of p-adic Representations in Bruhat-Tits Buildings*

A \(p\)-adic representation of a group \(G\) is a representation where the underlying vector space \(V\) is defined over a non-Archimedean local field. One aspect of such representations is the notion of stable lattices, namely lattices in the vector spaces which are stable under the action of \(G\). In a recent work of Junecue Suh, stable lattices of \(p\)-adic representations are studied and in particular, there is a formula for the class number (number of homothety classes of stable lattices) when the representation has regular reduction, and an estimate for the growth of class number along a tower of totally ramified extensions is given. In his work, the theory of Bruhat-Tits building plays an essential role: the class number is interpreted as a quantity related to the fixed point set in the Bruhat-Tits building of \(\mathrm{GL}(V)\) under the action of \(G\). The above story may be generalized: one may consider other variants of \(p\)-adic representations, namely the underlying space could be equipped with a bilinear form or is over a division ring. Then what generalizes the set of homothety classes of stable lattices is the fixed point set in the general Bruhat-Tits building. In this talk, I will briefly summarize Junecue's work and then talk a little about its generalization(s).

**Monday, November 29, 2021, 4pm**

**Nathan Marianovsky, UCSC**

*The Dynamics of Billiards in the Presence of a Magnetic Field*

For more than a century the study of billiards in the plane has captured the difficulties that arise whenever studying any dynamical system. Given a strictly convex set in the plane called the table and a homogeneous, stationary magnetic field in the plane, different types of setups for billiard systems arise. By setting distinct regions, formed by the table, with fixed values of the magnetic field we will observe several scenarios under which straight lines and circular arcs of a particular Larmor radius are concatenated to form the trajectory of a charged particle. Once a trajectory is understood, one is typically interested in understanding how many periodic orbits there are, if the periodic orbits can be classified, whether the system is integrable, and what can be said of the existence of caustics.