## Seminars and Colloquia Schedule

### Invariants of rational homology 3-spheres from the abelianization of the mod-p Torelli group (Virtual)

Series
Geometry Topology Seminar
Time
Monday, October 4, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Ricard Riba GarciaUAB Barcelona

Unlike the integral case, given a prime number p, not all Z/p-homology 3-spheres can be constructed as a Heegaard splitting with a gluing map an element of mod p Torelli group, M[p]. Nevertheless, letting p vary we can get any rational homology 3-sphere. This motivated us to study invariants of rational homology 3-spheres that comes from M[p]. In this talk we present an algebraic tool to construct invariants of rational homology 3-spheres from a family of 2-cocycles on M[p]. Then we apply this tool to give all possible invariants that are induced by a lift to M[p] of a family of 2-cocycles on the abelianization of M[p], getting a family of invariants that we will describe precisely.

### High-Order Multirate Explicit Time-Stepping Schemes for the Baroclinic-Barotropic Split Dynamics in Primitive Equations

Series
Applied and Computational Mathematics Seminar
Time
Monday, October 4, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
online
Speaker
Lili JuUniversity of South Carolina

To treat the multiple time scales of ocean dynamics in an efficient manner, the baroclinic-barotropic splitting technique has been widely used for solving the primitive equations for ocean modeling. In this paper, we propose second and third-order multirate explicit time-stepping schemes for such split systems based on the strong stability-preserving Runge-Kutta (SSPRK) framework. Our method allows for a large time step to be used for advancing the three-dimensional (slow) baroclinic mode and a small time step for the two-dimensional (fast) barotropic mode, so that each of the two mode solves only need satisfy their respective CFL condition to maintain numerical stability. It is well known that the SSPRK method achieves high-order temporal accuracy by utilizing a convex combination of forward-Euler steps. At each time step of our method, the baroclinic velocity is first computed by using the SSPRK scheme to advance the baroclinic-barotropic system with the large time step, then the barotropic velocity is specially corrected by using the same SSPRK scheme with the small time step to advance the barotropic subsystem with a barotropic forcing interpolated based on values from the preceding baroclinic solves. Finally, the fluid thickness and the sea surface height perturbation is updated by coupling the predicted baroclinic and barotropic velocities. Two benchmark tests drawn from the MPAS-Ocean" platform are used to numerically demonstrate the accuracy and parallel performance of the proposed schemes.

The bluejeans link for the seminar is https://bluejeans.com/457724603/4379

### Geometric equations for matroid varieties

Series
Algebra Seminar
Time
Tuesday, October 5, 2021 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ashley K. WheelerGeorgia Tech

Each point x in Gr(r, n) corresponds to an r × n matrix A_x which gives rise to a matroid M_x on its columns. Gel’fand, Goresky, MacPherson, and Serganova showed that the sets {y ∈ Gr(r, n)|M_y = M_x} form a stratification of Gr(r, n) with many beautiful properties. However, results of Mnëv and Sturmfels show that these strata can be quite complicated, and in particular may have arbitrary singularities. We study the ideals I_x of matroid varieties, the Zariski closures of these strata. We construct several classes of examples based on theorems from projective geometry and describe how the GrassmannCayley algebra may be used to derive non-trivial elements of I_x geometrically when the combinatorics of the matroid is sufficiently rich.

### Turán numbers of some complete degenerate hypergraphs

Series
Graph Theory Seminar
Time
Tuesday, October 5, 2021 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Xiaofan YuanGeorgia Institute of Technology

Note the unusual time!

Let $K^{(r)}_{s_1,s_2,\cdots,s_r}$ be the complete $r$-partite $r$-uniform hypergraph and $ex(n, K^{(r)}_{s_1,s_2,\cdots,s_r})$ be the maximum number of edges in any $n$-vertex $K^{(r)}_{s_1,s_2,\cdots,s_r}$-free $r$-uniform hypergraph. It is well-known in the graph case that $ex(n,K_{s,t})=\Theta(n^{2-1/s})$ when $t$ is sufficiently larger than $s$. We generalize the above to hypergraphs by showing that if $s_r$ is sufficiently larger than $s_1,s_2,\cdots,s_{r-1}$ then $$ex(n, K^{(r)}_{s_1,s_2,\cdots,s_r})=\Theta\left(n^{r-\frac{1}{s_1s_2\cdots s_{r-1}}}\right).$$ This is joint work with Jie Ma and Mingwei Zhang.

### Surfaces of Infinite Type

Series
Research Horizons Seminar
Time
Wednesday, October 6, 2021 - 12:30 for 1 hour (actually 50 minutes)
Location
ONLINE https://bluejeans.com/506659049/8073
Speaker
Yvon VerberneGeorgia Tech

The mapping class group of a surface is well understood for surfaces of finite type. In contrast, the study of mapping class groups of infinite type surfaces is a new field with many opportunities to establish new results. In this talk, we will introduce infinite type surfaces and their mapping class groups.

https://bluejeans.com/506659049/8073

### On Anosovity, divergence and bi-contact surgery

Series
Geometry Topology Student Seminar
Time
Wednesday, October 6, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006 (also on BlueJeans)
Speaker
Surena HozooriGeorgia Tech

I will revisit the relation between Anosov 3-flows and invariant volume forms, from a contact geometric point of view. Consequently, I will give a contact geometric characterization of when a flow with dominated splitting is Anosov based on its divergence, as well as a Reeb dynamical interpretation of when such flows are volume preserving. Moreover, I will discuss the implications of this study on the surgery theory of Anosov 3-flows. In particular, I will conclude that the Goodman-Fried surgery of Anosov flows can be reconstructed, using a bi-contact surgery of Salmoiraghi.

### Small breathers of nonlinear Klein-Gordon equations via exponentially small homoclinic splitting: Part 1 of 2

Series
CDSNS Colloquium
Time
Friday, October 8, 2021 - 13:00 for 1 hour (actually 50 minutes)
Location
Speaker
Chongchun ZengGeorgia Tech

<br />
This is a two-part talk- the continuation is to be given the following week.

Breathers are temporally periodic and spatially localized solutions of evolutionary PDEs. They are known to exist for integrable PDEs such as the sine-Gordon equation, but are believed to be rare for general nonlinear PDEs. When the spatial dimension is equal to one, exchanging the roles of time and space variables (in the so-called spatial dynamics framework), breathers can be interpreted as homoclinic solutions to steady solutions and thus arising from the intersections of the stable and unstable manifolds of the steady states. In this talk, we shall study small breathers of the nonlinear Klein-Gordon equation generated in an unfolding bifurcation as a pair of eigenvalues collide at the original when a parameter (temporal frequency) varies. Due to the presence of the oscillatory modes, generally the finite dimensional stable and unstable manifolds do not intersect in the infinite dimensional phase space, but with an exponentially small splitting (relative to the amplitude of the breather) in this singular perturbation problem of multiple time scales. This splitting leads to the transversal intersection of the center-stable and center-unstable manifolds which produces small amplitude generalized breathers with exponentially small tails. Due to the exponential small splitting, classical perturbative techniques cannot be applied. We will explain how to obtain an asymptotic formula for the distance between the stable and unstable manifold of the steady solutions. This is a joint work of O. Gomide, M. Guardia, T. Seara, and C. Zeng.

### Applications of Donaldson's Diagonlization Theorem

Series
Geometry Topology Working Seminar
Time
Friday, October 8, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jonathan SimoneGeorgia Tech

Donaldson’s Diagonalization Theorem has been used extensively over the past 15 years as an obstructive tool. For example, it has been used to obstruct: rational homology 3-spheres from bounding rational homology 4-balls; knots from being (smoothly) slice; and knots from bounding (smooth) Mobius bands in the 4-ball. In this multi-part series, we will see how this obstruction works, while getting into the weeds with concrete calculations that are usually swept under the rug during research talks.

### The degenerate Eulerian numbers and combinatorics behind them

Series
Combinatorics Seminar
Time
Friday, October 8, 2021 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Orli HerscoviciGeorgia Institute of Technology

In his works Carlitz defined and investigated a few generalizations of the Eulerian numbers and polynomials. For most of those generalizations he provided also a combinatorial interpretation. The classical Eulerian numbers and some of their generalizations are connected to combinatorial statistics on permutations. Carlitz intended to provide a combinatorial interpretation also to his degenerate Eulerian numbers. However since their introduction in 1979 these numbers had a pure analytic character. In this talk we consider a combinatorial model that generalizes the standard definition of permutations and show its relation to the degenerate Eulerian numbers.