## Seminars and Colloquia by Series

### Characterizing multigraded regularity on products of projective spaces

Series
Algebra Student Seminar
Time
Friday, October 15, 2021 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Mahrud SayrafiUniversity of Minnesota

Motivated by toric geometry, Maclagan-Smith defined the multigraded Castelnuovo-Mumford regularity for sheaves on a simplicial toric variety. While this definition reduces to the usual definition on a projective space, other descriptions of regularity in terms of the Betti numbers, local cohomology, or resolutions of truncations of the corresponding graded module proven by Eisenbud and Goto are no longer equivalent. I will discuss recent joint work with Lauren Cranton Heller and Juliette Bruce on generalizing Eisenbud-Goto's conditions to the "easiest difficult" case, namely products of projective spaces, and our hopes and dreams for how to do the same for other toric varieties.

### SLn skein algebra and quantum matrices

Series
Geometry Topology Student Seminar
Time
Wednesday, October 13, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006 (also in BlueJeans)
Speaker
Tao YuGeorgia Tech

Since Jones introduced his knot polynomial using representation theory, there has been a wide variety of invariants defined this way, e.g., HOMFLY-PT and Reshetikhin-Turaev. Recently, through the work of Bonahon-Wong and Constantino-Le, some of these invariants are reinterpreted as quantum matrices. In this talk, we will review the history of these representation theoretical knot invariants. Then we will discuss one particular connection to the quantum special linear group.

### Structure and computation of data-driven brain networks

Series
Research Horizons Seminar
Time
Wednesday, October 13, 2021 - 12:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hannah ChoiGeorgia Tech

Please Note: The seminar will also be streamed live at https://bluejeans.com/787128769/7101 . Questions will be fielded by the organizer.

The complex connectivity structure unique to the brain network is believed to underlie its robust and efficient coding capability. One of many unique features of the mammalian brain network is its spatial embedding and hierarchical organization. I will discuss effects of these structural characteristics on network dynamics as well as their computational implications with a focus on the flexibility between modular and global computations and predictive coding.

### Using simple baseline models to interpret developmental processes in C. elegans

Series
Mathematical Biology Seminar
Time
Wednesday, October 13, 2021 - 11:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Niall M. ManganNorthwestern University

Growth control establishes organism size, requiring mechanisms to sense and adjust growth. Studies of single cells revealed that size homeostasis uses distinct control methods: Size, Timer, and Adder. In multicellular organisms, mechanisms that regulate single cell growth must integrate control across organs and tissues during development to generate adult size and shape. We leveraged the roundworm Caenorhabditis elegans as a scalable and tractable model to collect precise growth measurements of thousands of individuals, measure feeding behavior, and quantify changes in animal size and shape. Using quantitative measurements and mathematical modeling, we propose two models of physical mechanisms by which C. elegans can control growth. First, constraints on cuticle stretch generate mechanical signals through which animals sense body size and initiate larval-stage transitions. Second, mechanical control of food intake drives growth rate within larval stages. These results suggest how physical constraints control developmental timing and growth rate in C. elegans.

https://www.biorxiv.org/content/10.1101/2021.04.01.438121v2

### 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.

### 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.

### 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

Please Note: Zoom link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09 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.

### 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.

### 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

### 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

Please Note: 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.