Georgia Institute of Technology College of Sciences

In this talk I will briefly describe link Floer homology toolbox and its usefulness. Then I will show how link Floer homology can detect links with small ranks, using a rank bound for fibered links by generalizing an existing result for knots. I will also show that stronger detection results can be obtained as the knot Floer homology can be shown to detect T(2,8) and T(2,10), and that link Floer homology detects (2,2n)-cables of trefoil and figure eight knot. This talk is based on a joint work with Fraser Binns (Boston College).

Large-scale machine learning models are trained by parallel (stochastic) gradient descent algorithms on distributed systems. The communications for gradient aggregation and model synchronization become the major obstacles for efficient learning as the number of nodes and the model's dimension scale up. In this talk, I will introduce several ways to compress the transferred data and reduce the overall communication such that the obstacles can be immensely mitigated. More specifically, I will introduce methods to reduce or eliminate the compression error without additional communication.

The $n$-th ordered configuration space of a graph parametrizes ways of placing $n$ distinct and labelled particles on that graph. The homology of the one-point compactification of such configuration space is equipped with commuting actions of a symmetric group and the outer automorphism group of a free group. We give a cellular decomposition of these configuration spaces on which the actions are realized cellularly and thus construct an efficient free resolution for their homology representations. As our main application, we obtain computer calculations of the top weight rational cohomology of the moduli spaces $\mathcal{M}_{2,n}$, equivalently the rational homology of the tropical moduli spaces $\Delta_{2,n}$, as a representation of $S_n$ acting by permuting point labels for all $n\leq 10$. This is joint work with Christin Bibby, Melody Chan, and Nir Gadish. Our paper can be found on arXiv with ID 2109.03302.

For a graphic degree sequence $\mathbf{d_n}= (d_1 , . . . , d_n)$ of graphs with vertices $v_1 , . . . , v_n$, $\mathbf{d_n}$-process is the random graph process that inserts one edge at a time at random with the restriction that the degree of $v_i$ is at most $d_i$ . In 1999, N. Wormald asked whether the final graph of random $\mathbf{d_n}$-process is "similar" to the uniform random graph with degree sequence $\mathbf{d_n}$ when $\mathbf{d_n}=(d,\dots, d)$. We answer this question for the $\mathbf{d_n}$-process when the degree sequence $\mathbf{d_n}$ that is not close to being regular. We used the method of switching for stochastic processes; this allows us to track the edge statistics of the $\mathbf{d_n}$-process. Joint work with Mike Molloy and Lutz Warnke.

In close collaboration with experimentalists and clinicians, mathematical models that are parameterized with experimental and clinical data can help estimate patient-specific disease dynamics and treatment success. This positions us at the forefront of the advent of ‘virtual trials’ that predict personalized optimized treatment protocols. I will discuss a couple of different projects to demonstrate how to integrate calculus into clinical decision making. I will present a variety of mathematical model that can be calibrated from early treatment response dynamics to forecast responses to subsequent treatment. This may help us to identify patient candidates for treatment escalation when needed, and treatment de-escalation without jeopardizing outcomes.

Recording link: https://bluejeans.com/s/dcDrDQuxm2W

Given a surface S, the Alexander method is a combinatorial tool used to determine whether two self-homeomorphisms of S are isotopic. This statement was formalized in the case of finite-type surfaces, which are surfaces with finitely generated fundamental groups. A version of the Alexander method was extended to infinite-type surfaces by Hernández-Morales-Valdez and Hernández-Hidber. We extend the remainder of the Alexander method to include infinite-type surfaces. 

I will discuss a general framework for studying what can be said about the rank of a matrix A over a field K if we only know certain crude features of A. For example, what can we say about rank(A) if we only know which entries are zero and which are nonzero? Or if K = R, what if we only know the signs of the entries of A? Or K is a normed field and we only know the absolute values? Or K=C and we only know the arguments? There are many partial answers to questions like this scattered throughout the literature, and I will explain how at least some of these results can be unified through a theory of ranks of matrices over hyperfields. This is work in progress with Tianyi Zhang.

From a perspective of an applied algebraic geometer, I will address several use cases of algorithmic machinery that goes hand in hand with the language of tropical geometry. One of the examples originates in dynamical systems and may shed (tropical) light on a long-standing conjecture in celestial mechanics.

 In their seminal paper on combinatorial Hodge theory, Adiprasito, Huh, and Katz showed, among other things, that a very specific set of toric varieties has Chow rings that satisfy Poincaré duality, even though the varieties are not compact. In joint work with Farbod Shokrieh, we generalize this statement to all toric varieties whose fans are supported on a tropically smooth set. This has several consequences in tropical intersection theory; most notably it allows us to prove the long-suspected duality between tropical cycles and cocycles.

In my talk I will assume no prior knowledge of tropical intersection theory. I will define tropical cycles and cocycles explicitly and explain how they are connected to the intersection theory of toric varieties and the Chow rings of fans appearing in combinatorial Hodge theory. Finally, we will see how to use the duality statement mentioned above to define the tropical intersection product.

Recall that, for a variety $X$ in a projective space $\mathbb{P}^d$, the $X$-rank of a point $p\in \mathbb{P}^d$ is the least number of points of $X$ whose span contains the point $p$. Studies about $X$-ranks include some well-known and important results about various tensor ranks. For example, 

• the rank of tensors is the rank with respect to Segre varieties,
• the rank of symmetric tensors, i.e. Waring rank, is the rank with respect to Veronese embeddings, and
• the rank of anti-symmetric tensors is the rank with respect to Grassmannians in its Plücker embedding.  

In this talk, we focus on ranks with respect to Veronese embeddings of a projective line $\mathbb{P}^1$. i.e. symmetric tensor ranks of binary forms. We will discuss how to associate points in $\mathbb{P}^d$ with binary forms and I will introduce apolarity for binary forms which gives an effective method to study Waring ranks of binary forms. We will discuss various ranks on the Veronese embedding and some results on the ranks.

My goal is to present a computer assisted proof of a non-trivial theorem in nonlinear dynamics, in full detail.  My (quite biased) definition of non-trivial is that there should be some infinite dimensional complications.  However, since I want to go through all the details, I need these complications to be as simple as possible.  So, I'll consider the Henon map, and prove that some 1 dimensional stable and unstable manifolds attached to a hyperbolic fixed point intersect transversally.  By Smale's theorem, this implies the existence of chaotic motions.  Recall that one can prove the existence chaotic dynamics for the Henon map more or less by hand using topological methods.  Yet transverse intersection of the manifolds is a stronger statement, and moreover the method I'll discuss generalizes to much more sophisticated examples where pen-and-paper fail.

The idea of the proof is to develop a high order polynomial expansion of the stable/unstable manifolds of the fixed point, to prove an a-posteriori theorem about the convergence and truncation error bounds for this expansion, and to check the hypotheses of this theorem using the computer.  All of this relies on the parameterization method of Cabre, Fontich, and de la Llave, and on finite numerical calculations using interval arithmetic to manage the inevitable roundoff errors. Once global enough representations of the local invariant manifolds are obtained and equipped with mathematically rigorous error bounds, it is a finite dimensional problem to establish that the manifolds intersect transversally.