Georgia Institute of Technology College of Sciences

Positroid varieties are subvarieties of the Grassmannian that arise in the study of total positivity. Knutson, Lam, and Speyer described a certain type of Gröbner degeneration called the Hodge degeneration as projections of order complexes of intervals in the Bruhat order, but their description does not give an explicit Gröbner basis nor initial ideal. We give an explicit, combinatorial description of the Gröbner basis and initial ideal corresponding to the Hodge degeneration for an arbitrary positroid variety. As an application, we show that promotion on rectangular-shaped semistandard tableaux gives a bijection between standard monomials of a positroid variety and its cyclic shifts. This is joint work with Shiliang Gao (UIUC) and Daoji Huang (Minnesota).

Inverse problems involve the reconstruction of hidden objects from possibly noisy indirect measurements and are ubiquitous in a variety of scientific and engineering applications. This kind of problems have two main features that make them interesting yet challenging to solve. First, they tend to be ill-posed: the reconstruction is very sensitive to perturbations in the measurements. Second, real-world applications are often large-scale: resulting in computationally demanding tasks. In this talk I will focus on discrete linear problems: giving a general overview of the well-established class of solvers called Krylov subspace methods and its regularizing properties; as well as flexible variants that make them suitable to solve more challenging optimization tasks. I will show results and examples in different imaging applications.

We consider the algorithmic problem of finding large balanced independent sets in sparse random bipartite graphs, and more generally the problem of finding independent sets with specified proportions of vertices on each side of the bipartition. In a bipartite graph it is trivial to find an independent set of density at least half (take one of the partition classes). In contrast, in a random bipartite graph of average degree d, the largest balanced independent sets (containing equal number of vertices from each class) are typically of density (2 + od(1)) log d/d . Can we find such large balanced independent sets in these graphs efficiently? By utilizing the overlap gap property and the low-degree algorithmic framework, we prove that local and low-degree algorithms (even those that know the bipartition) cannot find balanced independent sets of density greater than (1 + ε) log d/d for any ε > 0 fixed and d large but constant.

In recent decades there has been much interest and measured progress in the study of moments of the Riemann zeta-function and, more generally, of various L-functions. Despite a great deal of effort spanning over a century, asymptotic formulas for moments of L-functions remain stubbornly out of reach in all but a few cases. In this talk, we consider the problem for the family of all Dirichlet L-functions of even primitive characters of bounded conductor. I will outline how to harness the asymptotic large sieve to prove an asymptotic formula for the general 2kth moment of an approximation to this family. The result, which assumes the generalized Lindelöf hypothesis for large values of k, agrees with the prediction of Conrey, Farmer, Keating, Rubenstein, and Snaith. Moreover, it provides the first rigorous evidence beyond the so-called “diagonal terms” in their conjectured asymptotic formula for this family of L-functions

If the formal square root of an abelian surface over Q looks like an elliptic curve, it has to be an elliptic curve."

We discuss what such a proposition might mean, and prove the most straightforward version where the precise condition is simply that the L-function of the abelian surface possesses an entire holomorphic square root. The approach follows the Diophantine principle that algebraic numbers or zeros of L-functions repel each other, and is in some sense similar in spirit to the Gelfond--Linnik--Baker solution of the class number one problem.

We discuss furthermore this latter connection: the problems that it raises under a hypothetical presence of Siegel zeros, and a proven analog over finite fields. The basic remark that underlies and motivates these researches is the well-known principle (which is a consequence of the Deuring--Heilbronn phenomenon, to be taken with suitable automorphic forms $f$ and $g$): an exceptional character $\chi$ would cause the formal $\sqrt{L(s,f)L(s,f \otimes \chi)}L(s,g)L(s, g \otimes \chi)$ to have a holomorphic branch on an abnormally big part of the complex plane, all the while enjoying a Dirichlet series formal expansion with almost-integer coefficients. This leads to the kind of situation oftentimes amenable to arithmetic algebraization methods. The most basic (qualitative) form of our main tool is what we are calling the "integral converse theorem for GL(2)," and it is a refinement of a recent Unbounded Denominators theorem that we proved jointly with Frank Calegari and Yunqing Tang. 

Mapping class groups of surfaces in general have cohomology that is hard to compute. Meanwhile, within something called the cohomologically-stable range, a family of characteristic classes called the MMM classes (of surface bundles) is enough to generate this cohomology and thus plays an important role for understanding both the mapping class group and surface bundles. Moreover, constructing the so-called Atiyah-Kodaira manifold provides the setting to prove that these MMM classes are non-trivial. Most of this beginner-friendly talk will be dedicated to proving the non-triviality of the first MMM class. Maybe as a side quest, we will also give a crash course on the geometric viewpoint of (co)homology and then apply this viewpoint to understand the constructions and the proofs.

The clique chromatic number of a graph is the smallest number of colors in a vertex coloring so that no inclusion-maximal clique is monochromatic (ignoring isolated vertices). For the binomial random graph G_{n,p} the clique chromatic number has been studied in a number of works since 2016, but for sparse edge-probabilities in the range n^{-2/5} \ll p \ll 1 even the order of magnitude remained a technical challenge.

Resolving open problems of Alon and Krivelevich as well as Lichev, Mitsche and Warnke, we determine the clique chromatic number of the binomial random graph G_{n,p} in most of the missing regime: we show that it is of order (\log n)/p for edge-probabilities n^{-2/5+\eps} \ll p \ll n^{-1/3} and n^{-1/3+\eps} \ll p \ll 1, for any constant \eps > 0. Perhaps surprisingly for a result about random graphs, a key ingredient in the proof is an application of the probabilistic method (that hinges on careful counting and density arguments).

This talk is based on joint work with Lutz Warnke.

Discrete delay population models are often considered as a compromise between single-species models and more advanced age-structured population models, C.W. Clark, J. Math. Bio. 1976. This talk is based on a recent work (S. Streipert and G.S.K. Wolkowicz, 2023), where we provide a procedure for deriving discrete population models for the size of the adult population at the beginning of each breeding cycle and assume only adult individuals reproduce. This derivation technique includes delay to account for the number of breeding cycles a newborn individual remains immature and does not contribute to reproduction. These models include a survival probability (during the delay period) for the immature individuals, since these individuals have to survive to reach maturity and become members of what we consider the adult population. We discuss properties of this class of discrete delay population models and show that there is a critical delay threshold. The population goes extinct if the delay exceeds this threshold. We apply this derivation procedure to two well-known population models, the Beverton–Holt and the Ricker population model. We analyze their dynamics and compare it to existing delay models.

I will give an overview of recent work, joint with Jacques Verstraete, where we gave an improved lower bound for the off-diagonal Ramsey number $r(4,t)$, solving a long-standing conjecture of Erd\H{o}s. Our proof has a strong non-probabilistic component, in contrast to previous work. This approach was generalized in further work with David Conlon, Dhruv Mubayi and Jacques Verstraete to off-diagonal Ramsey numbers $r(H,t)$ for any fixed graph $H$. We will go over of the main ideas of these proofs and indicate some open problems.

In the early 60’s J. B. Keller and D. Levy discovered a fundamental property: the instability tongues in Mathieu-type equations lose sharpness with the addition of higher-frequency harmonics in the Mathieu potentials. Twenty years later, V. Arnold discovered a similar phenomenon on the sharpness of Arnold tongues in circle maps (and rediscovered the result of Keller and Levy). In this paper we find a third class of object where a similar type of behavior takes place: area-preserving maps of the cylinder. loosely speaking, we show that periodic orbits of standard maps are extra fragile with respect to added drift (i.e. non-exactness) if the potential of the map is a trigonometric polynomial. That is, higher-frequency harmonics make periodic orbits more robust with respect to “drift". This observation was motivated by the study of traveling waves in the discretized sine-Gordon equation which in turn models a wide variety of physical systems. This is a joint work with Mark Levi.