Georgia Institute of Technology College of Sciences

Suppose $f$ is a univariate polynomial with integer coefficients of absolute value at most $H$, exactly $t$ monomial terms, and degree $d$. Suppose also that $p,q$ are integers of absolute value at most $H$. Then one can determine the sign of $f(p/q)$ in time $(d^3 \log H)^{2+\epsilon}$, by combining work of Liouville from about 170 years ago, work of Mahler from about 60 years ago, work of Neff, Reif, and Pan from about 30 years ago, and more recent refinements.

However, when $t$ is small, one can do much better: When $t=2$, one can determine the sign of $f(p/q)$ in time $\log^5(dH)$, via work of Baker from about 55 years ago. Koiran asked, around 2016, about the $t=3$ case, after proving new bounds on the minimal separation of complex roots of univariate trinomials.

We make progress on Koiran's question by giving a new, dramatically faster algorithm for solving univariate trinomials over the real numbers. The key innovation is a new family of non-hypergeometric series for the roots of $f$ when $f$ is close to having a degenerate root. This is joint work with Emma Boniface and Weixun Deng.

Satellite operations are one of the most basic operations in knot theory. Many researchers have studied the behavior of knot Floer homology under satellite operations. Most of these results use Lipshitz, Ozsvath and Thurston's bordered Heegaard Floer theory. In this talk, we discuss a new technique for studying these operators, and we apply this technique to a family of operators called L-space operators. Using this theory, we are able in many cases to give a simple formula for the behavior of the concordance invariant tau under such operators. This formula generalizes a large number of existing formulas for the behavior of tau under satellite operations (such as cabling, 1-bridge braids and generalized Mazur patterns), and also has a number of topological applications. This is joint work with Daren Chen and Hugo Zhou.

Artificial intelligence (AI) has become a transformative force in scientific discovery---known as AI for Science---with profound impact on computational molecular design, as highlighted by the 2024 Nobel Prize in Chemistry. Due to their remarkable capability in analyzing complex structures, message-passing neural networks and diffusion- and flow-based generative models stand out as effective tools for molecular property prediction and structure generation. However, message-passing neural networks struggle to efficiently integrate multiscale molecular features and complex 3D geometry for accurate property prediction, and (2) the generative processes of generative models are often computationally intensive and error-prone. 

In this talk, I will present our recent advances toward overcoming these limitations: (1) multiscale graph representations and message-passing architectures for efficient and accurate molecular learning, and (2) one-step flow-based generative models that enable high-fidelity molecule generation with dramatically reduced computational cost.

Mean-field control with common noise and filtering problems naturally lead to second-order PDEs on Wasserstein space. In this talk, we analyze a class of such equations in which the second-order operator is finite-dimensional in nature. We establish comparison principles and apply them to obtain particle convergence rates in mean-field control. The talk is based on joint work with Erhan Bayraktar, Ibrahim Ekren, and Xihao He.

Suppose that a continuous-time, stochastic diffusion (i.e., the Susceptible-Infected process) spreads on an unknown graph. We only observe the time at which the diffusion reaches each vertex, i.e., the set of infection times. What can be learned about the unknown graph from the infection times? While there is far too little information to learn individual edges in the graph, we show that certain high-level properties -- such as the number of vertices of sufficiently high degree, or super-spreaders -- can surprisingly be determined with certainty. To achieve this goal, we develop a suite of algorithms that can efficiently detect vertices of degree asymptotically greater than sqrt(n) from infection times, for a natural and general class of graphs with n vertices. To complement these results, we show that our algorithms are information-theoretically optimal: there exist graphs for which it is impossible to tell whether vertices of degree larger than n^{1/2 - \epsilon} exist from vertices' infection times, for any \epsilon > 0. Finally, we discuss the broader implications of our ideas for change-point detection in non-stationary point processes. This talk is based on joint work with Anna Brandenberger (MIT) and Elchanan Mossel (MIT).

Given a connected finite graph $G$, an integer-valued function $f$ on $V(G)$ is called $M$-Lipschitz if the value of $f$ changes by at most $M$ along the edges of $G$. In 2013, Peled, Samotij, and Yehudayoff showed that random $M$-Lipschitz functions on graphs with sufficiently good expansion typically exhibit small fluctuations, giving sharp bounds on the typical range of such functions, assuming $M$ is not too large. We prove that the same conclusion holds under a relaxed expansion condition and for larger $M$, (partially) answering questions of Peled et al. Our approach combines Sapozhenko’s graph container method with entropy techniques from information theory.

This is joint work with Krueger and Park.

How hard is it to estimate a sequence of length N, satisfying some *unknown* linear recurrence relation of order S and observed in additive Gaussian noise? The class of all such sequences is extremely rich: it is formed by arbitrary (complex) exponential polynomials with total degree S. This includes the case of stationary sequences, a.k.a. harmonic oscillations, a.k.a. sequences with discrete​ Fourier spectra supported on S *arbitrary* frequencies. Strikingly, it turns out that one can estimate such sequences with almost the same statistical error as if the recurrence relation was known (and a simple least-squares estimator could be used). In particular, stationary sequences can be estimated with mean-squared error of order O(S/N) up to a polylogarithmic factor, without any assumption of spectral separation—despite what one might learn in a high-dimensional statistics class. Moreover, these methods are computationally tractable. 

In this talk, I will highlight some mathematics underlying this result, putting emphasis on analytical, rather than statistical, side of things. In particular, I will show how to invert a polynomial while ensuring that the result is a polynomial—rather than a reciprocal of a polynomial—and what this has to do with reproducing kernels. Then, I will pitch some accessible open problems in this area.

The Heisenberg projection problem asks whether there is an analogue of the Marstrand projection theorem in the first Heisenberg group, namely whether Hausdorff dimension of sets generically decreases under projection, for a natural family of projections arising from the group structure. This problem is still open, but I will discuss a recent improvement to the known bound obtained through a variable coefficient local smoothing inequality. 

Rather than going through the proof in detail, I will spend most of the talk introducing the problem and explaining the connection to averaging operators over curves, and explaining why these operators are Fourier integral operators satisfying Sogge's cinematic curvature condition. This condition was originally introduced by Sogge to generalise Bourgain's circular maximal theorem, but it turns out to have useful applications to projection theory. 

Two prominent questions in low dimensional topology are: which knots are slice, and which $\mathbb{Q}$-homology $S^3$'s bound $\mathbb{Q}$-homology $B^4$'s? These questions are connected by a theorem that states if a knot $K$ in $S^3$ is slice, then the 2-fold branch cover of $S^3$ over $K$ bounds a $\mathbb{Q}$-homology $B^4$. In this talk we introduce a generalization of $\chi$-sliceness of links to the rational homology context, generalize the earlier theorem to state that for a rationally $\chi$-slice link $L$, for all sufficiently large primes $p$, the $p$-fold cyclic branch cover of $S^3$ over $L$ bounds a $\mathbb{Q}$-homology $B^4$, and examine a connection to a number-theoretic obstruction on the Alexander polynomial.

We will start with a presentation by Griffin Edwards and continue with a free discussion.

First-passage percolation on the square lattice is a random growth model in which each edge of Z^2 is assigned an i.i.d. nonnegative weight.  The passage time between two points is the smallest total weight of a nearest-neighbor path connecting them, and a path achieving this minimum is called a geodesic.  Typically, the number of edges in a geodesic is comparable to the Euclidean distance between its endpoints.  However, when the edge-weights take the value 0 with probability exactly 1/2, a strikingly different behavior occurs: geodesics travel primarily on critical clusters of zero-weight edges, whose internal graph distance scales superlinearly with Euclidean distance.  Determining the precise degree of this superlinear scaling is a challenging and ongoing endeavor.  I will discuss recent progress on this front (joint with David Harper, Xiao Shen, and Evan Sorensen), along with complementary results on a dual problem, where we restrict path lengths and analyze passage times (joint with Jack Hanson and Daniel Slonim).