Tuesday,
October 6 |
- 13:15–14:15 Combinatorics Seminar, Kemeny 242
 
- Unbounded length minimal synchronizing words for quantum channels over qutrits and qubits
- Janani Lakshmanan
- A quantum variant of the longstanding Černy conjecture was addressed by Grudka et al., who constructed quantum channels with synchronizing words of length 3 for qutrits. We extend their result to arbitrary long minimal synchronizing words, then adjust the technique for the 2-dimensional qubit case.
- 13:30–14:30 Applied and Computational Mathematics Seminar, Kemeny 307
- The residual prior transform: from signal recovery to data assimilation
- Anne Gelb, Dartmouth College
- Recovering signals, images, and physical states from noisy or incomplete data is typically framed as a regularization problem: enforce that some transform of the unknown is sparse. Classical choices such as total variation and its higher-order variants fix a single assumed order of smoothness, which forces a trade-off between staircasing and ringing whenever the true variability changes across the domain—exactly the situation for piecewise smooth signals and for physical states governed by hyperbolic conservation laws, whose solutions develop shocks even from smooth initial data.
We introduce the residual transform operator , built from two distinct operators, , that nevertheless satisfy to the same order of approximation in the transform domain. Because and are matched in this way regardless of the underlying signal's local smoothness, acts as an annihilating operator: it is small wherever the underlying signal is smooth and large only at genuine discontinuities, without ever needing to know the signal's variability in advance. Placed inside a generalized sparse Bayesian learning framework, yields both a robust point estimate as well as uncertainty quantification, with the regularization strength learned automatically from the data rather than hand-tuned.
We then carry into the 3D-Var variational data assimilation setting for hyperbolic conservation laws, where the usual Gaussian background term is ill-suited to a shock's genuinely multimodal position uncertainty. Replacing this with residual-based regularization yields analyses that remain accurate even when the background-error covariance is misspecified, with the advantage over standard baselines growing precisely as observations become sparser—the regime where this structure-preserving approach earns its keep.
This research is done in collaboration with Yao Xiao and Adityvikram Viswanathan, and is partially supported by DOD (ONR MURI) #N00014-20-1-2595 and DOE ASCR #DE-SC0025555.
- 14:30–15:30 Algebra and Number Theory Seminar, Kemeny 343
- Recognizing zeta functions of K3 surfaces over finite fields
- Asher Auel, Dartmouth College
- How many solutions does a system of polynomial equations have over a finite field? This question is important, with direct applications to the construction of error-correcting codes. The Hasse-Weil zeta function of a variety over a finite field is a generating function for the number of points over all finite extensions. By the Weil conjectures, the zeta function is a rational function determined by an alternating product of characteristic polynomials of the action of Frobenius on the l-adic cohomology of the variety. Hence we gain information on the number of rational points by studying these characteristic polynomials.
In this talk, I'll give an update on the Honda-Tate program for K3 surfaces, which aims to determine the possible zeta functions of K3 surfaces over a finite field. I'll describe a recent breakthrough by Sair Shaikh, a recently graduated math major, and its applications to joint work with Engel, Kedlaya, and Shaikh to the problem of identifying K3 surfaces with no rational points.
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