All Seminars

Title: New applications of inexact Krylov methods
Seminar: CODES@Emory
Speaker: Malena Sabate Landman of Emory University
Contact: Matthias Chung, matthias.chung@emory.edu
Date: 2023-03-16 at 10:00AM
Venue: MSC W201
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Abstract:
In this talk I will present a new class of algorithms for separable nonlinear inverse problems based on inexact Krylov methods. In particular, I will focus on semi-blind deblurring applications, where we are interested in recovering an approximation of the original image and of a small number of parameters defining the blur. Classical methods in this setting involve solving a sequence of ill-posed and computationally expensive linear problems, and we propose using a new interpretation of inexact Krylov methods to solve this more efficiently. After giving a brief overview of the theoretical properties of these methods, as well as strategies to monitor the amount of inexactness that can be tolerated, the performance of the algorithms will be shown through numerical examples. Finally, I will also give an overview on current ongoing work on using inexact Krylov methods theory in a more general setting involving slowly varying linear systems.
Title: Stability and Statistical Inversion for Travel Time Tomography
Seminar: Analysis and Differential Geometry
Speaker: Hanming Zhou of University of California, Santa Barbara
Contact: Yiran Wang, yiran.wang@emory.edu
Date: 2023-03-16 at 4:00PM
Venue: MSC W301
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Abstract:
In this talk, we consider the travel time tomography for conformal metrics on a bounded domain which consists of determining the conformal factor of the metric from the length of geodesics joining boundary points. We establish forward and inverse stability estimates for simple conformal metrics under some a priori conditions. We then apply the stability estimates to show the consistency of the statistical inversion of the travel time tomography with discrete, noisy measurements. This is based on joint work with Ashwin Tarikere.
Title: Topics in arithmetic statistics
Defense: Dissertation
Speaker: Christopher Keyes of Emory University
Contact: Chris Keyes, christopher.keyes@emory.edu
Date: 2023-02-28 at 4:00PM
Venue: MSC W301
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Abstract:
Arithmetic statistics may be interpreted broadly to include questions in number theory and arithmetic geometry with a distinct quantitative flavor. To answer even simply stated such questions, we often employ diverse algebraic, analytic, or geometric techniques. This dissertation addresses several arithmetic statistical questions, and for its defense we focus on those related to superelliptic curves.\\ \\ A superelliptic curve is given by an affine algebraic equation of the form $C \colon y^m = f(x)$. For a fixed such curve $C$ and degree $n$, we ask how many number fields $K/\mathbb{Q}$ of degree $n$ arise as the minimal field of definition of an algebraic point on $C$, as counted by discriminant? For $n$ sufficiently large and subject to certain conditions, we find infinitely many of these fields, producing an asymptotic lower bound of the form $X^{\delta}$ for an explicit constant $\delta > 0$. In special cases, we are additionally able to count those extensions with prescribed Galois group.\\ \\ For certain degrees $n$, it is possible for a curve to have only finitely many points of degree $n$, or even none at all. Instead of fixing a curve $C$, one might ask how often a curve has (or lacks) points of certain degree, as it varies in some family. In the case of superelliptic curves, we make these questions precise by counting the defining polynomials $f$ by their coefficients. We then find that a positive proportion of superelliptic curves are everywhere locally soluble, a necessary condition for having a rational point, and pin down this proportion exactly in the trigonal genus 4 case. After placing conditions on the family, we also find that for certain degrees $n$, a positive proportion of curves have only finitely many points of degree $n$.
Title: Equivariant Enumerative Geometry
Seminar: Algebra
Speaker: Thomas Brazelton of University of Pennsylvania
Contact: Andrew Kobin, ajkobin@emory.edu
Date: 2023-02-21 at 4:00PM
Venue: MSC W301
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Abstract:
Classical enumerative geometry asks geometric questions of the form "how many?" and expects an integral answer. For example, how many circles can we draw tangent to a given three? How many lines lie on a cubic surface? The fact that these answers are well-defined integers, independent upon the initial parameters of the problem, is Schubert’s principle of conservation of number. In this talk we will outline a program of "equivariant enumerative geometry", which wields equivariant homotopy theory to explore enumerative questions in the presence of symmetry. Our main result is equivariant conservation of number, which states roughly that the orbits of solutions to an equivariant enumerative problem are conserved. We leverage this to compute the $S_{4}$ orbits of the 27 lines on any symmetric cubic surface.
Title: Uniform exponent bounds on the number of primitive extensions of number fields
Seminar: Algebra
Speaker: Robert Lemke Oliver of Tufts University
Contact: Andrew Kobin, ajkobin@emory.edu
Date: 2023-02-14 at 4:00PM
Venue: MSC W301
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Abstract:
A folklore conjecture asserts the existence of a constant $c_n > 0$ such that $N_n(X) \sim c_n X$ as $X\to \infty$, where $N_n(X)$ is the number of degree $n$ extensions $K/\mathbb{Q}$ with discriminant bounded by $X$. This conjecture is known if $n \leq 5$, but even the weaker conjecture that there exists an absolute constant $C\geq 1$ such that $N_n(X) \ll_n X^C$ remains unknown and apparently out of reach. Here, we make progress on this weaker conjecture (which we term the ``uniform exponent conjecture'') in two ways. First, we reduce the general problem to that of studying relative extensions of number fields whose Galois group is an almost simple group in its smallest degree permutation representation. Second, for almost all such groups, we prove the strongest known upper bound on the number of such extensions. These bounds have the effect of resolving the uniform exponent conjecture for solvable groups, sporadic groups, exceptional groups, and classical groups of bounded rank. This is forthcoming work that grew out of conversations with M. Bhargava.
Title: Matchings in hypergraphs defined by groups
Seminar: Combinatorics
Speaker: Alp Müyesser of University College London, UK
Contact: Liana Yepremyan, liana.yepremyan@emory.edu
Date: 2023-02-09 at 4:00PM
Venue: MSC W301
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Abstract:
When can we find perfect matchings in hypergraphs whose vertices represent group elements and edges represent solutions to systems of linear equations? A prototypical problem of this type is the Hall-Paige conjecture, which asks for a characterisation of the groups whose multiplication table (viewed as a Latin square) contains a transversal. Other problems expressible in this language include the toroidal n-queens problem, Graham-Sloane harmonious tree-labelling conjecture, Ringel's sequenceability conjecture, Snevily's subsquare conjecture, Tannenbaum's zero-sum conjecture, and many others. All of these problems have a similar flavour, yet until recently they have been approached in completely different ways, using algebraic tools ranging from the combinatorial Nullstellensatz to Fourier analysis. In this talk we discuss a unified approach to attack these problems, using tools from probabilistic combinatorics. In particular, we will see that a suitably randomised version of the Hall-Paige conjecture can be used as a black-box to settle many old problems in the area for sufficiently large groups. Joint work with Alexey Pokrosvkiy.
Title: Local-global principles over semi-global fields and applications to a generalized period-index problem
Seminar: Algebra
Speaker: Julia Hartmann of University of Pennsylvania
Contact: Parimala Raman, parimala.raman@emory.edu
Date: 2023-02-07 at 4:00PM
Venue: MSC W301
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Abstract:
It is a classical problem to relate the period and index of a Brauer class. Over semi-global fields, i.e., function fields over complete discretely valued fields, local-global principles have been a powerful tool in answering this question. In this talk, we consider an analog of the period-index problem for higher cohomology classes in place of Brauer classes. (Joint work with David Harbater and Daniel Krashen.)
Title: Minimal triangulations of manifolds
Job Talk: Combinatorics
Speaker: Sergey Avvakumov, Postdoctoral Fellow of University of Toronto
Contact: Liana Yepremyan, liana.yepremyan@emory.edu
Date: 2023-02-01 at 10:00AM
Venue: MSC W301
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Abstract:
Multiple results on face-vectors (numbers of faces of all dimension) of polytopes can be generalized to triangulated manifolds. They give good bounds on the number of facets. To the contrary, very little is known about the number of vertices in manifolds triangulations. I will describe how methods from combinatorics, topology, and metric geometry can tackle this problem yielding both new lower and upper bounds. Our go-to examples are going to be the n-dimensional real projective space and the n-dimensional torus.
Title: Convexity, color avoidance, and perfect hash codes
Seminar: Combinatorics
Speaker: Cosmin Pohoata of Institute of Advanced Study
Contact: Liana Yepremyan, liana.yepremyan@emory.edu
Date: 2023-01-31 at 4:00PM
Venue: MSC E408
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Abstract:
In this (informal) talk, I will discuss some favorite open problems which are related in some way or another with the Erd?s-Szekeres problem and the polynomial method.
Title: Plank Problems: Discrete Geometry and Convexity
Job Talk: Combinatorics
Speaker: Alexander Polyanskii, Senior Research Fellow of MIPT
Contact: Liana Yepremyan, liana.yepremyan@emory.edu
Date: 2023-01-27 at 10:00AM
Venue: https://emory.zoom.us/j/7744657281?pwd=TTFuLzYrVVUybkY4UlNmY0NINXNqdz09
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Abstract:
What is the smallest combined width of planks that cover a given convex region in the plane? What happens in higher dimensions? In the 50s, Thoger Bang answered this innocent question of Alfred Tarski and opened a box with many deceptively simple-looking problems. In my talk, I will overview progress in the area and its connection with other fields: theoretical computer science, number theory, and analysis. In particular, I will discuss a joint work with Zilin Jiang confirming Fejes Toth's long-standing zone conjecture and recent results with Alexey Glazyrin and Roman Karasev on a polynomial plank problem, a far-reaching generalization of Bang's theorem.