All Seminars
| Title: Topics in arithmetic statistics |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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. |
| Title: Continuous Combinatorics and Natural Quasirandomness |
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| Job Talk: Combinatorics |
| Speaker: Leonardo Nagami Coregliano, Postdoctoral Memb of Institute for Advanced Study |
| Contact: Liana Yepremyan, liana.yepremyan@emory.edu |
| Date: 2023-01-26 at 10:00AM |
| Venue: MSC W201 |
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| Abstract: The theory of graph quasirandomness studies graphs that "look like" samples of the Erd?s--Rényi random graph $G_{n,p}$. The upshot of the theory is that several ways of comparing a sequence with the random graph turn out to be equivalent. For example, two equivalent characterizations of quasirandom graph sequences is as those that are uniquely colorable or uniquely orderable, that is, all colorings (orderings, respectively) of the graphs "look approximately the same". Since then, generalizations of the theory of quasirandomness have been obtained in an ad hoc way for several different combinatorial objects, such as digraphs, tournaments, hypergraphs, permutations, etc. The theory of graph quasirandomness was one of the main motivations for the development of the theory of limits of graph sequences, graphons. Similarly to quasirandomness, generalizations of graphons were obtained in an ad hoc way for several combinatorial objects. However, differently from quasirandomness, for the theory of limits of combinatorial objects (continuous combinatorics), the theories of flag algebras and theons developed limits of arbitrary combinatorial objects in a uniform and general framework. In this talk, I will present the theory of natural quasirandomness, which provides a uniform and general treatment of quasirandomness in the same setting as continuous combinatorics. The talk will focus on the first main result of natural quasirandomness: the equivalence of unique colorability and unique orderability for arbitrary combinatorial objects. Although the theory heavily uses the language and techniques of continuous combinatorics from both flag algebras and theons, no familiarity with the topic is required as I will also briefly cover all definitions and theorems necessary. This talk is based on joint work with Alexander A. Razborov. |
| Title: The Problem of Polytope Reconstruction |
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| Job Talk: Combinatorics |
| Speaker: Hailun Zheng, Assistant Professor of University of Houston-Downtown |
| Contact: Liana Yepremyan, liana.yepremyan@emory.edu |
| Date: 2023-01-24 at 10:00AM |
| Venue: MSC W201 |
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| Abstract: What partial information about a convex d-polytope is enough to uniquely determine its combinatorial type? This problem, known as the problem of polytope reconstruction, has been extensively studied since the sixties. For instance, a famous result of Perles asserts that simplicial d-polytopes are determined by their \lfloor d/2 \floor-skeletons. In this talk, I will survey recent advances in this field, from mainly two perspectives. 1) realizability: can a certain simplicial complex be realized as the (\lfloor d/2 \rfloor-1)-skeleton of a simplicial d-polytope or a simplicial (d-1)-sphere? 2) sufficiency: can the i-skeleton (where i < \lfloor d/2 \rfloor), together with some additional information such as affine (i+1)-stresses, determine the combinatorial or even affine type of the polytope? This is joint work with Satoshi Murai and Isabella Novik. |