All Seminars
| 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. |
| Title: Randomness in Ramsey Theory and Coding Theory |
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| Job Talk: Combinatorics |
| Speaker: Xiaoyu He, NSF Postdoctoral Research Fellow of Princeton University |
| Contact: Liana Yepremyan, liana.yepremyan@emory.edu |
| Date: 2023-01-19 at 10:00AM |
| Venue: MSC W201 |
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| Abstract: Two of the most influential theorems in discrete mathematics state, respectively, that diagonal Ramsey numbers grow exponentially and that error-correcting codes for noisy channels exist up to the information limit. The former, proved by Erd?s in 1947 using random graphs, led to the development of the probabilistic method in combinatorics. The latter, proved by Shannon in 1948 using random codes, is one of the founding results of coding theory. Since then, the probabilistic method has been a cornerstone in the development of both Ramsey theory and coding theory. In this talk, we highlight a few important applications of the probabilistic method in these two parallel but interconnected worlds. We then present new results on Ramsey numbers of graphs and hypergraphs and codes correcting deletion errors, all based on probabilistic ideas. |
| Title: Measure growth in groups and the Kemperman inverse problem |
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| Job Talk: Math |
| Speaker: Yifan Jing of University of Oxford |
| Contact: Liana Yepremyan, liana.yepremyan@emory.edu |
| Date: 2023-01-17 at 10:00AM |
| Venue: MSC W201 |
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| Abstract: Two perennially studied questions in arithmetic combinatorics are: (i) Given two sets $A,B$ of a given size, how small can $AB$ be? (ii) What structure must $A$ and $B$ have when $AB$ is as small as possible, or nearly as small as possible? Theorems addressing (i) are called direct theorems, and those addressing (ii) are called inverse theorems. The direct theorem for locally compact groups was obtained by Kemperman (well-known special cases include Kneser's inequality and the Cauchy-Davenport inequality). The Kemperman inverse problem (proposed by Kemperman in 1964, also by Griesmer and Tao) corresponds to question (ii) when the ambient group is connected. In this talk, I will discuss the recent solution to this problem, highlighting the new-developed measure growth phenomenon: if $G$ is connected compact equipped with a normalized measure $\mu$, and $G$ is ``sufficiently non-abelian'', $A\subseteq G$ has a sufficiently small measure, then there is a constant gap between $\mu(AA)$ and $2\mu(A)$. We then discuss a few other applications of this phenomenon, including a Brunn-Minkowski inequality in non-abelian groups. This is based on joint work with Chieu-Minh Tran and Ruixiang Zhang. |