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Abstract: We consider a multivariate polynomial Q_n defined over the unitary cube, endowed with a symmetrical behaviour, and we look for its maximum. This is required in order to obtain an estimate for the discriminant of a totally real number field in terms of the regulator and the degree. Instead of studying local analytical properties, we obtain the result by means of a global approach, which considers the sign of the variables and transforms the problem into the research of signs patterns for an array: in particular, there is a procedure which transforms any considered array into a specific configuration which provides the desired maximum value. This confirms a conjecture by Pohst on the value of the maximum, which was precisely proved just up to n=11.
Link to paper: https://arxiv.org/pdf/2101.06163.pdf
Subject codes: 11R80, 11Y40
Speaker's Contact Information:
Email: francesco.battistoni@univ-fcomte.fr
Website: https://sites.google.com/view/fbattistoni
Abstract: Using Kuperberg's B2/C2 webs, and following Elias and Libedinsky, we describe a "light leaves" algorithm to construct a basis of morphisms between arbitrary tensor products of fundamental representations for the Lie algebra of type C2 (and the associated quantum group). Our argument has very little dependence on the base field. As a result, we prove that when quantum two is invertible, the Karoubi envelope of the C2 web category is equivalent to the category of tilting modules for the divided powers quantum group.
Link to paper: https://arxiv.org/abs/2009.13786
Speaker's Contact Information:
Email: ebodish@uoregon.edu
Website: https://sites.google.com/view/elijahbodish/home
Abstract: Additive combinatorics studies the presence of patterns such as arithmetic progressions in subsets of groups and rings. These structures can be examined using ideas from combinatorics, number theory, analysis and dynamics. In this talk, we estimate the number of certain polynomial configurations in subsets of finite fields. An example of configurations that we look at is x, x+y^2, x+2y^2, a three-term arithmetic progression with square differences. As a corollary, we derive upper bounds for the size of the subsets of finite finites lacking these configurations. This is connected to the polynomial extension of Szemerédi theorem proved by Bergelson and Leibman.
Link to paper: https://arxiv.org/pdf/1907.08446.pdf, https://arxiv.org/pdf/2001.05220.pdf
Speaker's Contact Information:
Email: boryskuca@gmail.com
Website: https://sites.google.com/view/boryskuca
Abstract: Schubert calculus connects problems in algebraic geometry to combinatorics, classically resolving the question of counting points in the intersection of certain subvarieties of the Grassmannian with Young tableaux. Subsequent research has been dedicated to carrying out a similar program in more intricate settings. A recent breakthrough in the Schubert calculus program concerning the homology of the affine Grassmannian and quantum cohomology of flags was made by identifying k-Schur functions with a new class of symmetric functions called Catalan functions. In this talk, we will discuss a K-theoretic refinement of this theory and how it sheds light on K-k-Schur functions, the Schubert representatives for the K homology of the affine Grassmannian.
Subject codes: 05E05, 14N15
Link to paper: https://arxiv.org/abs/2010.01759
Speaker's Contact Information:
Email: ghs9ae@virginia.edu
Website: https://ghseeli.github.io/
Abstract: Let X be an elliptic curve over C with one point removed, and consider the unordered configuration spaces Conf^n(X)={(x_1,...,x_n): x_i\neq x_j for i\neq j} / S_n. We present a rational function in two variables from whose coefficients we can read off the i-th Betti numbers of Conf^n(X) for all i and n. The key of the proof is a property called "purity", which was known to Kim for (ordered or unordered) configuration spaces of the complex plane with r \geq 0 points removed. We show that the unordered configuration spaces of X also have purity (but with different weights). This is a joint work with G. Cheong.
Link to paper: https://arxiv.org/abs/2009.07976
Speaker's Contact Information:
Email: huangyf@umich.edu
Abstract: I will describe some of the results of a recent paper joint with Kathrin Bringmann and Ben Kane. First introducing the notion of t-core partitions, the motivation and some history of their study. I'll then give a brief description of our new results that relate certain 7-cores with Hurwitz class numbers and a genus of quadratic forms in the class group.
Link to paper: https://arxiv.org/abs/2005.07020
Speaker's Contact Information:
Email: jmales@math.uni-koeln.de
Website: http://www.mi.uni-koeln.de/~jmales/
Abstract: The snake graph construction from Musiker-Schiffler-Williams provides an expansion formula for cluster variables in cluster algebras from surfaces. We generalize this construction to generalized cluster algebras from orbifolds. This is joint work with Elizabeth Kelley.
Link to Paper: https://arxiv.org/abs/2003.13872
Speaker's Contact Information:
Email: banai003@umn.edu
Abstract: The Mehta integral is the canonical partition function for 1-dimensional log-Coulomb gas in a harmonic potential well. Mehta and Dyson showed that it also determines the joint probability densities for the eigenvalues of Gaussian random matrix ensembles, and Bombieri later found its explicit form. We introduce the p-adic analogue of the Mehta integral as the canonical partition function for a p-adic log-Coulomb gas, discuss its underlying combinatorial structure, and find its explicit formula and domain.
Link to paper: https://arxiv.org/abs/2001.03892
Speaker's Contact Information:
Email: jwebster@uoregon.edu
Website: https://pages.uoregon.edu/jwebster/
Abstract: A theorem of Brauer shows that there exists a positive integer B(r, k) such that any colouring of the set {1, 2, ..., B(r, k)} with r colours produces a monochromatic k-term arithmetic progression which receives the same colour as its common difference. We obtain a quantitative version of this result. In particular, we show that the quantity B(r, k) can be taken to be double exponential in the number of colours r, and quintuple exponential in the length k. This talk is based on joint work with Sean Prendiville.
Link to paper: https://doi.org/10.1112/blms.12327
Subject Codes: 11B30, 05D10
Speaker's Contact Information:
Abstract: The Sandpile Matrix-Tree Theorem shows that the size of the sandpile group of a graph is equal to the number of spanning trees. This result generalizes to higher dimensions where the size of the sandpile group of a large class of representable matroids is a weighted sum of their bases depending on multiplicity. We give a family of maps from the sandpile group to the bases of these matroids such that the size of each preimage is a very particular size. This generalizes a bijection given by Backman, Baker, and Yuen and extends work by Duval, Klivans, and Martin.
Link to paper: https://arxiv.org/abs/2007.09501
Speaker's Contact Information:
Email: amcd@math.brown.edu
Website: https://www.math.brown.edu/~amcd/
Abstract: In 1995 Kim famously proved the Ramsey bound R(3,t) \ge c t^2/\log t by constructing an n-vertex graph that is triangle-free and has independence number at most C \sqrt{n \log n}. We extend this celebrated result, which is best possible up to the value of the constants, by approximately decomposing the complete graph K_n into a packing of such nearly optimal Ramsey R(3,t) graphs.
More precisely, for any \epsilon>0 we find an edge-disjoint collection (G_i)_i of n-vertex graphs G_i \subseteq K_n such that (a) each G_i is triangle-free and has independence number at most C_\epsilon \sqrt{n \log n}, and (b) the union of all the G_i contains at least (1-\epsilon)\binom{n}{2} edges. Our algorithmic proof proceeds by sequentially choosing the graphs G_i via a semi-random (i.e., Rodl nibble type) variation of the triangle-free process.
As an application, we prove a conjecture in Ramsey theory by Fox, Grinshpun, Liebenau, Person, and Szabo (concerning a Ramsey-type parameter introduced by Burr, Erdos, Lovasz in 1976). Namely, denoting by s_r(H) the smallest minimum degree of r-Ramsey minimal graphs for H, we close the existing logarithmic gap for H=K_3 and establish that s_r(K_3) = \Theta(r^2 \log r).
Subject codes: 05C55, 05C80, 05D10, 60C05
Link to paper: https://link.springer.com/article/10.1007/s00493-019-3921-7
Speaker's Contact Information:
Email: he.guo@gatech.edu