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Noise stability of ranked choice voting (Steven Heilman, USC)

Davidson Lecture Hall, CMC 340 E 9th St, Claremont, CA, United States

Given votes for candidates, what is the best way to determine the winner of the election, if some of the votes have been corrupted or miscounted?  As we saw in Florida in 2000, where a difference of 537 votes determined the president of the United States, the electoral college system does not seem to be […]

Discrete Calculus through generating functions (Wai Yan Pong, Cal State Dominguez Hills)

Davidson Lecture Hall, CMC 340 E 9th St, Claremont, CA, United States

Discrete Calculus studies discrete structures, such as sequences and graphs, using techniques similar to those used in Calculus for continuous functions. The basic idea of generating functions is to associate a function with a sequence so that the coefficients of the power series expansion of the function represent the terms of the sequence. They provide […]

Systems of homogeneous polynomials over finite fields with maximum number of common zeros (Sudhir Ghorpade, IIT Bombay)

Davidson Lecture Hall, CMC 340 E 9th St, Claremont, CA, United States

It is elementary and well known that a nonzero polynomial in one variable of degree d with coefficients in a field F has at most d zeros in F. It is meaningful to ask similar questions for systems of several polynomials in several variables of a fixed degree, provided the base field F is finite. […]

Bias in cubic Gauss sums: Patterson’s conjecture (Alex Dunn, CalTech)

Davidson Lecture Hall, CMC 340 E 9th St, Claremont, CA, United States

We prove, in this joint work with Maksym Radziwill, a 1978 conjecture of S. Patterson (conditional on the Generalized Riemann Hypothesis) concerning the bias of cubic Gauss sums. This explains a well-known numerical bias in the distribution of cubic Gauss sums first observed by Kummer in 1846. One important byproduct of our proof is that […]

Towers and elementary embeddings in total relatively hyperbolic groups (Christopher Perez, Loyola University New Orleans)

Davidson Lecture Hall, CMC 340 E 9th St, Claremont, CA, United States

In a remarkable series of papers Zlil Sela classified the first-order theories of free groups and torsion-free hyperbolic groups using geometric structures he called towers, and independently Olga Kharlampovich and Alexei Myasnikov did the same using equivalent structures they called regular NTQ groups. It was later proved by Chloé Perin that if H is an […]

Quantum money from Brandt operators (Shahed Sharif, CSU San Marcos)

Roberts North 102, CMC

Public key quantum money is a replacement for paper money which has cryptographic guarantees against counterfeiting. We propose a new idea for public key quantum money. In the abstract sense, our bills are encoded as a joint eigenstate of a fixed system of commuting unitary operators. We show that the proposal is secure against black […]

Numerical semigroups, minimal presentations, and posets (Chris O’Neill, SDSU)

Roberts North 102, CMC

A numerical semigroup is a subset S of the natural numbers that is closed under addition.  One of the primary attributes of interest in commutative algebra are the relations (or trades) between the generators of S; any particular choice of minimal trades is called a minimal presentation of S (this is equivalent to choosing a […]

Biquandle power brackets (Sam Nelson, CMC)

Roberts North 102, CMC

Biquandle brackets are skein invariants of biquandle-colored knots, with skein coefficients that are functions of the colors at a crossing. Biquandle power brackets take this idea a step further with state component values that also depend on biquandle colors. This is joint work with Neslihan Gügümcü (IYTE).

Chromatic numbers of abelian Cayley graphs (Michael Krebs, Cal State LA)

Roberts North 102, CMC

A classic problem in graph theory is to find the chromatic number of a given graph: that is, to find the smallest number of colors needed to assign every vertex a color such that whenever two vertices are adjacent, they receive different colors.  This problem has been studied for many families of graphs, including cube-like […]

Cellular resolutions of the diagonal and exceptional collections for toric D-M stacks (Reginald Anderson, CMC)

Roberts North 102, CMC

Beilinson gave a resolution of the diagonal for complex projective space, which gives a strong, full exceptional collection of line bundles as a generating set for the derived category of coherent sheaves. Bayer-Popescu-Sturmfels generalized Beilinson's resolution of the diagonal by giving a cellular resolution of the diagonal for a proper subclass of smooth toric varieties […]

Deep hole lattices and isogenies of elliptic curves (Lenny Fukshansky, CMC)

Roberts North 102, CMC

For a lattice L in the plane, we define the affiliated deep hole lattice H(L) to be spanned by a shortest vector of L and the furthest removed vector from the lattice contained in the triangle with sides corresponding to the shortest basis vectors. We study the geometric and arithmetic properties of deep hole lattices, […]

On the spectra of syntactic structures (Isabella Senturia, Yale University)

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We explore the application of spectral graph theory to the problem of characterizing linguistically-significant classes of tree structures. We focus on various classes of syntactically-defined tree graphs, and show that the spectral properties of different matrix representations of these classes of trees provide insight into the linguistic properties that characterize these classes. More generally, our […]