## September 2018

### Small representations of integers by integral quadratic form (Lenny Fukshansky, CMC)

Given an isotropic integral quadratic form which assumes a value t, we investigate the distribution of integer points at which this value is assumed. Building on the previous work about the distribution of small-height zeros of quadratic forms, we produce bounds on height of points outside of some algebraic sets in a quadratic space at which the form assumes the value t. Our bounds on height are explicit in terms of heights of the form, the space, the algebraic set…

Find out more »### Inversions for reduced words (Sami Assaf, USC)

The number of inversions of a permutation is an important statistic that arises in many contexts, including as the minimum number of simple transpositions needed to express the permutation and, equivalently, as the rank function for weak Bruhat order on the symmetric group. In this talk, I’ll describe an analogous statistic on the reduced expressions for a given permutation that turns the Coxeter graph into a ranked poset with unique maximal element. This statistic simplifies greatly when shifting our paradigm…

Find out more »### Quandle coloring quivers (Sam Nelson, CMC)

Given a finite quandle $X$, a set $S \subset \mathrm{Hom}(X,X)$ of quandle endomoprhisms, and an oriented knot or link $L$, we construct a quiver-valued invariant of oriented knots and links. This quiver categorifies the quandle counting invariant in the most literal sense and can be used to define many enhancements of the counting invariant. This is joint work with Harvey Mudd College student Karina Cho.

Find out more »## October 2018

### An Introduction to the Sato-Tate Conjecture (Edray Goins, Pomona College)

In 1846, Ernst Eduard Kummer conjectured a distribution of values of a cubic Gauss sum after computing a few values by hand. This was forgotten about for nearly 100 years until John von Neumann and Herman Goldstine attempted to verify the conjecture as a way to test the new ENIAC machine in 1953. They found evidence that the conjecture was false, but trusted Kummer more than they did their digital computer. The conjecture would hold until 1979, when Roger Heath-Brown…

Find out more »### State Polytopes of Combinatorial Neural Codes (Rob Davis, HMC)

Combinatorial neural codes are 0/1 vectors that are used to model the co-firing patterns of a set of place cells in the brain. One wide-open problem in this area is to determine when a given code can be algorithmically drawn in the plane as a Venn diagram-like figure. A sufficient condition to do so is for the code to have a property called k-inductively pierced. Gross, Obatake, and Youngs recently used toric algebra to show that a code on three…

Find out more »### The Bateman—Horn Conjecture, Part I: heuristic derivation (Stephan Garcia, Pomona)

The Bateman—Horn Conjecture is a far-reaching statement about the distribution of the prime numbers. It implies many known results, such as the Green—Tao theorem, and a variety of famous conjectures, such as the Twin Prime Conjecture. In this expository talk, we start from basic principles and provide a heuristic argument in favor of the conjecture. This talk should be accessible to undergraduates with a background in modular arithmetic.

Find out more »### Uniform asymptotic growth of symbolic powers (Robert Walker, University of Michigan)

Algebraic geometry (AG) is a major generalization of linear algebra which is fairly influential in mathematics. Since the 1980's with the development of computer algebra systems like Mathematica, AG has been leveraged in areas of STEM as diverse as statistics, robotic kinematics, computer science/geometric modeling, and mirror symmetry. Part one of my talk will be a brief introduction to AG, to two notions of taking powers of ideals (regular vs symbolic) in Noetherian commutative rings, and to the ideal containment problem…

Find out more »## November 2018

### Turning probability into polynomials (Mark Huber, CMC)

Moment generating functions (Laplace transforms) are a means for transforming probability problems into problems involving polynomials. Here I will concentrate on the binomial distribution, and use the mgf to link this distributions probabilities directly to the binomial theorem. The mgf is also a key ingredient in Chernoff bounds, which give upper bounds on the tail probabilities of binomial distributions (aka partial sums of the binomial theorem). By employing the method of smoothing and tilting, it is possible to attain bounds…

Find out more »### Cayley digraphs of matrix rings over finite fields (Yesim Demiroglu, HMC)

In this talk we use the unit-graphs and the special unit-digraphs on matrix rings to show that every n x n nonzero matrix over F_q can be written as a sum of two SL_n-matrices when n>1. We compute the eigenvalues of these graphs in terms of Kloosterman sums and study their spectral properties; and prove that if X is a subset of Mat_2 (F_q) with size |X| > (2 q^3 \sqrt{q})/(q - 1), then X contains at least two distinct…

Find out more »### Weil sums of binomials: properties and applications (Daniel Katz, CSUN)

We consider sums in which an additive character of a finite field F is applied to a binomial whose individual terms (monomials) become permutations of F when regarded as functions. These Weil sums characterize the nonlinearity of power permutations of interest in cryptography. They also tell us about the correlation of linear recursive sequences over finite fields that are used in digital communications and remote sensing. In these applications, one is interested in the spectrum of Weil sum values that are obtained as the coefficients in the…

Find out more »## December 2018

### Sperner’s lemma: generalizations and applications (Oleg Musin, UT Rio Grande Valley)

The classical Sperner - KKM (Knaster - Kuratowski - Mazurkiewicz) lemma has many applications in combinatorics, algorithms, game theory and mathematical economics. In this talk we consider generalizations of this lemma as well as Gale's colored KKM lemma and Shapley's KKMS theorem. It is shown that spaces and covers can be much more general and the boundary KKM rules can be substituted by more weaker boundary assumptions. These generalizations of Sperner's lemma rely on homotopy invariants of covers that in…

Find out more »### The Bateman—Horn conjecture II: applications (Stephan Garcia, Pomona)

We begin with a review of the Bateman—Horn conjecture, which sheds light on the intimate relationship between polynomials and prime numbers. In this expository talk, we survey a host of applications of the conjecture. For example, Landau’s conjecture, the twin prime conjecture, and the Green—Tao theorem are all consequences of the Bateman—Horn conjecture. Moreover, the conjecture also illuminates the mysterious patterns observed in the Ulam spiral.

Find out more »## January 2019

### Niebrzydowski tribrackets and algebras (Sam Nelson, CMC)

In this talk we will survey recent work on Niebzydowski Tribrackets and Niebrydowski Algebras, algebraic structures related to region colorings the planar complements of knots and trivalent spatial graphs.

Find out more »### Discrete compressed sensing: lattices and frames (Josiah Park, Georgia Tech)

Lattice valued vector systems have taken an important role in packing, coding, cryptography, and signal processing problems. In compressed sensing, improvements in sparse recovery methods can be reached with an additional assumption that the signal of interest is lattice valued, as demonstrated by A. Flinth and G. Kutyniok. Equiangular tight frames are particular systems of unit vectors with minimal coherence, a measure of how well distributed the vectors are, and have provable guarantees for recovery of sparse vectors in standard…

Find out more »## February 2019

### Lattices from group frames and vertex transitive graphs (Lenny Fukshansky, CMC)

Tight frames in Euclidean spaces are widely used convenient generalizations of orthonormal bases. A particularly nice class of such frames is generated as orbits under irreducible actions of finite groups of orthogonal matrices: these are called irreducible group frames. Integer spans of rational irreducible group frames form Euclidean lattices with some very nice geometric properties, called strongly eutactic lattices. We discuss this construction, focusing on an especially interesting infinite family in arbitrarily large dimensions, which comes from vertex transitive graphs.…

Find out more »### Subgraph statistics (Benny Sudakov, ETH Zurich)

Given integers $k,l$ and a graph $G$, how large can be the fraction of $k$-vertex subsets of $G$ which span exactly $l$ edges? The systematic study of this very natural question was recently initiated by Alon, Hefetz, Krivelevich and Tyomkyn who also proposed several interesting conjectures on this topic. In this talk we discuss a theorem which proves one of their conjectures and implies an asymptotic version of another. We also make some first steps towards analogous question for hypergraphs. Our proofs involve…

Find out more »### Knowledge, strategies, and know-how (Pavel Naumov, CMC)

An agent comes to a fork in a road. There is a sign that says that one of the two roads leads to prosperity and another to death. The agent must take the fork, but she does not know which road leads where. Does the agent have a strategy to get to prosperity? On one hand, since one of the roads leads to prosperity, such a strategy clearly exists. On the other, the agent does not know what the strategy…

Find out more »### When is the product of Siegel eigenforms an eigenform? (Jim Brown, Occidental College)

Modular forms are ubiquitous in modern number theory. For instance, showing that elliptic curves are secretly modular forms was the key to the proof of Fermat's Last Theorem. In addition to number theory, modular forms show up in diverse areas such as coding theory and particle physics. Roughly speaking, a modular form is a complex-valued function defined on the complex upper half-plane that satisfies a large number of symmetries. A modular form has two invariants: weight and level. If one…

Find out more »## March 2019

### Nonvanishing minors and uncertainty principles for Fourier analysis over finite fields (Daniel Katz, CSUN)

Chebotarev's theorem on roots of unity says that every minor of a discrete Fourier transform matrix of prime order is nonzero. We present a generalization of this result that includes analogues for discrete cosine and discrete sine transform matrices as special cases. This leads to a generalization of the Biro-Meshulam-Tao uncertainty principle to functions with symmetries that arise from certain group actions, with some of the simplest examples being even and odd functions. This new uncertainty principle gives a bound…

Find out more »### Indiana Pols Forced to Eat Humble Pi: The Curious History of an Irrational Number (Edray Goins, Pomona)

In 1897, Indiana physician Edwin J. Goodwin believed he had discovered a way to square the circle, and proposed a bill to Indiana Representative Taylor I. Record which would secure Indiana's the claim to fame for his discovery. About the time the debate about the bill concluded, Purdue University professor Clarence A. Waldo serendipitously came across the claimed discovery, and pointed out its mathematical impossibility to the lawmakers. It had only be shown just 15 years before, by the German…

Find out more »### Refinements of metrics (Wai Yan Pong, CSUDH)

I will talk about a few graph-theoretic metrics then introduce the concept of refinements on a class of functions that include all metrics. As a case study, we will construct various refinements on the shortest-path distance. Consequently, we obtain a few "better" versions of the Erdos number. In the course of our investigation, we realized various construction of metrics can be unified under a rather natural concept that we called monotonic monoid norm. This is a joint work with Kayla Lock and Alex…

Find out more »## April 2019

### Fibonacci and Lucas analogues of binomial coefficients and what they count (Curtis Bennett, CSULB)

A Fibonomial is what is obtained when you replace each term of the binomial coefficients $ {n \choose k}$ by the corresponding Fibonacci number. For example, the Fibonomial $${ 6\brace 3 } = \frac{F_6 \cdot F_5 \cdot \dots \cdot F_1}{(F_3\cdot F_2 \cdot F_1)(F_3\cdot F_2 \cdot F_1)} = \frac{8\cdot5\cdot3\cdot2\cdot1\cdot1}{(2\cdot1\cdot1)(2\cdot1\cdot1)} = 60$$ since the first six Fibonacci numbers are 1, 1, 2, 2, 5, and 8. Curiously the Fibonomials are always integers, raising the combinatorial question: what do they count? In this…

Find out more »### Matrix multiplication: the hunt for $\omega$ (Mark Huber, CMC)

For centuries finding the determinant of a matrix was considered to be something that took $\Theta(n^3)$ steps. Only in 1969 did Strassen discover that there was a faster method. In this talk I'll discuss his finding, how the Master Theorem for divide-and-conquer plays into it, and how it was shown that finding determinants, inverting matrices, and Gaussian elimination are the same time complexity as to matrix multiplication.

Find out more »### Chow rings of heavy/light Hassett spaces via tropical geometry (Dagan Karp, HMC)

In this talk, I will try to give a fun introduction to tropical geometry and Hassett spaces, and show how tropical geometry can be used to compute the Chow rings of Hassett spaces combinatorially. This is joint work with Siddarth Kannan and Shiyue Li.

Find out more »### Theory of vertex Ho-Lee-Schur graphs (Sin-Min Lee, SJSU)

A triple of natural numbers (a,b,c) is an S-set if a+b=c. I. Schur used the S-sets to show that for n >3, there exists s(n) such that for prime p > s(n), x^p + y^p = z^p (mod p) has a nontrivial solution. A (p,q)-graph G is said to be vertex Ho-Lee-Schur graph if there exists a bijection f: V(G) --> {1,2,…,p} such that for each C3 subgraph of G with vertices {x,y,z} the triple (f(x),f(y),f(z)) is an S-set. The VHLS deficiency of…

Find out more »### What Did Ada Do? Digging into the Mathematical Work of Ada Lovelace (Gizem Karaali, Pomona)

Augusta Ada Byron King Lovelace (1815-1852) is today celebrated as the first computer programmer in history. This might be confusing to some because in 1852 there were no machines that looked like what we call computers today. In this talk I attempt to explain what Ada really did, and delineate the mathematics involved. Bernoulli numbers will definitely come into play, but there may also be other fun distractions along the way, possibly including some juicy gossip about Ada’s life.

Find out more »## May 2019

### Notions of stability in algebraic geometry (Jason Lo, CSUN)

One of the main drivers of current research in geometry is the classification of Calabi-Yau threefolds. Towards this effort, a particular approach in algebraic geometry is via the study of stability conditions. In this talk, I will explain what constitutes a notion of stability in algebraic geometry, and what the challenges are in studying them.

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