In 1900, the mathematician David Hilbert gave a lecture at the International Congress of Mathematicians in which he proposed 23 research problems for the next century (and beyond). This is probably the most famous list of research problems in mathematics. Some of Hilbert's problems have been solved and some are still open.

This first talk is an overview of some of these problems: their statement, some background, and current status. All the later math club talks this semester will be about one of the Hilbert problems.

Note: Free pizza and drinks!

In 1874, Georg Cantor discovered that infinite sets can come in different sizes: there are "just as many'' natural numbers as rational numbers but "more'' real numbers than natural numbers. Every infinite set of real numbers that Cantor knew was either the "same size'' as the natural numbers or the "same size'' as the real numbers. This led to the question of whether there could be an infinite set of real numbers that is "bigger'' than the natural numbers but "smaller'' than the real numbers. Cantor believed such sets did not exist, but he could not prove it. This became known as the Contintuum Hypothesis or Hilbert's First Problem.

The surprising solution to this problem (to the extent that there is a solution) came through a combination of work by Goedel in 1937 and Cohen in 1963. This talk will tell the story of this problem and the unexpected course it has taken since Cantor's initial explorations of sizes of infinity.


Note: Free pizza and drinks!

Polygons that can be cut up into congruent polygonal pieces are called equidecomposable. Two equidecomposable polygons have the same area, and it turns out the other direction is true too: two polygons with the same area are equidecomposable. This lets us describe what "equal area'' means for polygons without using calculus.

Hilbert asked if a similar result is true for polyhedra (the 3D-analogue of polygons, like a cube): is equal volume the same as being decomposable into congruent polyhedra? The answer is no, and this is due to Hilbert's PhD student Max Dehn.

In this talk we'll prove equal area = equidecomposable for polygons and see why the same approach doesn't work in 3 dimensions using a clever numerical invariant found by Dehn, now called the Dehn invariant.

Note: Free pizza and drinks!

A polynomial $$f(x,y)$$ in $$x$$ and $$y$$ is called symmetric if it is unchanged by swapping $$x$$ and $$y$$: $$f(y,x) = f(x,y)$$. Examples include the sum and product $$x+y$$ and $$xy$$, and
it turns out that every symmetric polynomial in $$x$$ and $$y$$ is a polynomial in $$x+y$$ and $$xy$$.
For example, $$x^4+y^4$$ is symmetric and here it is in terms of $$x+y$$ and $$xy$$:
\[
x^4 + y^4 = (x+y)^4 - 4(xy)(x+y)^2 + 2(xy)^2.
\]
We say the set of all symmetric polynomials in $$x$$ and $$y$$ is finitely generated, with
generators $$x+y$$ and $$xy$$.

Hilbert's 14th problem essentially asks whether something like this (finite generatedness) is true for the polynomials in $$n$$ variables (not just $$n = 2$$) that are unchanged by any set of invertible transformations (not just swapping $$x$$ and $$y$$).


The answer to this question is sometimes yes, but in general is no. We will describe some important cases where the answer is yes and some counterexamples for the general case.

Note: Free pizza and drinks!

A real number that is $$\geq 0$$ has to be a square of some real number, but a one-variable polynomial whose values are always $$\geq 0$$ need not be the square of a polynomial, e.g., $$x^2 - x + 1 \geq 0$$ for all $$x$$ but we can't write $$x^2 - x + 1 = f(x)^2$$ for a polynomial $$f(x)$$. However, $$x^2 - x + 1$$ is a sum of squares of polynomials:

\[
x^2 - x + 1 = (x-1/2)^2 + 3/4 = (x-1/2)^2 + (\sqrt{3}/2)^2.
\]

A sum of squares of polynomials has values that are always $$\geq 0$$. Conversely, if a polynomial has all of its values $$\geq 0$$, must it in fact be a sum of squares of polynomials? And what about polynomials in two or more variables? This question is the setting of Hilbert's 17th problem.


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A number that is the root of a nonconstant polynomial with integer coefficients is called algebraic (it is related to integers by algebraic operations) and numbers that are not algebraic are called transcendental (they "transcend" the tools of algebra). For example, $$\sqrt{2}$$ is algebraic since it is a root of $$x^2-2$$, while $$\pi$$ is transcendental. Showing $$\pi$$ is transcendental is very hard!

Hilbert's 7th problem asks about the transcendence of certain exponential expressions $$a^b$$, such as $$2^{\sqrt{2}}$$. We will explain the background to this problem, how the solution turned out, and sketch the argument that a particular number (not $$\pi$$) is transcendental.

Note: Free pizza and drinks!

Prime numbers are the building blocks of the integers. Much like molecules are built from atoms, every integer breaks down into a product of prime numbers and you cannot break down a prime number into a product of smaller integers. In this talk, we will discuss some ways to study the structure of prime numbers, with special attention to three prime number questions included by Hilbert in his 8th problem: the Riemann Hypothesis, Goldbach's Conjecture, and the Twin Prime Conjecture. We will include recent developments, such as the proof of bounded gaps between primes.


Note: Free pizza and drinks!

Pythagorean Triples are integer solutions to $$x^2+y^2=z^2$$, like 3, 4, and 5. The study of this equation goes back to ancient Greece. Number theorists are interested in similar quadratic equations, allowing other coefficients (like $$3x^2+2y^2=5z^2$$) or more variables (like $$x^2+y^2-z^2=5w^2$$). Before the 20th century, people like Lagrange, Legendre, Gauss, and Minkowski had developed a theory that predicted when such equations have nonzero rational solutions.

In his 11th Problem, Hilbert proposed totally resolving the theory of such quadratic equations that allow any number of variables and coefficients that are not just rational numbers but algebraic numbers. His problem was ultimately solved by Hasse using the idea of a "local-global principle.''

In this talk I'll give examples of variants of the Pythagorean Triples problem, illustrate how to approach it, and give a sense of what a "local-global principle'' is.

Note: Free pizza and drinks!

For some differential equations, the solutions may not be as "nice" as the equation itself: a differential equation having coefficients that are infinitely differentiable may have a solution that is differentiable only once or twice. Hilbert's 19th problem asks whether the solutions of certain partial differential equations are as "nice" as the coefficients of the equation. We will describe what this means in some examples and then discuss an important iterative procedure discovered independently by De Giorgi and Nash in their solution of Hilbert's 19th problem.

It will be assumed that the audience has seen partial derivatives, at least at the level of a multivariable calculus course.


Note: Free pizza and drinks!

In mathematics, a proof is a sequence of logical steps that ultimately start with axioms, which are the basic mathematical "facts'' considered elementary enough to require no further justification Hilbert’s second problem asks if certain commonly accepted axioms are consistent, meaning that they do not lead to contradictory results (like proving the Pythagorean theorem is both true and not true). This problem was an earnest attempt at making sure we can be confident in mathematical results.

There is good news and bad news. The good news is that since Hilbert posed his second problem almost 120 years ago, the axioms of arithmetic have not been found to be inconsistent. The bad news is that no proof of this consistency in the form that Hilbert envisioned is known and, in a certain
sense, no proof can ever exist. This follows from Goedel's 2nd incompleteness theorem, which extends beyond arithmetic to many axiomatic systems used throughout mathematics. What does it mean to prove that no proof of something can exist? For this, and pizza, come to my talk.



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A diophantine equation is a polynomial equation with integer coefficients. For example, $$x^2+3y^2=7$$ and $$x^2 - 3y^2 = 7$$ are diophantine equations and the first one has a solution in integers, namely $$(2,1)$$, while the second does not. In his 10th problem, Hilbert asked if there is a general algorithm that decides whether or not each diophantine equation has a solution in integers.

It turns out there is no such algorithm: this was shown in 1970 by Yuri Matiyasevich,
building on work by Martin Davis, Hilary Putnam, and Julia Robinson. However, one can still hope that there are algorithms for certain types of diophantine equations. In this talk we will discuss how to find integer solutions of certain diophantine equations, as well as other generalizations of Hilbert's 10th problem.


Note: Free pizza and drinks!

The math club will host a panel discussion on undergraduate mathematics research opportunities. This panelists will include both faculty and students. They will discuss what the research process is like and how to become involved in mathematics research as an undergraduate.