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2019-12-02We will explore the foundations and motivations of Justification Logic and its relationship to Provability Logic, noting in particular its relationship to Godels original conception for a logic of proofs. Time permitting, we will look into some original results in Topological Semantics for LP(CS) and the relationship between modality, topology, and intuitionism.
2019-11-25Suppose we are given a quadratic polynomial Q and some initial data, and we wanted to study the recurrence relation defined by Q. Unfortunately the quadratic-ness gives two possible next terms, so partial solutions to this recurrence should be indexed by a 2-branching tree; full solutions to the recurrence should be indexed by a choice of branch. All of these branches are made equally, because they are governed by the same initial data and the same Q. It turns out that sometimes, some branches are more equal than others. I hope to explain to you why this is the case. If time permits, I hope to outline some joint work with Sergey Fomin.
2019-11-18How hard could it possibly be to solve 4x^3-g_2x-g_3=0 for x a complex number.
2019-11-11In 1980, Rabin modified Millers primality test to obtain a polynomial-time probabilistic primality test. This was one of the first discoveries of a non-deterministic algorithm that is asymptotically faster than its best known deterministic analog. We will discuss several modern algorithms and data structures that employ nondeterminism for speed, and explore how randomness helps them run faster. Additionally, it is currently an open problem whether Turing machines with access to randomness can run strictly faster than those without; if time permits, we will go into a brief discussion of this.
Cornell Math Department
2019-10-28Discover the many opportunities in math, including spring class offerings, study abroad, the major, and the minor. Meet fellow math sutents and faculty. Snacks will be served. Drop in any time before 5:00pm.
2019-10-21Bayesian optimization is a surrogate optimization method for expensive, black-box functions. It uses Gaussian process regression to quantify the uncertainty in the surrogate and continuously updates the GP with each new function evaluation. Key components are the kernel function -- which measures similarity between data points -- and the acquisition function -- which strikes a balance between exploration and exploitation. In this talk, we take a look at the underlying machinery and give live demos using MATLAB.
2019-10-07When L/K is a finite Galois extension of fields, we have a beautiful inverse correspondence between the intermediate fields of L/K with the subgroups of the automorphism group Gal(L/K). We can attempt to extend this notion to infinite algebraic field extensions, but the previous correspondence fails to hold. We will see an example of why not; nevertheless, there fortunately is still a correspondence. In this talk, we will quickly revisit the finite theory and discuss the correct way to think about the infinite Galois correspondence by the way of profinite groups. It is useful to know elementary topology and basic definitions from field theory. Given enough time, details may be provided, but this talk is more to provide enough language to show the results.
2019-09-30Broadly, analytic number theory is the study of the analytic aspects of the objects of number theory. Counting solutions of polynomial equations, estimating the number of prime divisors of an integer $\sim N$, and bounding the first prime in an arithmetic progression are all examples of analytic number theory. This talk will outline some of the main techniques and problems in analytic number theory, with a focus on exposition and ideas instead of proofs.
2019-09-23In this talk we will talk about Ordinary Differential Equations. By using the Fundamental Theorem of Calculus as the main uniqueness theorem, we will re-derive properties of trig and exponential functions including their derivatives, addition formulas, power series expansions, and Eulers identity. We will then think more about uniqueness and existence. Consider the equations dx/dt=x^2 and dx/dt=sqrt(abs(x)). The first has solutions that go to infinity in finite time and the second has multiple solutions for a given initial condition. We will think about these, along with 2d linear systems. Putting this all together, we will think about resonance, and explain why forced oscillators get so crazy.
2019-09-11Boolean circuits are a restricted model of computation where one can find many interesting lower bound results. One such result is that, for n large enough, the parity of n bits cannot be computed by constant-depth, polynomial-size circuits with unbounded fan-in. Ill discuss the Razborov-Smolensky proof tof this lower bound using the method of approximation after reviewing some requisite circuit complexity. Please note the unusual schedule - this talk with be on a Wednesday.
Cornell Math Club
2019-08-29Welcome back to school! Come to Mallot 532 this Thursday for board games, pizza, and people.
2019-05-06Given a commutative ring R, we can define the unitary Cayley graph of R as the graph with vertices labeled by the elements of R, with x adjacent to y if and ony if x-y is a unit in R. These graphs are full of symmetry and structure-which often makes computing graph parameters for them quite nice. We will talk about why these graphs are important, investigate some connections to number theory, and then play around with various graph parameters. This talk should require no background to understand and will end with some fun problems to try.
2019-04-29In 2001, we had a breakthrough: Shors algorithm was used to factor 15 in polynomial time. We will begin with a brief rundown of the mathematics involvded in quantum computation, such as qubits and entanglement, and see why linear algebra lets us have our cake and eat it too. The talk will focus on the construction of several quantum logic gates, such as the Toffoli and Hadamard gates, which we will leverage to prove that quantum circuits are actually capable of faster computation than classical ones. If time permits, we willhopefully get to an overview of Shors algorithm.
2019-04-22A falling cat tends to land on its feet, if given enough time, even for non-upright, non-rotating initial conditions. This poses an apparent paradox. The cat has access to no external torques, and therefore angular momentum is conserved and zero during the cats fall. How can the cat flip itself over?
2019-04-17The surface of the earth is curved, a fact one can notice intrinsically by taking a friend to the equator and both walking straight north: initially you two begin your journey walking parallel to one another, but eventually collide at the north pole. Mathematically speaking, the positive curvature of the earth causes the straight line (geodesic) paths you two are walking on to converge. While harder to visualize, curvature is an important property of 3-dimensional spaces as well, and gaining intuition about curved 3-dimensional worlds leads to some fun mathematical thought experiments!
Math Majors Committee
2019-04-15Please come and tell me what we are doing right, what we are doing wrong, and what else we might be doing. Oh and there will be FREE PIZZA!
Cornell Math Department
2019-04-10Discover the many opportunities in math, including fall class offerings, summer programs, the major, and the minor. Meet fellow math students and faculty. Snacks will be served. The reception will start at 4, and one can drop by any time before 5:00 p.m.
2019-04-08We will begin with a cursory overview of modal logic, the meaning of soundness and completeness, and the traditional axiomatic extensions. The provability logic will be motivated by Lobs theorem, and we will explore the axiomatic extension known as GL, and discover the extent to which CL is able to encapsulate provability statements in Peano arithmetic, such that the Second Incompleteness Theorem. Time permitting, we will get into the details of Solovay’s proof of the completeness of GL with respect to Peano arithmetic. The presentation will be heavily motivated by The Unprovability of Consistency by George Boolos. Please note the unusual location.
2019-03-25A fundamental idea in number theory is to, given some equation, reduce it modulo some number. This is a fertile source of information: this is how one tells that, for example every prime greater than 2 that is the sum of two squares must be 1 mod 4. The subject of this talk will be the p-adic integers, which give us a way to leverage all of this information about an equation modulo some numbers. I will go over a few definitions, some basic properties and hopefully the above picture, which depicts the metric on the 3-adic integers.
2019-03-18The theory of optimal control provides methods to solve dynamical optimization problems in discrete and continuous time. Somewhat surprisingly, there are similarities between continuous-time control problems and static problems that arise in the study of optimal nonlinear income taxation. I will provide an introduction to continuous-time control problems and the classic solution method, Pontryagin’s Maximum Principle, and demonstrate an application of this method to optimal taxation.
Cornell Math Club
2019-03-14The math club is hosting a weekly meeting. This week we will be e having a pi day themed puzzle session and as always it will have pizza.
Cornell Math Department
2019-03-14On Pi day the math department will be hosting a pi eating contest at 1:59, to see who can eat the most pie in 3:14. For other festivities and so on, come to mallot 532.
2019-03-11Julia sets are one of the most famous and beautiful examples of fractals, arising from a simple iterative rule applied to the complex numbers. I will discuss basic properties of Julia sets from the dynamical systems perspective. To explore some topological properties of Julia sets and the connection with the famous Mandelbrot set, I will prove the ~fundamental theorem of the Mandelbrot set~. Time permitting, I will explain why the Mandelbrot set is connected and discuss a very recent and amusing result on Julia sets.
Cornell Math Club
2019-03-07We will be hosting our weekly meetings, this week we will have lolots of board games. As usual catering will be provided by Papa Johns.
2019-03-04Calculus of Variations derives conditions for finding maxima and minima of functionals (functions of functions) by slightly varying a function (here it is the input) to see how the functional value reacts. We will derive the Euler-Lagrange equation and see it in practice, finding the curve of a soup bubble over two rings and the brachistochrone. If time allows, we will look at generalizations of the Euler-Lagrange equation in higher dimensions.
Cornell Math Club
2019-02-28Puzzle session in 532 with pizza at 6:00?
2019-02-18We will explore the reflectional and rotational transformations of the Riemann Sphere and derive the functions in the complex plane which correspond to them.
2019-02-11Combinatorics concerns itself with counting. To count more complicated things, one might try to reduce the problem to counting a simpler thing by finding a bijection between the complicated thing and the simple thing. I want to convince you that bijections are fun, and that combinatorics is like a playground. I will talk about Catalan numbers and tell you about some go the things they count; if time allows, I want to introduce root polytopes, their triangulations, and some relations to pipe dream complexes.
Undergraduate Math Club
2019-02-07This thursday the math club will be hosting a game session in Mallot 532 from 6-7pm. There will be pizza!
Robert S. Strichartz and Andy Borum
2019-02-04Wondering what to do this summer? Interested in math? Want funding to support yourself? We will talk about the Cornell summer program for undergraduate research (SPUR) and other REU programs throughout the country. Afterwards there will be an opportunity to chat with students who attended such programs in the past, both at Cornell and elsewhere.
2018-12-03A formal proof is a proof in which every logical inference has been checked all the way back to the fundamental axioms of mathematics. Although there is a long way to go for formalization to be practical for working mathematicians, existing theories and technologies of formal verification are already capable of formalizing a large body of modern mathematics. In this talk, I will introduce the building blocks of formal verification, including logical foundations, proof assistants, and expressing mathematics formally in such systems. In particular, I will illustrate the relevant concepts by giving a brief overview of type theories, the Coq proof assistant, and the Flyspeck project that formally verified the 400 years old Keplers Conjecture on sphere packing. Furthermore, I will discuss common challenges in formalization endeavors.
Professor Bob Connelly
2018-11-26Put a bunch of circular disks in a container and squeeze the container until they jam. What does the packing look like? What can you say about the density of the packing? When the disks are the same size and the container is a flat torus, the answer is known. If the Radii in ratio 1:2:3, density = 7π/24=0.92015.. sizes are random as with granular materials, for existence, there will be a minimum number of contacts. If the graph of contacts is a triangulation, often the density of the packing is quite large. Evan Solomonides and Maria Yampolskaya will demonstrate a simulation of packings as the container contracts until they jam.
2018-11-19Using the tools of smooth manifold theory, we propose a generalized framework for olfactory reception, learning, and processing. Inspection of the tangent bundle to a manifold yields vector fields which allow for quantification of changes. We utilize group actions to discover fibre bundles over the manifold and discover various properties related to learning. Under this paradigm, we develop a method for categorization as well as analytical tools to model changes in the category. We end with a quick discussion of searching for data on the manifold in a way that beats nearest neighbour.
2018-11-12The Weil Conjectures are four statements about an analogue of the Zeta function over finite fields. I’m going to talk about the roots of polynomial equations in a mod p setting, define the zeta function for a projective variety, and with the new vocabulary Ill state the Weil Conjectures at the end. Familiarity of finite fields and projective geometry is useful but not required.
Cornell Math Club
2019-02-2-28Puzzle session in 532 with pizza at 6:00!