Tom Peters and Evelyn Nitch-Griffin will speak.

A curve over a field of characteristic $$p$$ is called ordinary if the $$p$$ torsion of its Jacobian is the largest possible. But what is the probability that a curve is ordinary? I will talk about various perspectives on this question, including some heuristics about how random $$p$$-divisible groups behave. I will also talk about cases for which the answer is known, namely Artin-Schreier and superelliptic curves.

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.


Note: Free pizza and drinks!

Nonlinear patterns and waves appear in many different systems and on vastly different scales in both time and space: despite this variability, their dynamic behavior is similar across these systems. Mathematical techniques can help identify the origins and common properties of patterns and waves across different applications. Despite many advances, understanding patterns and waves still poses significant mathematical challenges. I will show how a combination of geometric dynamical-systems ideas combined with PDE approaches can shed light on the existence, stability, and dynamical properties of nonlinear waves and will also discuss applications and open problems.

t-SNE (t-distributed stochastic embedding) is a widely used algorithm for embedding high dimensional data into two or three dimensional space. I will outline how this algorithm works, and in the process discuss some general aspects of the problem of visualizing and interpreting high dimensional data.

Abstract: I will introduce a class of spaces that can be seen as an infinite-dimensional analogue of Carnot groups and, on the other hand, as a noncommutative generalization of Banach spaces. We will see examples of these spaces as well as a geometric characterization. I will also show that any Lipschitz map with infinite-dimensional Carnot group domain has a point of Gâteaux differentiability (joint work with Enrico Le Donne and Sean Li) and that these spaces naturally appear as direct limits of classical Carnot groups (in preparation, with Enrico Pasqualetto).

Join us for a talk by Ellen Lehet on category theory.

Ellen Lehet (University of Notre Dame)

"The Explanatory Value of Category Theory"

Abstract

Category theory has proven to be applicable across all of mathematics. In some sense this is not surprising because category theory was created for the purpose of application (specifically, application to algebraic topology). But I will argue that the significance of category theory extends past its applicability — in particular, there is a significant explanatory benefit. The question of what constitutes a mathematical explanation is of perennial interest to philosophers. Reflection on category theory’s unique role in mathematics unearths some features of mathematical explanation that are not often made explicit and that philosophers have tended not to notice.

There are many ways that category theoretic methods provide explanations. For instance, important results in different areas of mathematics are unified by the fact that they are all corollaries of the same category theoretic theorem, such as the theorem that right adjoints preserve limits. Or consider the ways of defining structures in category theory with universal properties — the whole perspective sheds light on how constructions from different domains are related to one another. The categorical product for instance, unites many seemingly unrelated mathematical constructions such as the Cartesian product, union, and conjunction. Such examples introduce both generalization and unification within mathematics. Moreover, this unification allows for meaningful and surprising mathematical analogies to arise. These gen- eralizations and analogies are explanatory and result from the structural features of category theory.

In order to highlight the explanatory value of category theory, I will first provide a characterization of the structure unique to category theory. It is this structure that makes category theory apt for producing explanations. With a clear picture of category theoretic structure, I will present a few examples that illustrate how category theory proves to be explanatory — in particular, how the structural features of category theory are explanatory.

All welcome!

https://logic.uconn.edu/calendar/

seminar

The axiom of choice is widely accepted in mathematics, but in analysis, it lets you construct a number of "monsters": non-measurable sets, discontinuous linear functionals, paradoxical hat guessing strategies, and so on. In this talk, we'll look at some alternative axioms that result in a "dream universe" where these monsters don't exist, though perhaps a few other things are a little different. Along the way, we'll see some interesting tidbits from the foundations of mathematics, and start to get a sense for "how much choice" is used in various parts of analysis.

In this talk, I give a brief introduction to Bitcoin and Bitcoin inverse futures. Next, I study the risk factors (mean and variance of returns) and optimal hedging of Bitcoin inverse futures.

The goal of this talk is to give a characterization of hyperbolic space using a geometric invariant that is defined for a class of manifolds asymptotic to hyperbolic space. This result is inspired by a characterization of Euclidean space called the rigidity of the Riemannian positive mass theorem in the context of general relativity. We discuss the history and context of our result, as well as some of the interesting elements of its proof

Topics covered will be:
General requirements of 3D printing,
Meeting general requirements of 3D printing with Mathematica,
Exporting STL files from Mathematica,
Basic use of TinkerCad,
Importing STL files into TinkerCad,
Exporting STL files from TinkerCad.

What will not be covered:
Sending an STL file to the 3D printer.
(This will be an additional workshop and requires additional online training for information about hazardous materials.
)

Arrangements can be made to print interesting 3D models.

Continuing series of presentations on clustering by participants.

A fundamental problem in arithmetic geometry is to determine which abelian varieties arise as Jacobians of (smooth) curves. In positive characteristic p, we study this problem from the moduli perspective by asking which Newton strata intersect the Torelli locus in the moduli of abelian varieties. In this talk, I will introduce a general picture where we try to answer this question by replacing $$A_g$$ with a Shimura variety of PEL-type, and $$M_g$$ with a Hurwitz space of cyclic covers of $$P^1$$. Using an inductive method, when $$p = 2 \pmod 3$$, for all $$g$$, we prove the existence of a smooth curve of genus $$g$$ whose Newton polygon has about $$2g/3$$ slopes of $$1/2$$. This work is joint with Mantovan, Pries and Tang.​​

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!

Riemannian geometry deals with generalizing the notion of Euclidean space into spaces with curvature. This requires many new ideas to make sense of the natural notions of length, vector fields, derivatives, and many others. We will give an overview of these concepts, culminating with the notion of affine connections and parallel transport.

Abstract: Liouville's theorem, in its strong form, states that harmonic functions of polynomial growth are polynomials. I will describe a new proof of this well-known theorem on Abelian and nilpotent groups. The method arises from the study of convexity properties of harmonic functions on discrete spaces. Joint work with Dan Mangoubi.

Join us for a talk in the Logic Colloquium!


Ethan Brauer (OSU):

A Modal Theory of Free Choice Sequences

Free choice sequences (also called 'infinitely proceeding sequences') are a concept from intuitionistic mathematics that are central to the development of the intuitionistic theory of the continuum. Free choice sequences have, however, been largely rejected or ignored both by classical and other constructive mathematicians. In this talk I argue that the objections to free choice sequences can be overcome by grounding the concept in our temporal intuition and formalizing the theory in modal logic. I will present a theory of free choice sequences as a modal extension of classical second-order arithmetic. The resulting theory is able to prove modal versions of the intuitionist's axioms for so-called lawless sequences, suffices for the development of a theory of real number generators, and captures many results distinctive of intuitionistic analysis including the non-existence of functions on real numbers with definable discontinuities.

All welcome.

https://logic.uconn.edu/

seminar

The theory of rough path can be briefly described as a non-linear extension of the classical theory of controlled differential equations which is robust enough to allow a deterministic treatment of stochastic differential equations, or even differential equations driven by much rougher signals than semi-martingales. In this talk, I will present the most fundamental ideas of the theory and discuss some examples.

For a coherent risk measure $$\rho: L^\infty \to \mathbb{R}$$, Delbaen (2002) proved that $$\rho$$ can be represented as the worst expectation over a class of probability measures whenever it has the Fatou property. Later, it has been asked whether Delbaen's representation theorem holds on more general model spaces containing unbounded positions. In this talk, we will present a comprehensive investigation on this problem for risk measures on Orlicz spaces. We firest characterize the Orlicz spaces over which the representation holds. We also show that the representation holds on general Orlicz spaces if the risk measures possess additional properties,
e.g., law-invariance or surplus-invariance.

The talk is based on several joint papers with C. Munari, D. Leung,
and F. Xanthos.

TThe asymptotic expansion of the complete K-E metric on the punctured sphere is the dimension one case of the asymptotic expansion on a quasi-projective manifold M\D, which was proposed by S. T. Yau. Several people have worked on this problem by using techniques from partial differential equations. In this talk, I will use analytic properties of the covering map, the Schwarzian derivative and the modular form to derive a precise asymptotic expansion on the punctured sphere. More precisely, the coefficients in the expansion will be uniquely determined up to two parameters. In a recent work joint with G. Cho, the Kobayashi-Royden metric can be given as a consequence.

Iddo Ben-Ari and Jeremy Teitelbaum
Mixture Models and Topological Data Analysis

TBA

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!

Partial differential equations posed on surfaces arise in mathematical models for many natural phenomena: diffusion along grain boundaries, lipid interactions in biomembranes, pattern formation, and transport of surfactants on fluidic interfaces to mention a few. Numerical methods for solving PDEs posed on manifolds recently received considerable attention. In this talk, we discuss finite element methods to solve PDES on both stationary surfaces and surface with prescribed evolution. The focus of the talk is on geometrically unfitted methods, i.e. methods that avoid parametrization and triangulation of surfaces in a common sense. We explain how unfitted discretizations facilitate the development of a fully Eulerian numerical framework and enable easy handling of time-dependent surfaces including the case of topological transitions. We consider two methods falling in this category: a method based on PDE extensions off the surface and its `dual’ that uses the restrictions of outer finite element spaces to solve PDE on the surface. The application of the latter technique known as Trace FEM or Cut FEM is further demonstrated for a sequence of problems of increasing complexity, ranging from the Laplace-Beltrami equation on a fixed domain to the evolving surface Cahn-Hilliard equation, which models lateral phase separation in plasma membranes undergoing deformation and fusion.

From vegetation patterns in an ecological system to propagating waves in a nerve conduction system, patterns emerge everywhere in nature. From a mathematical perspective, reaction-
diffusion models are widely used to understand the mechanism underlying pattern formation. In
this talk, we introduce reaction-diffusion models with a skew gradient structure which
encompasses an important class of activator-inhibitor type systems that exhibit localized
patterns. A variational framework is proposed to show the existence of standing pulse solutions to a skew-gradient system.

Abstract: We will define caloric measure by first looking at the definition of harmonic measure (the steady
state analogue of caloric measure) and some interpretations of harmonic measure. We look at extending
known results for harmonic measure to caloric measure, and discuss where some difficulties arise. We then
will give a brief overview of some recent results, including a Bourgain-type estimate, a criterion for nondoubling caloric measure to satisfy a weak reverse H¨older inequality, and that BMO-solvability implies scale
invariant quantitative absolute continuity of caloric measure with respect to surface measure.

seminar

Equipping a Riemannian manifold with an affine connection allows for a notion of parallel transport of vector fields. Considering loops based at a fixed point, the parallel transport induces a subgroup of the orthogonal transformations of the tangent space of vector fields known as the holonomy group of the manifold. The list of possible holonomy groups (classified by Cartan and Berger) is remarkably short, and there is a strong relationship between the holonomy group of a manifold and its global geometry, which we will explore in this talk.

The Bartnik quasi-local mass is defined to measure the mass of a bounded manifold with boundary, where a collection of geometric boundary data — the so-called Bartnik boundary data— plays a key role. Bartnik proposed the open problem whether, on a given manifold with boundary, there exists a stationary vacuum metric so that the Bartnik boundary conditions are realized. In the effort to answer this question, it is important to prove the ellipticity of Bartnik boundary conditions for stationary vacuum metrics.
In this talk, I will start with an introduction to the Bartnik quasi-local mass and the moduli space of stationary vacuum metrics. Then I will explain the ellipticity result for the Bartnik boundary conditions and, as an application, derive a local result to the existence question.

Abstract: tSNE has become the standard method of dimensionality
reduction in the medical sciences (clustering cell types, gene
expressions, etc.); amusingly (or maybe not amusingly), the
mathematical theory did not exist until recently. Even a basic
understanding of the mathematics immediately implies a series
of improvements -- I will survey the existing arguments and discuss
a number of interesting problems; given the prevalence of tSNE
in medicine, even small improvements can affect a lot of research
within a very short period of time. Joint work with George Linderman.

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!

An important principle of the Newtonian physics is that a
full knowledge of a given system at some moment of time together with
the equations of motion should uniquely determine the future of the
system. We describe the basic questions in this direction for
equations describing incompressible fluid flows, where our knowledge
is still incomplete. We will also discuss some model equations which
are similar in certain aspects and for which our understanding is
better.

There are 1-dimensional models inspired by equations of fluid mechanic
which exhibit quite interesting behavior. Lesson learned from some of
the 1d models can be important for the original equations. We will
discuss a few examples, focusing mostly on the De Gregorio
modification of the Constantin-Lax-Majda mod

Abstract: We discuss the method of picking a convenient coordinate system adapted to vector fields. Let $$X_1,\ldots,X_q$$ be either real or complex $$C^1$$ vector fields. We discuss the question of when there is a coordinate system in which the vector fields are smoother (e.g., $$C^m$$, or $$C^\infty$$, or real analytic). By answering this in a quantitative way, we obtain coordinate charts which can be used as generalized scaling maps. When the vector fields are real this is joint work with Stovall, and continues in the line of quantitative sub-Riemannian geometry initiated by Nagel, Stein, and Wainger. When the vector fields are complex one obtains a geometry with more structure which can be thought of as "sub-Hermitian".

UConn History of Analytical Philosophy Workshop

Join us for a talk by Philip A. Ebert (Stirling) on Frege!

Details t.b.a.

https://rossberg.philosophy.uconn.edu/histanalytic/

Abstract: The Eisenbud-Green-Harris (EGH) conjecture states that a homogeneous ideal in a polynomial ring K[x1, . . . , xn] over a field K that contains a regular sequence with given degrees a1, . . . , an has the same Hilbert function as a lex-plus-powers ideal containing the powers of the variables xi with the degrees ai. We discuss a case of the EGH conjecture for homogeneous ideals generated by quadrics containing a regular sequence of full length and talk about our result that gives an affirmative answer to EGH when n = 5. This is a joint work with Mel Hochster.

TBA

We will introduce d-bar Neumann operator and subelliptic estimates on pseudoconvex domains in $$\mathbb{C}^n$$ and we will try to see geometric applications that are very closely related to complex geometry, sub-Riemannian geometry, and CR-geometry and algebraic geometry.

Join us in the Logic Colloquium for a talk:

Peter Pagin (Stockholm)

Indexicals, Time, and Compositionality

Abstract:

Kaplan’s official argument in “Demonstratives” for Temporalism, the view that some English sentences express propositions that can vary in truth value across time, is the so-called Operator Argument: temporal operators, such as “sometimes”, would be vacuous without such propositions.

Equally important is the argument from compositionality. Without temporal propositions, the sentences

(1) It is raining where John is
(2) It is raining where John is now

would express the same proposition. But they embed differently:

(3) Sometimes, it is raining where John is
(4) Sometimes it is raining where John is now

(3) and (4) express distinct propositions, so if they both are of the form “Sometimes, p”, and if (1) and (2) express the same proposition, we have a violation of compositionality.

In this talk it is shown that with Switcher Semantics, which allows for a generalized form of compositionality, we can have the result that (1) and (2) agree in content when unembedded (assertoric content) but differ in content when embedded under temporal operators (ingredient sense).

We can also show that Switcher Semantics, over Kaplan’s models, preserves the validities in the Logic of Demonstratives. All in all, the arguments for Temporalism are substantially undermined.

https://logic.uconn.edu/calendar/

All welcome!

TBA

In papers of 1944 and 1946, McKinsey and Tarski initiated the study of closure algebras—Boolean algebras equipped with an operation obeying a version of Kuratowski’s axioms—and in 1948 they applied their results to reach conclusions about intuitionistic logic and the modal logic S4. In 1982 Lipparini found non-elementary axioms for existentially closed (e.c.) closure algebras and showed that they do not form an elementary class. After outlining Lipparini’s results from a different perspective, this talk will provide new information about closure algebras that are e.c., (finitely or infinitely) generic, or algebraically closed.

Continuing series of presentations on clustering by participants.

Abstract: Both discrete and continuous Dirac-type systems play an important role in several domains of mathematics. Direct and inverse problems for these systems are closely connected with structured operators. Moreover, discrete Dirac systems may be considered as analogs and sometimes alternatives of the famous Szeg\"o recurrencies from the theory of orthogonal polynoimials. The talk is dedicated to Dirac systems and mentioned above interconnections.

TBA

TBA

TBA

Note: Free pizza and drinks!